["peano-lab-bundle-v1",215,["and",["and",["forall",["forall",["forall",["and",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]],["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]]],["imp",["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]]],["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["bot"]]]]]]]]],["and",["forall",["forall",["forall",["and",["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["or",["and",["eq",["var",2],["zero"]],["eq",["var",1],["var",0]]],["and",["eq",["var",1],["zero"]],["eq",["var",2],["var",0]]]]],["imp",["or",["and",["eq",["var",2],["zero"]],["eq",["var",1],["var",0]]],["and",["eq",["var",1],["zero"]],["eq",["var",2],["var",0]]]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]]]]]]],["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]],["bot"]]]]]]]]],[[152,["forall",["eq",["add",["zero"],["var",0]],["var",0]]],[],["ind",["eq",["add",["zero"],["var",0]],["var",0]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["axiom","PA3"],["zero"]]],["eq_refl",["zero"]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["succ",["var",1]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["zero"]],["var",0]]],["eq_subst",["eq",["succ",["var",0]],["succ",["var",1]]],["eq_sym",["hyp",0]],["eq_refl",["succ",["var",0]]]]]]]]],[232,["forall",["forall",["eq",["add",["succ",["var",1]],["var",0]],["succ",["add",["var",1],["var",0]]]]]],[],["forall_intro",["ind",["eq",["add",["succ",["var",1]],["var",0]],["succ",["add",["var",1],["var",0]]]],["eq_subst",["eq",["var",0],["succ",["add",["var",1],["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["succ",["var",0]]]],["eq_subst",["eq",["succ",["var",1]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_refl",["succ",["var",0]]]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["succ",["add",["var",2],["succ",["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["var",1]]],["var",0]]],["eq_subst",["eq",["succ",["add",["succ",["var",2]],["var",1]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["var",0]]],["eq_subst",["eq",["succ",["var",0]],["succ",["succ",["add",["var",2],["var",1]]]]],["eq_sym",["hyp",0]],["eq_refl",["succ",["succ",["add",["var",1],["var",0]]]]]]]]]]]]],[248,["forall",["forall",["eq",["add",["var",1],["var",0]],["add",["var",0],["var",1]]]]],[0,1],["imp_intro",["imp_intro",["forall_intro",["ind",["eq",["add",["var",1],["var",0]],["add",["var",0],["var",1]]],["eq_subst",["eq",["var",0],["add",["zero"],["var",1]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_subst",["eq",["var",1],["var",0]],["eq_sym",["forall_elim",["hyp",1],["var",0]]],["eq_refl",["var",0]]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["add",["succ",["var",1]],["var",2]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["var",0]]],["eq_subst",["eq",["succ",["add",["var",2],["var",1]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["hyp",1],["var",0]],["var",1]]],["eq_subst",["eq",["succ",["var",0]],["succ",["add",["var",1],["var",2]]]],["eq_sym",["hyp",0]],["eq_refl",["succ",["add",["var",0],["var",1]]]]]]]]]]]]]],[280,["forall",["forall",["forall",["eq",["add",["add",["var",2],["var",1]],["var",0]],["add",["var",2],["add",["var",1],["var",0]]]]]]],[],["forall_intro",["forall_intro",["ind",["eq",["add",["add",["var",2],["var",1]],["var",0]],["add",["var",2],["add",["var",1],["var",0]]]],["eq_subst",["eq",["var",0],["add",["var",2],["add",["var",1],["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["add",["var",1],["var",0]]]],["eq_subst",["eq",["add",["var",2],["var",1]],["add",["var",2],["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_refl",["add",["var",1],["var",0]]]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["add",["var",3],["add",["var",2],["succ",["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["add",["var",2],["var",1]]],["var",0]]],["eq_subst",["eq",["succ",["add",["add",["var",3],["var",2]],["var",1]]],["add",["var",3],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["var",0]]],["eq_subst",["eq",["succ",["add",["add",["var",3],["var",2]],["var",1]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",2]],["add",["var",1],["var",0]]]],["eq_subst",["eq",["succ",["var",0]],["succ",["add",["var",3],["add",["var",2],["var",1]]]]],["eq_sym",["hyp",0]],["eq_refl",["succ",["add",["var",2],["add",["var",1],["var",0]]]]]]]]]]]]]]],[184,["forall",["eq",["mul",["zero"],["var",0]],["zero"]]],[],["ind",["eq",["mul",["zero"],["var",0]],["zero"]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_refl",["zero"]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["zero"]],["var",0]]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["zero"],["var",0]]]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["hyp",0]],["eq_refl",["zero"]]]]]]]]],[544,["forall",["forall",["eq",["mul",["succ",["var",1]],["var",0]],["add",["mul",["var",1],["var",0]],["var",0]]]]],[2,3],["imp_intro",["imp_intro",["forall_intro",["ind",["eq",["mul",["succ",["var",1]],["var",0]],["add",["mul",["var",1],["var",0]],["var",0]]],["eq_subst",["eq",["mul",["succ",["var",1]],["zero"]],["var",0]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["var",0],["zero"]]]],["eq_subst",["eq",["var",0],["mul",["var",1],["zero"]]],["eq_sym",["forall_elim",["axiom","PA5"],["succ",["var",0]]]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]],["eq_refl",["zero"]]]]],["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["eq_subst",["eq",["mul",["succ",["var",2]],["succ",["var",1]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["var",1],["succ",["var",0]]]],["var",0]]],["eq_subst",["eq",["var",0],["succ",["add",["mul",["var",2],["succ",["var",1]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["var",1]]],["var",0]]],["eq_subst",["eq",["var",0],["succ",["add",["mul",["var",2],["succ",["var",1]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",1]],["var",0]]],["var",1]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["var",2]],["var",1]],["var",2]]],["succ",["add",["var",0],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",1]],["var",0]]],["eq_subst",["eq",["succ",["add",["var",0],["var",2]]],["succ",["add",["add",["mul",["var",2],["var",1]],["var",2]],["var",1]]]],["eq_sym",["hyp",2]],["eq_subst",["eq",["succ",["var",0]],["succ",["add",["add",["mul",["var",2],["var",1]],["var",2]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",1],["var",0]]],["var",0]],["var",1]]],["eq_subst",["eq",["succ",["add",["mul",["var",2],["var",1]],["add",["var",1],["var",2]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",1],["var",0]]],["var",1]],["var",0]]],["eq_subst",["eq",["succ",["add",["mul",["var",2],["var",1]],["add",["var",1],["var",2]]]],["succ",["add",["mul",["var",2],["var",1]],["var",0]]]],["eq_sym",["hyp",0]],["eq_refl",["succ",["add",["mul",["var",1],["var",0]],["add",["var",0],["var",1]]]]]]]]]]]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",2],["var",1]]]]]]]]]],[248,["forall",["forall",["eq",["mul",["var",1],["var",0]],["mul",["var",0],["var",1]]]]],[4,5],["imp_intro",["imp_intro",["forall_intro",["ind",["eq",["mul",["var",1],["var",0]],["mul",["var",0],["var",1]]],["eq_subst",["eq",["var",0],["mul",["zero"],["var",1]]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["hyp",1],["var",0]]],["eq_refl",["zero"]]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["mul",["succ",["var",1]],["var",2]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",1]],["var",0]]],["eq_subst",["eq",["add",["var",0],["var",2]],["mul",["succ",["var",1]],["var",2]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",2]],["var",2]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["hyp",1],["var",0]],["var",1]]],["eq_refl",["add",["mul",["var",0],["var",1]],["var",1]]]]]]]]]]]]],[368,["forall",["forall",["forall",["eq",["mul",["var",2],["add",["var",1],["var",0]]],["add",["mul",["var",2],["var",1]],["mul",["var",2],["var",0]]]]]]],[3],["imp_intro",["forall_intro",["forall_intro",["ind",["eq",["mul",["var",2],["add",["var",1],["var",0]]],["add",["mul",["var",2],["var",1]],["mul",["var",2],["var",0]]]],["eq_subst",["eq",["mul",["var",2],["var",0]],["add",["mul",["var",2],["var",1]],["mul",["var",2],["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_subst",["eq",["mul",["var",2],["var",1]],["add",["mul",["var",2],["var",1]],["var",0]]],["eq_sym",["forall_elim",["axiom","PA5"],["var",1]]],["eq_subst",["eq",["mul",["var",2],["var",1]],["var",0]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["var",1],["var",0]]]],["eq_refl",["mul",["var",1],["var",0]]]]]],["forall_intro",["imp_intro",["eq_subst",["eq",["mul",["var",3],["var",0]],["add",["mul",["var",3],["var",2]],["mul",["var",3],["succ",["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["var",0]]],["eq_subst",["eq",["var",0],["add",["mul",["var",3],["var",2]],["mul",["var",3],["succ",["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",2]],["add",["var",1],["var",0]]]],["eq_subst",["eq",["add",["mul",["var",3],["add",["var",2],["var",1]]],["var",3]],["add",["mul",["var",3],["var",2]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",2]],["var",0]]],["eq_subst",["eq",["add",["var",0],["var",3]],["add",["mul",["var",3],["var",2]],["add",["mul",["var",3],["var",1]],["var",3]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["var",0],["add",["mul",["var",3],["var",2]],["add",["mul",["var",3],["var",1]],["var",3]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",2],["var",1]]],["mul",["var",2],["var",0]]],["var",2]]],["eq_refl",["add",["mul",["var",2],["var",1]],["add",["mul",["var",2],["var",0]],["var",2]]]]]]]]]]]]]]]],[328,["forall",["forall",["forall",["eq",["mul",["mul",["var",2],["var",1]],["var",0]],["mul",["var",2],["mul",["var",1],["var",0]]]]]]],[7],["imp_intro",["forall_intro",["forall_intro",["ind",["eq",["mul",["mul",["var",2],["var",1]],["var",0]],["mul",["var",2],["mul",["var",1],["var",0]]]],["eq_subst",["eq",["var",0],["mul",["var",2],["mul",["var",1],["zero"]]]],["eq_sym",["forall_elim",["axiom","PA5"],["mul",["var",1],["var",0]]]],["eq_subst",["eq",["zero"],["mul",["var",2],["var",0]]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",1]]],["eq_refl",["zero"]]]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["mul",["var",3],["mul",["var",2],["succ",["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["mul",["var",2],["var",1]]],["var",0]]],["eq_subst",["eq",["add",["mul",["mul",["var",3],["var",2]],["var",1]],["mul",["var",3],["var",2]]],["mul",["var",3],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",1]],["var",0]]],["eq_subst",["eq",["add",["var",0],["mul",["var",3],["var",2]]],["mul",["var",3],["add",["mul",["var",2],["var",1]],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",3],["mul",["var",2],["var",1]]],["mul",["var",3],["var",2]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",2]],["mul",["var",1],["var",0]]],["var",1]]],["eq_refl",["add",["mul",["var",2],["mul",["var",1],["var",0]]],["mul",["var",2],["var",1]]]]]]]]]]]]]]],[224,["forall",["eq",["mul",["succ",["zero"]],["var",0]],["var",0]]],[],["ind",["eq",["mul",["succ",["zero"]],["var",0]],["var",0]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["axiom","PA5"],["succ",["zero"]]]],["eq_refl",["zero"]]],["forall_intro",["imp_intro",["eq_subst",["eq",["var",0],["succ",["var",1]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["zero"]]],["var",0]]],["eq_subst",["eq",["var",0],["succ",["var",1]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["zero"]],["var",0]]],["zero"]]],["eq_subst",["eq",["succ",["var",0]],["succ",["var",1]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["zero"]],["var",0]]]],["eq_subst",["eq",["succ",["var",0]],["succ",["var",1]]],["eq_sym",["hyp",0]],["eq_refl",["succ",["var",0]]]]]]]]]]],[144,["forall",["eq",["mul",["var",0],["succ",["zero"]]],["var",0]]],[0],["imp_intro",["forall_intro",["eq_subst",["eq",["var",0],["var",1]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",0]],["zero"]]],["eq_subst",["eq",["add",["var",0],["var",1]],["var",1]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]],["eq_subst",["eq",["var",0],["var",1]],["eq_sym",["forall_elim",["hyp",0],["var",0]]],["eq_refl",["var",0]]]]]]]],[232,["forall",["forall",["forall",["eq",["mul",["add",["var",2],["var",1]],["var",0]],["add",["mul",["var",2],["var",0]],["mul",["var",1],["var",0]]]]]]],[6,7],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["eq_subst",["eq",["var",0],["add",["mul",["var",3],["var",1]],["mul",["var",2],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",1],["add",["var",2],["var",1]]],["var",0]]],["eq_subst",["eq",["mul",["var",1],["add",["var",3],["var",2]]],["add",["var",0],["mul",["var",2],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",0]]],["eq_subst",["eq",["mul",["var",1],["add",["var",3],["var",2]]],["add",["mul",["var",1],["var",3]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",0]]],["eq_subst",["eq",["var",0],["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",0]],["var",2]],["var",1]]],["eq_refl",["add",["mul",["var",0],["var",2]],["mul",["var",0],["var",1]]]]]]]]]]]]]],[24,["forall",["imp",["eq",["succ",["var",0]],["zero"]],["bot"]]],[],["axiom","PA1"]],[80,["forall",["exists",["eq",["add",["var",0],["var",1]],["var",1]]]],[0],["imp_intro",["forall_intro",["exists_intro",["zero"],["eq_subst",["eq",["var",0],["var",1]],["eq_sym",["forall_elim",["hyp",0],["var",0]]],["eq_refl",["var",0]]]]]]],[208,["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",3]],["var",2]]],["imp",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["exists",["eq",["add",["var",0],["var",3]],["var",1]]]]]]]],[3],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["var",0],["var",1]],["eq_subst",["eq",["var",0],["var",3]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",0]],["var",1]],["var",4]]],["eq_subst",["eq",["add",["var",1],["var",0]],["var",3]],["eq_sym",["hyp",1]],["eq_subst",["eq",["var",0],["var",3]],["eq_sym",["hyp",0]],["eq_refl",["var",2]]]]]]]]]]]]]]],[256,["forall",["forall",["imp",["eq",["add",["succ",["var",1]],["var",0]],["var",0]],["bot"]]]],[],["forall_intro",["ind",["imp",["eq",["add",["succ",["var",1]],["var",0]],["var",0]],["bot"]],["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["var",0]],["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["var",0],["zero"]],["forall_elim",["axiom","PA3"],["succ",["var",0]]],["hyp",0]]]]],["forall_intro",["imp_intro",["imp_intro",["imp_elim",["hyp",1],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["add",["succ",["var",1]],["var",0]]],["var",0]],["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["var",0],["succ",["var",1]]],["forall_elim",["forall_elim",["axiom","PA4"],["succ",["var",1]]],["var",0]],["hyp",0]]]]]]]]]]],[384,["forall",["forall",["forall",["imp",["eq",["add",["add",["var",1],["var",2]],["var",0]],["var",0]],["eq",["add",["var",2],["var",0]],["var",0]]]]]],[0,1,15],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["ind",["forall",["imp",["eq",["add",["add",["var",1],["var",2]],["var",0]],["var",0]],["eq",["add",["var",2],["var",0]],["var",0]]]],["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["add",["var",0],["var",1]],["var",1]],["hyp",0],["hyp",1]]]],["forall_elim",["hyp",3],["var",1]]]]],["forall_intro",["imp_intro",["forall_intro",["imp_intro",["bot_elim",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["add",["var",0],["var",1]],["var",1]],["hyp",0],["hyp",4]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",5],["var",1]]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",2],["add",["var",1],["var",2]]]]]]]]]]]]]]],[392,["forall",["forall",["forall",["forall",["imp",["eq",["add",["var",3],["var",1]],["var",0]],["imp",["eq",["add",["var",2],["var",0]],["var",1]],["eq",["var",1],["var",0]]]]]]]],[3,16],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["eq_sym",["eq_subst",["eq",["var",0],["var",2]],["hyp",1],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["eq_subst",["eq",["var",0],["var",2]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["var",3],["var",0]],["var",2]],["eq_sym",["hyp",7]],["eq_subst",["eq",["var",0],["var",2]],["eq_sym",["hyp",6]],["eq_refl",["var",1]]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",3]]]],["forall_elim",["hyp",6],["var",2]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",2],["var",3]]]]]]]]]]]]]],[160,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["imp",["exists",["eq",["add",["var",0],["var",1]],["var",2]]],["eq",["var",1],["var",0]]]]]],[17],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",0]],["var",3]],["var",2]],["hyp",1]],["hyp",0]]]]]]]]]],[408,["forall",["forall",["or",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["exists",["eq",["add",["var",0],["var",1]],["var",2]]]]]],[],["ind",["forall",["or",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["exists",["eq",["add",["var",0],["var",1]],["var",2]]]]],["forall_intro",["or_intro_l",["exists_intro",["var",0],["eq_subst",["eq",["var",0],["var",1]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_refl",["var",0]]]]]],["forall_intro",["imp_intro",["ind",["or",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["exists",["eq",["add",["var",0],["var",1]],["succ",["var",2]]]]],["or_intro_r",["exists_intro",["succ",["var",0]],["eq_subst",["eq",["var",0],["succ",["var",1]]],["eq_sym",["forall_elim",["axiom","PA3"],["succ",["var",0]]]],["eq_refl",["succ",["var",0]]]]]],["forall_intro",["imp_intro",["imp_elim",["imp_intro",["or_elim",["hyp",0],["exists_elim",["hyp",0],["or_intro_l",["exists_intro",["var",0],["eq_subst",["eq",["var",0],["succ",["var",2]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",2]]],["cong_s",["hyp",0]]]]]],["exists_elim",["hyp",0],["or_intro_r",["exists_intro",["var",0],["eq_subst",["eq",["var",0],["succ",["var",3]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",1]]],["cong_s",["hyp",0]]]]]]]],["forall_elim",["hyp",1],["var",0]]]]]]]]]],[168,["forall",["forall",["imp",["eq",["add",["var",1],["var",0]],["zero"]],["eq",["var",0],["zero"]]]]],[],["forall_intro",["ind",["imp",["eq",["add",["var",1],["var",0]],["zero"]],["eq",["var",0],["zero"]]],["imp_intro",["eq_refl",["zero"]]],["forall_intro",["imp_intro",["imp_intro",["bot_elim",["imp_elim",["forall_elim",["axiom","PA1"],["add",["var",1],["var",0]]],["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["var",0],["zero"]],["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["var",0]],["hyp",0]]]]]]]]]]],[240,["forall",["forall",["imp",["eq",["mul",["var",1],["var",0]],["zero"]],["or",["eq",["var",1],["zero"]],["eq",["var",0],["zero"]]]]]],[20],["imp_intro",["forall_intro",["ind",["imp",["eq",["mul",["var",1],["var",0]],["zero"]],["or",["eq",["var",1],["zero"]],["eq",["var",0],["zero"]]]],["imp_intro",["or_intro_r",["eq_refl",["zero"]]]],["forall_intro",["imp_intro",["imp_intro",["or_intro_l",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["var",0],["zero"]],["forall_elim",["forall_elim",["axiom","PA6"],["var",1]],["var",0]],["hyp",2]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",2],["mul",["var",1],["var",0]]]]]]]]]]]],[80,["forall",["or",["eq",["var",0],["zero"]],["exists",["eq",["var",1],["succ",["var",0]]]]]],[],["ind",["or",["eq",["var",0],["zero"]],["exists",["eq",["var",1],["succ",["var",0]]]]],["or_intro_l",["eq_refl",["zero"]]],["forall_intro",["imp_intro",["or_intro_r",["exists_intro",["var",0],["eq_refl",["succ",["var",0]]]]]]]]],[104,["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["exists",["eq",["var",1],["succ",["var",0]]]]]],[],["ind",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["exists",["eq",["var",1],["succ",["var",0]]]]],["imp_intro",["bot_elim",["imp_elim",["hyp",0],["eq_refl",["zero"]]]]],["forall_intro",["imp_intro",["imp_intro",["exists_intro",["var",0],["eq_refl",["succ",["var",0]]]]]]]]],[344,["forall",["forall",["forall",["imp",["eq",["add",["var",2],["var",0]],["add",["var",1],["var",0]]],["eq",["var",2],["var",1]]]]]],[],["forall_intro",["forall_intro",["ind",["imp",["eq",["add",["var",2],["var",0]],["add",["var",1],["var",0]]],["eq",["var",2],["var",1]]],["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["var",2],["var",0]],["forall_elim",["axiom","PA3"],["var",0]],["hyp",0]]]],["eq_subst",["eq",["var",0],["add",["var",1],["zero"]]],["forall_elim",["axiom","PA3"],["var",1]],["hyp",0]]]],["forall_intro",["imp_intro",["imp_intro",["imp_elim",["hyp",1],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["add",["var",2],["var",0]]],["add",["var",1],["var",0]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["succ",["add",["var",3],["var",1]]],["var",0]],["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["var",0]],["hyp",0]]]],["eq_subst",["eq",["var",0],["add",["var",2],["succ",["var",1]]]],["forall_elim",["forall_elim",["axiom","PA4"],["var",2]],["var",0]],["hyp",0]]]]]]]]]]]],[248,["forall",["forall",["forall",["imp",["eq",["add",["var",2],["var",1]],["add",["var",2],["var",0]]],["eq",["var",1],["var",0]]]]]],[2,24],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_trans",["forall_elim",["forall_elim",["hyp",5],["var",1]],["var",2]],["eq_trans",["hyp",3],["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",0]]]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",1],["var",1]]]]]]]]]],[128,["forall",["exists",["eq",["add",["var",0],["var",1]],["succ",["var",1]]]]],[0,1],["imp_intro",["imp_intro",["forall_intro",["exists_intro",["succ",["zero"]],["eq_subst",["eq",["var",0],["succ",["var",1]]],["eq_sym",["forall_elim",["forall_elim",["hyp",0],["zero"]],["var",0]]],["eq_subst",["eq",["succ",["var",0]],["succ",["var",1]]],["eq_sym",["forall_elim",["hyp",1],["var",0]]],["eq_refl",["succ",["var",0]]]]]]]]]],[96,["forall",["imp",["exists",["eq",["add",["var",0],["var",1]],["zero"]]],["eq",["var",0],["zero"]]]],[20],["imp_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",1]],["hyp",0]]]]]]],[176,["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["succ",["zero"]]],["var",1]]]]],[],["ind",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["succ",["zero"]]],["var",1]]]],["imp_intro",["bot_elim",["imp_elim",["hyp",0],["eq_refl",["zero"]]]]],["forall_intro",["imp_intro",["imp_intro",["exists_intro",["var",0],["eq_subst",["eq",["var",0],["succ",["var",1]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["zero"]]],["eq_subst",["eq",["succ",["var",0]],["succ",["var",1]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_refl",["succ",["var",0]]]]]]]]]]],[144,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",2]]],["succ",["var",1]]]],["exists",["eq",["add",["var",0],["var",2]],["var",1]]]]]],[],["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_intro",["var",0],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["add",["var",0],["var",2]]],["var",1]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",2]]],["hyp",0]]]]]]]]],[152,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["exists",["eq",["add",["var",0],["var",2]],["var",1]]]]]],[1],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_intro",["succ",["var",0]],["eq_trans",["eq_trans",["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",2]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",2]]]],["hyp",0]]]]]]]]],[224,["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",1]]],["bot"]]],[1,15],["imp_intro",["imp_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_trans",["eq_trans",["forall_elim",["forall_elim",["hyp",5],["var",0]],["var",1]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",1]]]],["hyp",2]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",2],["var",0]]]]]]]]],[384,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["or",["eq",["var",1],["var",0]],["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]]]]]],[22,0,1],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["imp_elim",["imp_intro",["or_elim",["hyp",0],["or_intro_l",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["eq_subst",["eq",["var",0],["var",2]],["hyp",0],["hyp",1]]]],["forall_elim",["hyp",6],["var",2]]]],["eq_subst",["eq",["add",["var",0],["var",3]],["var",2]],["hyp",0],["hyp",2]]]],["exists_elim",["hyp",0],["or_intro_r",["exists_intro",["var",0],["eq_trans",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",3]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["var",0]],["var",3]]]],["eq_subst",["eq",["add",["var",0],["var",4]],["var",3]],["hyp",0],["hyp",3]]]]]]]],["forall_elim",["hyp",4],["var",0]]]]]]]]]]],[200,["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",3]]],["var",2]]],["imp",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["exists",["eq",["add",["var",0],["succ",["var",3]]],["var",1]]]]]]]],[14],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",4]],["hyp",3]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",2],["succ",["var",2]]]]]]]]]]],[216,["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",3]],["var",2]]],["imp",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["exists",["eq",["add",["var",0],["succ",["var",3]]],["var",1]]]]]]]],[3],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["var",0],["var",1]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",0]],["var",1]],["succ",["var",4]]],["eq_subst",["eq",["add",["var",1],["var",0]],["var",3]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["var",4]]],["eq_subst",["eq",["add",["var",1],["succ",["var",0]]],["var",3]],["eq_sym",["hyp",1]],["hyp",0]]]]]]]]]]]]]],[400,["forall",["forall",["or",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]]]]],[],["ind",["forall",["or",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]]]],["forall_intro",["or_intro_l",["exists_intro",["var",0],["forall_elim",["axiom","PA3"],["var",0]]]]],["forall_intro",["imp_intro",["ind",["or",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["succ",["var",2]]]]],["or_intro_r",["exists_intro",["var",0],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["var",0]]]]]],["forall_intro",["imp_intro",["imp_elim",["imp_intro",["or_elim",["hyp",0],["or_intro_l",["exists_elim",["hyp",0],["exists_intro",["var",0],["eq_subst",["eq",["var",0],["succ",["var",2]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",2]]],["cong_s",["hyp",0]]]]]],["or_intro_r",["exists_elim",["hyp",0],["exists_intro",["var",0],["eq_subst",["eq",["var",0],["succ",["var",3]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["succ",["var",1]]]],["cong_s",["hyp",0]]]]]]]],["forall_elim",["hyp",1],["var",0]]]]]]]]]],[464,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["imp",["exists",["eq",["add",["var",0],["var",1]],["var",2]]],["bot"]]]]],[],["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["ind",["forall",["forall",["imp",["eq",["var",2],["add",["var",1],["add",["var",0],["succ",["var",2]]]]],["bot"]]]],["forall_intro",["forall_intro",["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["add",["var",1],["add",["var",0],["zero"]]]],["eq_sym",["eq_subst",["eq",["zero"],["var",0]],["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["add",["var",0],["zero"]]],["eq_subst",["eq",["zero"],["add",["var",2],["var",0]]],["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["zero"]],["hyp",0]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["var",2]],["add",["var",1],["add",["var",0],["succ",["var",2]]]]],["eq_subst",["eq",["succ",["var",3]],["var",0]],["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["add",["var",0],["succ",["var",2]]]],["eq_subst",["eq",["succ",["var",3]],["add",["var",2],["var",0]]],["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["succ",["var",2]]],["hyp",0]]]]]]]]]]],["var",3]],["var",0]],["var",1]],["eq_sym",["eq_subst",["eq",["add",["var",1],["var",0]],["var",4]],["eq_sym",["hyp",1]],["hyp",0]]]]]]]]]]],[560,["forall",["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",4]]],["var",3]]],["imp",["eq",["var",3],["add",["add",["var",1],["var",2]],["var",0]]],["bot"]]]]]]],[1],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["ind",["forall",["forall",["forall",["imp",["eq",["var",3],["add",["add",["var",2],["add",["var",1],["succ",["var",3]]]],["var",0]]],["bot"]]]]],["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["add",["add",["var",2],["add",["var",1],["zero"]]],["var",0]]],["eq_sym",["eq_subst",["eq",["zero"],["var",0]],["forall_elim",["forall_elim",["hyp",4],["add",["var",2],["add",["var",1],["zero"]]]],["var",0]],["eq_subst",["eq",["zero"],["add",["var",0],["var",1]]],["forall_elim",["forall_elim",["axiom","PA4"],["var",2]],["add",["var",1],["zero"]]],["eq_subst",["eq",["zero"],["add",["add",["var",3],["var",0]],["var",1]]],["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["zero"]],["hyp",0]]]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",1]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["var",3]],["add",["add",["var",2],["add",["var",1],["succ",["var",3]]]],["var",0]]],["eq_subst",["eq",["succ",["var",4]],["var",0]],["forall_elim",["forall_elim",["hyp",5],["add",["var",2],["add",["var",1],["succ",["var",3]]]]],["var",0]],["eq_subst",["eq",["succ",["var",4]],["add",["var",0],["var",1]]],["forall_elim",["forall_elim",["axiom","PA4"],["var",2]],["add",["var",1],["succ",["var",3]]]],["eq_subst",["eq",["succ",["var",4]],["add",["add",["var",3],["var",0]],["var",1]]],["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["succ",["var",3]]],["hyp",0]]]]]]]]]]]]],["var",4]],["var",2]],["var",0]],["var",1]],["eq_subst",["eq",["var",5],["add",["add",["var",3],["var",0]],["var",2]]],["eq_sym",["hyp",0]],["hyp",1]]]]]]]]]]]],[152,["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",3]],["var",2]]],["exists",["eq",["add",["var",0],["mul",["var",1],["var",3]]],["mul",["var",1],["var",2]]]]]]]],[7],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_intro",["mul",["var",1],["var",0]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]],["var",3]]],["cong_mul",["eq_refl",["var",1]],["hyp",0]]]]]]]]]]],[152,["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",3]],["var",2]]],["exists",["eq",["add",["var",0],["mul",["var",3],["var",1]]],["mul",["var",2],["var",1]]]]]]]],[11],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_intro",["mul",["var",0],["var",1]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",3]],["var",1]]],["cong_mul",["hyp",0],["eq_refl",["var",1]]]]]]]]]]]],[264,["forall",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["succ",["mul",["var",2],["var",1]]]],["mul",["var",2],["succ",["var",1]]]]]]]],[2],["imp_intro",["ind",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["succ",["mul",["var",2],["var",1]]]],["mul",["var",2],["succ",["var",1]]]]]]],["forall_intro",["imp_intro",["bot_elim",["imp_elim",["hyp",0],["eq_refl",["zero"]]]]]],["forall_intro",["imp_intro",["forall_intro",["imp_intro",["exists_intro",["var",1],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["var",1]],["mul",["succ",["var",1]],["var",0]]],["eq_trans",["cong_s",["forall_elim",["forall_elim",["hyp",2],["var",1]],["mul",["succ",["var",1]],["var",0]]]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",1]],["var",0]]],["var",1]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["var",1]]],["var",0]]]]]]]]]]]]]],[1032,["forall",["forall",["exists",["exists",["and",["eq",["var",2],["add",["mul",["succ",["var",3]],["var",1]],["var",0]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["succ",["var",4]]]]]]]]],[0,1,22],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["ind",["exists",["exists",["and",["eq",["var",2],["add",["mul",["succ",["var",3]],["var",1]],["var",0]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["succ",["var",4]]]]]]],["exists_intro",["zero"],["exists_intro",["zero"],["and_intro",["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["var",0]],["zero"]]]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["succ",["var",0]]]],["eq_refl",["zero"]]]],["exists_intro",["var",0],["eq_subst",["eq",["var",0],["succ",["var",1]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["zero"]]],["eq_subst",["eq",["succ",["var",0]],["succ",["var",1]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_refl",["succ",["var",0]]]]]]]]],["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_elim",["and_elim_r",["hyp",0]],["or_elim",["forall_elim",["hyp",4],["var",0]],["exists_intro",["succ",["var",2]],["exists_intro",["zero"],["and_intro",["eq_subst",["eq",["succ",["var",0]],["add",["mul",["succ",["var",5]],["succ",["var",3]]],["zero"]]],["eq_sym",["and_elim_l",["hyp",2]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["var",5]],["var",3]],["var",0]]],["add",["mul",["succ",["var",5]],["succ",["var",3]]],["zero"]]],["eq_sym",["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["var",1]],["var",4]],["eq_subst",["eq",["var",0],["succ",["var",5]]],["forall_elim",["hyp",7],["succ",["var",1]]],["eq_subst",["eq",["add",["var",0],["succ",["var",2]]],["succ",["var",5]]],["hyp",0],["hyp",1]]]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["var",5]],["var",3]],["var",5]]],["var",0]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["var",4]],["succ",["var",2]]]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["var",5]],["var",3]],["var",5]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["var",4]]],["var",2]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["var",5]],["var",3]],["var",5]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",4]],["var",2]]],["var",4]]],["eq_refl",["succ",["add",["mul",["succ",["var",4]],["var",2]],["var",4]]]]]]]]],["exists_intro",["var",4],["eq_subst",["eq",["var",0],["succ",["var",5]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",4]],["zero"]]],["eq_subst",["eq",["succ",["var",0]],["succ",["var",5]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",4]]],["eq_refl",["succ",["var",4]]]]]]]]],["exists_elim",["hyp",0],["exists_intro",["var",3],["exists_intro",["succ",["var",2]],["and_intro",["eq_subst",["eq",["succ",["var",0]],["add",["mul",["succ",["var",6]],["var",4]],["succ",["var",3]]]],["eq_sym",["and_elim_l",["hyp",3]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["var",6]],["var",4]],["var",3]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",5]],["var",3]]],["var",2]]],["eq_refl",["succ",["add",["mul",["succ",["var",5]],["var",3]],["var",2]]]]]],["exists_intro",["var",0],["eq_subst",["eq",["add",["var",1],["succ",["succ",["var",3]]]],["var",0]],["hyp",2],["eq_subst",["eq",["add",["var",1],["succ",["succ",["var",3]]]],["add",["var",0],["succ",["var",3]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["var",0],["add",["succ",["var",1]],["succ",["var",3]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["succ",["var",2]]]],["eq_subst",["eq",["succ",["var",0]],["add",["succ",["var",1]],["succ",["var",3]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",2]]],["eq_subst",["eq",["succ",["succ",["add",["var",1],["var",3]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["var",0]]],["var",2]]],["eq_subst",["eq",["succ",["succ",["add",["var",1],["var",3]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["hyp",7],["var",0]],["var",2]]],["eq_refl",["succ",["succ",["add",["var",0],["var",2]]]]]]]]]]]]]]]]]]]]]]]]]]]],[272,["forall",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["exists",["exists",["and",["eq",["var",2],["add",["mul",["var",3],["var",1]],["var",0]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]],[22,41],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["or_elim",["hyp",0],["bot_elim",["imp_elim",["hyp",2],["hyp",0]]],["exists_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["eq_subst",["exists",["exists",["and",["eq",["var",4],["add",["mul",["var",2],["var",1]],["var",0]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",6]]]]]],["eq_sym",["hyp",2]],["eq_subst",["exists",["exists",["and",["eq",["var",4],["add",["mul",["succ",["var",3]],["var",1]],["var",0]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",3]]]]]],["eq_sym",["hyp",2]],["hyp",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",4],["var",0]]]]]],["forall_elim",["hyp",2],["var",1]]]]]]]]],[784,["forall",["forall",["forall",["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",3]]],["var",6]]],["imp",["eq",["add",["succ",["var",0]],["var",4]],["var",3]],["imp",["eq",["add",["mul",["var",5],["var",4]],["var",2]],["add",["mul",["var",5],["var",3]],["var",1]]],["bot"]]]]]]]]]],[2,3,7,25,37],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",6]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_trans",["hyp",7],["eq_subst",["eq",["add",["mul",["var",6],["var",0]],["var",2]],["add",["mul",["var",6],["var",5]],["add",["add",["mul",["var",6],["var",1]],["var",6]],["var",2]]]],["hyp",8],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["eq_subst",["eq",["add",["mul",["var",6],["var",0]],["var",2]],["add",["mul",["var",6],["var",5]],["add",["add",["mul",["var",6],["var",1]],["var",6]],["var",2]]]],["eq_sym",["hyp",0]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["eq_subst",["eq",["add",["var",0],["var",2]],["add",["mul",["var",6],["var",5]],["add",["add",["mul",["var",6],["var",1]],["var",6]],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["add",["mul",["var",6],["var",5]],["var",0]],["var",2]],["add",["mul",["var",6],["var",5]],["add",["add",["mul",["var",6],["var",1]],["var",6]],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",5]],["var",0]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["add",["mul",["var",5],["var",0]],["var",5]]]]],["forall_elim",["hyp",18],["mul",["var",5],["var",4]]]]]]],["forall_elim",["hyp",0],["succ",["var",0]]]]],["forall_elim",["hyp",0],["var",4]]]],["forall_elim",["hyp",14],["var",5]]]]],["forall_elim",["hyp",0],["var",4]]]],["forall_elim",["hyp",14],["succ",["var",0]]]]]]]],["forall_elim",["hyp",0],["add",["add",["mul",["var",5],["var",0]],["var",5]],["var",1]]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",8],["mul",["var",5],["var",4]]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["mul",["var",5],["var",0]]]]],["forall_elim",["hyp",0],["var",5]]]],["forall_elim",["hyp",3],["var",2]]]]]]]]]]]]]]]]]],[976,["forall",["forall",["forall",["forall",["forall",["forall",["imp",["eq",["var",4],["add",["mul",["var",5],["var",3]],["var",2]]],["imp",["exists",["eq",["add",["var",0],["succ",["var",3]]],["var",6]]],["imp",["eq",["var",4],["add",["mul",["var",5],["var",1]],["var",0]]],["imp",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",6]]],["and",["eq",["var",3],["var",1]],["eq",["var",2],["var",0]]]]]]]]]]]]],[0,19,22,25,43],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["or_elim",["forall_elim",["forall_elim",["hyp",7],["var",3]],["var",1]],["exists_elim",["hyp",0],["or_elim",["forall_elim",["hyp",8],["var",0]],["and_intro",["eq_subst",["eq",["var",0],["var",3]],["forall_elim",["hyp",11],["var",4]],["eq_subst",["eq",["add",["var",0],["var",5]],["var",3]],["hyp",0],["hyp",1]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["mul",["var",6],["var",4]]],["var",3]],["var",1]],["eq_subst",["eq",["add",["mul",["var",7],["var",5]],["var",4]],["add",["mul",["var",7],["var",0]],["var",2]]],["eq_sym",["eq_subst",["eq",["var",0],["var",3]],["forall_elim",["hyp",11],["var",4]],["eq_subst",["eq",["add",["var",0],["var",5]],["var",3]],["hyp",0],["hyp",1]]]],["eq_trans",["eq_sym",["hyp",6]],["hyp",4]]]]],["exists_elim",["hyp",0],["bot_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",7]],["var",5]],["var",3]],["var",4]],["var",2]],["var",0]],["hyp",6]],["eq_subst",["eq",["add",["var",0],["var",6]],["var",4]],["hyp",0],["hyp",2]]],["eq_trans",["eq_sym",["hyp",7]],["hyp",5]]]]]]],["exists_elim",["hyp",0],["or_elim",["forall_elim",["hyp",8],["var",0]],["and_intro",["eq_sym",["eq_subst",["eq",["var",0],["var",5]],["forall_elim",["hyp",11],["var",2]],["eq_subst",["eq",["add",["var",0],["var",3]],["var",5]],["hyp",0],["hyp",1]]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["mul",["var",6],["var",4]]],["var",3]],["var",1]],["eq_subst",["eq",["add",["mul",["var",7],["var",5]],["var",4]],["add",["mul",["var",7],["var",0]],["var",2]]],["eq_subst",["eq",["var",0],["var",5]],["forall_elim",["hyp",11],["var",2]],["eq_subst",["eq",["add",["var",0],["var",3]],["var",5]],["hyp",0],["hyp",1]]],["eq_trans",["eq_sym",["hyp",6]],["hyp",4]]]]],["exists_elim",["hyp",0],["bot_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",7]],["var",3]],["var",5]],["var",2]],["var",4]],["var",0]],["hyp",4]],["eq_subst",["eq",["add",["var",0],["var",4]],["var",6]],["hyp",0],["hyp",2]]],["eq_sym",["eq_trans",["eq_sym",["hyp",7]],["hyp",5]]]]]]]]]]]]]]]]]]]]]]]]],[176,["forall",["forall",["imp",["eq",["add",["var",1],["var",0]],["zero"]],["eq",["var",1],["zero"]]]]],[2,20],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_trans",["forall_elim",["forall_elim",["hyp",4],["var",0]],["var",1]],["hyp",2]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",1],["var",0]]]]]]]]],[896,["forall",["forall",["imp",["eq",["mul",["var",1],["var",0]],["succ",["zero"]]],["and",["eq",["var",1],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]]]],[4,20,9],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["forall",["imp",["eq",["mul",["var",1],["var",0]],["succ",["zero"]]],["and",["eq",["var",1],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]]],["forall_intro",["imp_intro",["bot_elim",["imp_elim",["forall_elim",["axiom","PA1"],["zero"]],["eq_sym",["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["hyp",3],["var",0]],["hyp",0]]]]]]],["forall_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["imp",["eq",["mul",["succ",["var",2]],["var",0]],["succ",["zero"]]],["and",["eq",["succ",["var",2]],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]],["imp_intro",["bot_elim",["imp_elim",["forall_elim",["axiom","PA1"],["zero"]],["eq_sym",["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["axiom","PA5"],["succ",["var",1]]],["hyp",0]]]]]],["forall_intro",["imp_intro",["imp_intro",["and_intro",["cong_s",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["succ",["var",2]],["var",0]]],["var",2]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["add",["mul",["succ",["var",2]],["var",0]],["var",2]]],["zero"]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",2]],["var",0]]],["var",2]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA6"],["succ",["var",2]]],["var",0]],["hyp",0]]]]]],["cong_s",["eq_subst",["eq",["var",0],["zero"]],["forall_elim",["hyp",3],["var",0]],["eq_subst",["eq",["var",0],["zero"]],["forall_elim",["axiom","PA3"],["mul",["succ",["zero"]],["var",0]]],["eq_subst",["eq",["add",["mul",["succ",["zero"]],["var",1]],["var",0]],["zero"]],["imp_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["succ",["var",2]],["var",0]]],["var",2]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["add",["mul",["succ",["var",2]],["var",0]],["var",2]]],["zero"]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",2]],["var",0]]],["var",2]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA6"],["succ",["var",2]]],["var",0]],["hyp",0]]]]],["eq_subst",["eq",["add",["mul",["succ",["var",0]],["var",1]],["var",3]],["zero"]],["imp_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["succ",["var",2]],["var",0]]],["var",2]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["add",["mul",["succ",["var",2]],["var",0]],["var",2]]],["zero"]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",2]],["var",0]]],["var",2]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA6"],["succ",["var",2]]],["var",0]],["hyp",0]]]]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["add",["mul",["succ",["var",2]],["var",0]],["var",2]]],["zero"]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",2]],["var",0]]],["var",2]],["eq_subst",["eq",["var",0],["succ",["zero"]]],["forall_elim",["forall_elim",["axiom","PA6"],["succ",["var",2]]],["var",0]],["hyp",0]]]]]]]]]]]]]],["var",0]]]]]],["var",0]]]]]]],[160,["forall",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["mul",["var",1],["var",0]],["zero"]],["bot"]]]]]],[21],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",3],["var",1]],["var",0]],["hyp",0]],["imp_elim",["hyp",3],["hyp",0]],["imp_elim",["hyp",2],["hyp",0]]]]]]]]]],[576,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["mul",["var",2],["var",1]],["mul",["var",2],["var",0]]],["eq",["var",1],["var",0]]]]]]],[21,47,24,12],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["ind",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["mul",["var",2],["var",1]],["mul",["var",2],["var",0]]],["eq",["var",1],["var",0]]]]],["forall_intro",["imp_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",1]],["var",0]],["eq_sym",["eq_subst",["eq",["var",0],["mul",["var",2],["var",1]]],["forall_elim",["axiom","PA5"],["var",1]],["hyp",0]]]],["bot_elim",["imp_elim",["hyp",2],["hyp",0]]],["eq_sym",["hyp",0]]]]]],["forall_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["imp",["imp",["eq",["var",3],["zero"]],["bot"]],["imp",["eq",["mul",["var",3],["succ",["var",2]]],["mul",["var",3],["var",0]]],["eq",["succ",["var",2]],["var",0]]]],["imp_intro",["imp_intro",["bot_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",2]],["succ",["var",1]]],["hyp",1]],["forall_elim",["hyp",3],["var",1]]],["eq_subst",["eq",["mul",["var",3],["succ",["var",2]]],["var",0]],["forall_elim",["axiom","PA5"],["var",2]],["hyp",0]]]]]],["forall_intro",["imp_intro",["imp_intro",["imp_intro",["cong_s",["imp_elim",["imp_elim",["forall_elim",["hyp",3],["var",0]],["hyp",1]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",3],["var",2]]],["mul",["var",3],["var",0]]],["var",3]],["eq_subst",["eq",["add",["mul",["var",4],["var",3]],["var",4]],["var",0]],["forall_elim",["forall_elim",["axiom","PA6"],["var",3]],["var",0]],["eq_subst",["eq",["var",0],["mul",["var",4],["succ",["var",1]]]],["forall_elim",["forall_elim",["axiom","PA6"],["var",3]],["var",2]],["hyp",0]]]]]]]]]]],["var",0]]]]]]]]]]]],[272,["forall",["forall",["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["mul",["var",2],["var",0]],["mul",["var",1],["var",0]]],["eq",["var",2],["var",1]]]]]]],[6,48],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",4]],["eq_trans",["forall_elim",["forall_elim",["hyp",6],["var",0]],["var",2]],["eq_trans",["hyp",3],["forall_elim",["forall_elim",["hyp",6],["var",1]],["var",0]]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",2],["var",0]]]]]]]]]]],[376,["forall",["forall",["imp",["eq",["succ",["succ",["zero"]]],["mul",["succ",["succ",["var",1]]],["succ",["succ",["var",0]]]]],["bot"]]]],[5,45,21],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",1],["succ",["var",1]]],["succ",["succ",["var",0]]]],["imp_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["succ",["var",1]],["succ",["succ",["var",0]]]]],["var",0]],["eq_sym",["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["zero"]],["add",["mul",["succ",["var",1]],["succ",["succ",["var",0]]]],["var",0]]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["succ",["zero"]]],["succ",["add",["mul",["succ",["var",1]],["succ",["succ",["var",0]]]],["var",0]]]],["eq_subst",["eq",["succ",["succ",["zero"]]],["succ",["var",0]]],["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",1]],["succ",["succ",["var",0]]]]],["var",0]],["eq_subst",["eq",["succ",["succ",["zero"]]],["var",0]],["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["var",1]],["succ",["succ",["var",0]]]]],["succ",["var",0]]],["eq_subst",["eq",["succ",["succ",["zero"]]],["var",0]],["forall_elim",["forall_elim",["hyp",3],["succ",["var",1]]],["succ",["succ",["var",0]]]],["hyp",0]]]]]]]]],["imp_elim",["forall_elim",["axiom","PA1"],["var",1]],["hyp",0]],["imp_elim",["forall_elim",["axiom","PA1"],["succ",["var",0]]],["hyp",0]]]]]]]]]],[488,["and",["imp",["eq",["succ",["succ",["zero"]]],["succ",["zero"]]],["bot"]],["forall",["forall",["imp",["eq",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]],["or",["eq",["var",1],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]]]]],[4,50],["imp_intro",["imp_intro",["and_intro",["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["zero"]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["succ",["zero"]]],["zero"]],["hyp",0]]]],["forall_intro",["ind",["imp",["eq",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]],["or",["eq",["var",1],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]],["imp_intro",["bot_elim",["imp_elim",["forall_elim",["axiom","PA1"],["succ",["zero"]]],["eq_subst",["eq",["succ",["succ",["zero"]]],["var",0]],["forall_elim",["axiom","PA5"],["var",0]],["hyp",0]]]]],["forall_intro",["imp_intro",["forall_elim",["ind",["imp",["eq",["succ",["succ",["zero"]]],["mul",["var",2],["succ",["var",0]]]],["or",["eq",["var",2],["succ",["zero"]]],["eq",["succ",["var",0]],["succ",["zero"]]]]],["imp_intro",["or_intro_r",["eq_refl",["succ",["zero"]]]]],["forall_intro",["imp_intro",["forall_elim",["ind",["imp",["eq",["succ",["succ",["zero"]]],["mul",["var",0],["succ",["succ",["var",1]]]]],["or",["eq",["var",0],["succ",["zero"]]],["eq",["succ",["succ",["var",1]]],["succ",["zero"]]]]],["imp_intro",["bot_elim",["imp_elim",["forall_elim",["axiom","PA1"],["succ",["zero"]]],["eq_subst",["eq",["succ",["succ",["zero"]]],["var",0]],["forall_elim",["hyp",4],["succ",["succ",["var",0]]]],["hyp",0]]]]],["forall_intro",["imp_intro",["forall_elim",["ind",["imp",["eq",["succ",["succ",["zero"]]],["mul",["succ",["var",0]],["succ",["succ",["var",2]]]]],["or",["eq",["succ",["var",0]],["succ",["zero"]]],["eq",["succ",["succ",["var",2]]],["succ",["zero"]]]]],["imp_intro",["or_intro_l",["eq_refl",["succ",["zero"]]]]],["forall_intro",["imp_intro",["imp_intro",["bot_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",0]],["var",2]],["hyp",0]]]]]]],["var",0]]]]],["var",2]]]]],["var",0]]]]]]]]]],[72,["forall",["exists",["eq",["zero"],["mul",["var",1],["var",0]]]]],[],["forall_intro",["exists_intro",["zero"],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]],["eq_refl",["zero"]]]]]],[64,["forall",["exists",["eq",["var",1],["mul",["var",1],["var",0]]]]],[10],["imp_intro",["forall_intro",["exists_intro",["succ",["zero"]],["eq_sym",["forall_elim",["hyp",0],["var",0]]]]]]],[192,["forall",["forall",["forall",["imp",["exists",["eq",["var",2],["mul",["var",3],["var",0]]]],["imp",["exists",["eq",["var",1],["mul",["var",3],["var",0]]]],["exists",["eq",["add",["var",2],["var",1]],["mul",["var",3],["var",0]]]]]]]]],[7],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["var",1],["var",0]],["eq_subst",["eq",["add",["var",0],["var",3]],["mul",["var",5],["add",["var",2],["var",1]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["add",["mul",["var",5],["var",2]],["var",0]],["mul",["var",5],["add",["var",2],["var",1]]]],["eq_sym",["hyp",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",4]],["var",1]],["var",0]]]]]]]]]]]]]]],[136,["forall",["forall",["forall",["imp",["exists",["eq",["var",2],["mul",["var",3],["var",0]]]],["exists",["eq",["mul",["var",2],["var",1]],["mul",["var",3],["var",0]]]]]]]],[8],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_intro",["mul",["var",0],["var",1]],["eq_subst",["eq",["mul",["var",0],["var",2]],["mul",["var",4],["mul",["var",1],["var",2]]]],["eq_sym",["hyp",0]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",3]],["var",0]],["var",1]]]]]]]]]]],[184,["forall",["forall",["forall",["imp",["exists",["eq",["var",1],["mul",["var",3],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",2],["var",0]]]],["exists",["eq",["var",1],["mul",["var",2],["var",0]]]]]]]]],[8],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["mul",["var",0],["var",1]],["eq_subst",["eq",["var",0],["mul",["var",4],["mul",["var",1],["var",2]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["mul",["var",0],["var",2]],["mul",["var",4],["mul",["var",1],["var",2]]]],["eq_sym",["hyp",0]],["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",3]],["var",0]],["var",1]]]]]]]]]]]]]],[112,["forall",["imp",["exists",["eq",["succ",["zero"]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]],[46],["imp_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["and_elim_l",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]],["eq_sym",["hyp",0]]]]]]]]],[672,["forall",["forall",["forall",["forall",["imp",["eq",["mul",["var",3],["var",2]],["add",["mul",["var",3],["var",1]],["var",0]]],["exists",["eq",["var",1],["mul",["var",4],["var",0]]]]]]]]],[0,20,24,3,2],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["ind",["forall",["forall",["imp",["eq",["mul",["var",3],["var",2]],["add",["mul",["var",3],["var",1]],["var",0]]],["exists",["eq",["var",1],["mul",["var",4],["var",0]]]]]]],["forall_intro",["forall_intro",["imp_intro",["exists_intro",["zero"],["eq_subst",["eq",["var",0],["mul",["var",3],["zero"]]],["eq_sym",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",2],["var",1]]],["var",0]],["eq_sym",["eq_subst",["eq",["var",0],["add",["mul",["var",3],["var",2]],["var",1]]],["forall_elim",["axiom","PA5"],["var",2]],["hyp",0]]]]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",2]]],["eq_refl",["zero"]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["forall",["imp",["eq",["mul",["var",4],["succ",["var",3]]],["add",["mul",["var",4],["var",1]],["var",0]]],["exists",["eq",["var",1],["mul",["var",5],["var",0]]]]]],["forall_intro",["imp_intro",["exists_intro",["succ",["var",2]],["eq_sym",["eq_subst",["eq",["mul",["var",4],["succ",["var",3]]],["var",0]],["forall_elim",["hyp",6],["var",0]],["eq_subst",["eq",["mul",["var",4],["succ",["var",3]]],["add",["var",0],["var",1]]],["forall_elim",["axiom","PA5"],["var",3]],["hyp",0]]]]]]],["forall_intro",["imp_intro",["forall_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",4],["var",3]]],["add",["mul",["var",4],["var",1]],["var",0]]],["var",4]],["eq_trans",["eq_subst",["eq",["add",["mul",["var",5],["var",4]],["var",5]],["add",["var",0],["var",1]]],["forall_elim",["forall_elim",["axiom","PA6"],["var",4]],["var",1]],["eq_subst",["eq",["var",0],["add",["mul",["var",5],["succ",["var",2]]],["var",1]]],["forall_elim",["forall_elim",["axiom","PA6"],["var",4]],["var",3]],["hyp",0]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",4],["var",1]]],["var",4]],["var",0]],["eq_trans",["cong_add",["eq_refl",["mul",["var",4],["var",1]]],["forall_elim",["forall_elim",["hyp",3],["var",4]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",4],["var",1]]],["var",0]],["var",4]]]]]]]]]]]]],["var",0]]]]]]]]]]]]],[384,["forall",["forall",["forall",["forall",["forall",["imp",["exists",["eq",["var",4],["mul",["var",5],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",5],["var",0]]]],["imp",["eq",["var",3],["add",["mul",["var",2],["var",1]],["var",0]]],["exists",["eq",["var",1],["mul",["var",5],["var",0]]]]]]]]]]]],[8,58],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",2],["exists_elim",["hyp",2],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_trans",["eq_sym",["hyp",5]],["eq_trans",["hyp",6],["cong_add",["eq_subst",["eq",["mul",["var",0],["var",4]],["mul",["var",7],["mul",["var",1],["var",4]]]],["eq_sym",["hyp",4]],["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",6]],["var",0]],["var",3]]],["eq_refl",["var",2]]]]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",0],["mul",["var",0],["var",3]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",5],["var",6]]]]]]]]]]]]]]]],[264,["forall",["forall",["forall",["forall",["imp",["exists",["eq",["var",3],["mul",["var",4],["var",0]]]],["imp",["exists",["eq",["var",1],["mul",["var",4],["var",0]]]],["exists",["eq",["add",["mul",["var",3],["var",2]],["var",1]],["mul",["var",4],["var",0]]]]]]]]]],[8,7],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["mul",["var",1],["var",3]],["var",0]],["eq_subst",["eq",["add",["mul",["var",0],["var",4]],["var",3]],["mul",["var",6],["add",["mul",["var",2],["var",4]],["var",1]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["add",["mul",["mul",["var",6],["var",2]],["var",4]],["var",0]],["mul",["var",6],["add",["mul",["var",2],["var",4]],["var",1]]]],["eq_sym",["hyp",0]],["eq_trans",["cong_add",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",5]],["var",1]],["var",3]],["eq_refl",["mul",["var",5],["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",5]],["mul",["var",1],["var",3]]],["var",0]]]]]]]]]]]]]]]]]],[168,["forall",["and",["and",["exists",["eq",["var",1],["mul",["var",1],["var",0]]]],["exists",["eq",["zero"],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["zero"],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]]],[53,52],["imp_intro",["imp_intro",["forall_intro",["and_intro",["and_intro",["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",1],["var",0]]],["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",0],["var",0]]]],["forall_intro",["imp_intro",["imp_intro",["hyp",1]]]]]]]]],[184,["forall",["forall",["forall",["imp",["and",["and",["exists",["eq",["var",2],["mul",["var",3],["var",0]]]],["exists",["eq",["var",1],["mul",["var",3],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",4],["mul",["var",1],["var",0]]]]]]]],["exists",["eq",["var",2],["mul",["var",3],["var",0]]]]]]]],[],["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",1]],["and_elim_r",["hyp",2]]]],["and_elim_l",["hyp",1]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]],[184,["forall",["forall",["forall",["imp",["and",["and",["exists",["eq",["var",2],["mul",["var",3],["var",0]]]],["exists",["eq",["var",1],["mul",["var",3],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",4],["mul",["var",1],["var",0]]]]]]]],["exists",["eq",["var",1],["mul",["var",3],["var",0]]]]]]]],[],["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["and_elim_r",["hyp",2]]]],["and_elim_l",["hyp",1]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]],[208,["forall",["forall",["forall",["forall",["imp",["and",["and",["exists",["eq",["var",3],["mul",["var",4],["var",0]]]],["exists",["eq",["var",2],["mul",["var",4],["var",0]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["exists",["eq",["var",5],["mul",["var",1],["var",0]]]]]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",4],["mul",["var",1],["var",0]]]]]]]]]]],[],["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",4]],["hyp",3]]],["forall_elim",["hyp",0],["var",0]]]],["and_elim_r",["hyp",3]]]],["and_elim_l",["hyp",2]]]]]]]]]]],[736,["forall",["forall",["forall",["forall",["forall",["imp",["eq",["var",3],["add",["mul",["var",2],["var",1]],["var",0]]],["imp",["and",["and",["exists",["eq",["var",3],["mul",["var",5],["var",0]]]],["exists",["eq",["var",1],["mul",["var",5],["var",0]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",6],["mul",["var",1],["var",0]]]]]]]],["and",["and",["exists",["eq",["var",4],["mul",["var",5],["var",0]]]],["exists",["eq",["var",3],["mul",["var",5],["var",0]]]]],["forall",["imp",["exists",["eq",["var",5],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",6],["mul",["var",1],["var",0]]]]]]]]]]]]]]],[59,60],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["and_intro",["and_intro",["eq_subst",["exists",["eq",["var",1],["mul",["var",6],["var",0]]]],["eq_sym",["hyp",5]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",5]],["hyp",4]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",6],["var",4]]]],["hyp",1]],["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",1]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["hyp",0],["hyp",7]],["hyp",6]],["hyp",13]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",0],["var",3]]]],["forall_elim",["hyp",0],["var",4]]]],["forall_elim",["hyp",10],["var",0]]]]],["forall_elim",["hyp",4],["var",0]]]]]]]],["and_elim_r",["hyp",2]]]],["and_elim_l",["hyp",1]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]]]]]]],[736,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",1]],["var",2]]],["forall",["exists",["and",["and",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",3],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]]]]]]],[53,27,32,29,42,65],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["forall",["imp",["exists",["eq",["add",["var",0],["var",1]],["var",2]]],["forall",["exists",["and",["and",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",3],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["exists_intro",["var",0],["and_intro",["and_intro",["forall_elim",["hyp",6],["var",0]],["exists_intro",["zero"],["eq_trans",["imp_elim",["forall_elim",["hyp",5],["var",1]],["hyp",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]]]]],["forall_intro",["imp_intro",["imp_intro",["hyp",1]]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["imp_intro",["forall_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",1]],["succ",["var",2]]],["hyp",0]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",0]],["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["var",2]],["eq_trans",["eq_sym",["hyp",1]],["hyp",0]]]]],["exists_elim",["hyp",0],["exists_elim",["forall_elim",["imp_elim",["forall_elim",["hyp",4],["var",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",0]],["var",4]],["eq_subst",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["hyp",2],["and_elim_r",["hyp",0]]]]],["var",3]],["exists_intro",["var",0],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",0]],["var",3]],["var",4]],["var",2]],["var",1]],["and_elim_l",["hyp",1]]],["hyp",0]]]]]],["forall_elim",["imp_elim",["forall_elim",["hyp",2],["var",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",1]],["var",2]],["hyp",0]]],["var",0]]]]]]]]],["var",0]]]]]]]]]],[104,["forall",["forall",["exists",["and",["and",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]]]]],[13,66],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",0],["var",0]],["var",0]],["forall_elim",["hyp",1],["var",0]]],["var",1]]]]]]],[136,["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]],[],["forall_intro",["forall_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",1]],["hyp",2]]],["forall_elim",["hyp",2],["var",0]]]]]]]]]],[152,["forall",["forall",["imp",["and",["and",["exists",["eq",["var",2],["mul",["succ",["zero"]],["var",0]]]],["exists",["eq",["var",1],["mul",["succ",["zero"]],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["succ",["zero"]],["mul",["var",1],["var",0]]]]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]],[57],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["forall_elim",["hyp",3],["var",0]],["imp_elim",["imp_elim",["forall_elim",["and_elim_r",["hyp",2]],["var",0]],["hyp",1]],["hyp",0]]]]]]]]]]],[360,["forall",["forall",["forall",["forall",["eq",["add",["add",["var",3],["var",2]],["add",["var",1],["var",0]]],["add",["add",["var",1],["var",2]],["add",["var",3],["var",0]]]]]]]],[3,2],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",2]],["add",["var",1],["var",0]]],["eq_trans",["cong_add",["eq_refl",["var",3]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",1]],["var",0]]]],["eq_trans",["cong_add",["eq_refl",["var",3]],["cong_add",["forall_elim",["forall_elim",["hyp",0],["var",2]],["var",1]],["eq_refl",["var",0]]]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["add",["var",1],["var",2]]],["var",0]]],["eq_trans",["cong_add",["forall_elim",["forall_elim",["hyp",0],["var",3]],["add",["var",1],["var",2]]],["eq_refl",["var",0]]],["forall_elim",["forall_elim",["forall_elim",["hyp",1],["add",["var",1],["var",2]]],["var",3]],["var",0]]]]]]]]]]]]]],[864,["forall",["forall",["forall",["forall",["forall",["forall",["forall",["forall",["forall",["imp",["eq",["var",8],["add",["mul",["var",7],["var",6]],["var",5]]],["imp",["eq",["add",["mul",["var",7],["var",3]],["mul",["var",5],["var",2]]],["add",["var",4],["add",["mul",["var",7],["var",1]],["mul",["var",5],["var",0]]]]],["eq",["add",["mul",["var",8],["var",2]],["mul",["var",7],["add",["var",3],["mul",["var",6],["var",0]]]]],["add",["var",4],["add",["mul",["var",8],["var",0]],["mul",["var",7],["add",["var",1],["mul",["var",6],["var",2]]]]]]]]]]]]]]]]]],[3,2,7,8,11,70],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["eq_subst",["eq",["add",["mul",["var",0],["var",3]],["mul",["var",8],["add",["var",4],["mul",["var",7],["var",1]]]]],["add",["var",5],["add",["mul",["var",9],["var",1]],["mul",["var",8],["add",["var",2],["mul",["var",7],["var",3]]]]]]],["eq_sym",["hyp",1]],["eq_trans",["cong_add",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",7],["var",6]]],["var",5]],["var",2]],["eq_refl",["mul",["var",7],["add",["var",3],["mul",["var",6],["var",0]]]]]],["eq_trans",["cong_add",["eq_refl",["add",["mul",["mul",["var",7],["var",6]],["var",2]],["mul",["var",5],["var",2]]]],["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",7]],["var",3]],["mul",["var",6],["var",0]]]],["eq_trans",["cong_add",["eq_refl",["add",["mul",["mul",["var",7],["var",6]],["var",2]],["mul",["var",5],["var",2]]]],["cong_add",["eq_refl",["mul",["var",7],["var",3]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",7]],["var",6]],["var",0]]]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["mul",["var",7],["var",6]],["var",2]]],["mul",["var",5],["var",2]]],["mul",["var",7],["var",3]]],["mul",["mul",["var",7],["var",6]],["var",0]]],["eq_trans",["cong_add",["eq_refl",["add",["mul",["var",7],["var",3]],["mul",["var",5],["var",2]]]],["forall_elim",["forall_elim",["hyp",6],["mul",["mul",["var",7],["var",6]],["var",2]]],["mul",["mul",["var",7],["var",6]],["var",0]]]],["eq_trans",["cong_add",["hyp",0],["eq_refl",["add",["mul",["mul",["var",7],["var",6]],["var",0]],["mul",["mul",["var",7],["var",6]],["var",2]]]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",4]],["add",["mul",["var",7],["var",1]],["mul",["var",5],["var",0]]]],["add",["mul",["mul",["var",7],["var",6]],["var",0]],["mul",["mul",["var",7],["var",6]],["var",2]]]],["eq_trans",["cong_add",["eq_refl",["var",4]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",7],["var",1]]],["mul",["var",5],["var",0]]],["mul",["mul",["var",7],["var",6]],["var",0]]],["mul",["mul",["var",7],["var",6]],["var",2]]]],["eq_trans",["cong_add",["eq_refl",["var",4]],["cong_add",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",7],["var",6]]],["var",5]],["var",0]]],["eq_refl",["add",["mul",["var",7],["var",1]],["mul",["mul",["var",7],["var",6]],["var",2]]]]]],["eq_trans",["cong_add",["eq_refl",["var",4]],["cong_add",["eq_refl",["mul",["add",["mul",["var",7],["var",6]],["var",5]],["var",0]]],["cong_add",["eq_refl",["mul",["var",7],["var",1]]],["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",7]],["var",6]],["var",2]]]]],["cong_add",["eq_refl",["var",4]],["cong_add",["cong_mul",["eq_sym",["hyp",1]],["eq_refl",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",7]],["var",1]],["mul",["var",6],["var",2]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[1360,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",1]],["var",2]]],["forall",["exists",["and",["and",["and",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",3],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]],["exists",["exists",["exists",["exists",["eq",["add",["mul",["var",5],["var",3]],["mul",["var",6],["var",2]]],["add",["var",4],["add",["mul",["var",5],["var",1]],["mul",["var",6],["var",0]]]]]]]]]]]]]]],[0,27,32,29,42,61,65,71],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["forall",["imp",["exists",["eq",["add",["var",0],["var",1]],["var",2]]],["forall",["exists",["and",["and",["and",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",3],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]],["exists",["exists",["exists",["exists",["eq",["add",["mul",["var",5],["var",3]],["mul",["var",6],["var",2]]],["add",["var",4],["add",["mul",["var",5],["var",1]],["mul",["var",6],["var",0]]]]]]]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["exists_intro",["var",0],["and_intro",["eq_subst",["and",["and",["exists",["eq",["var",2],["mul",["var",2],["var",0]]]],["exists",["eq",["var",1],["mul",["var",2],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",3],["mul",["var",1],["var",0]]]]]]]],["eq_sym",["imp_elim",["forall_elim",["hyp",7],["var",1]],["hyp",0]]],["eq_subst",["and",["and",["exists",["eq",["var",2],["mul",["var",2],["var",0]]]],["exists",["eq",["zero"],["mul",["var",2],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",3],["mul",["var",1],["var",0]]]]]]]],["eq_sym",["imp_elim",["forall_elim",["hyp",7],["var",1]],["hyp",0]]],["forall_elim",["hyp",3],["var",0]]]],["exists_intro",["succ",["zero"]],["exists_intro",["zero"],["exists_intro",["zero"],["exists_intro",["zero"],["eq_subst",["eq",["add",["mul",["var",1],["succ",["zero"]]],["mul",["var",0],["zero"]]],["add",["var",1],["add",["mul",["var",1],["zero"]],["mul",["var",2],["zero"]]]]],["eq_sym",["imp_elim",["forall_elim",["hyp",7],["var",1]],["hyp",0]]],["eq_subst",["eq",["add",["mul",["var",1],["succ",["zero"]]],["var",0]],["add",["var",1],["add",["mul",["var",1],["zero"]],["mul",["var",2],["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["var",0],["add",["var",1],["add",["mul",["var",1],["zero"]],["mul",["var",2],["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["var",0],["succ",["zero"]]]]],["eq_subst",["eq",["mul",["var",1],["succ",["zero"]]],["add",["var",1],["add",["var",0],["mul",["var",2],["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]],["eq_subst",["eq",["mul",["var",1],["succ",["zero"]]],["add",["var",1],["add",["zero"],["var",0]]]],["eq_sym",["forall_elim",["axiom","PA5"],["var",1]]],["eq_subst",["eq",["mul",["var",1],["succ",["zero"]]],["add",["var",1],["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["zero"]]],["eq_subst",["eq",["mul",["var",1],["succ",["zero"]]],["var",0]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_subst",["eq",["var",0],["var",1]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",0]],["zero"]]],["eq_subst",["eq",["add",["var",0],["var",1]],["var",1]],["eq_sym",["forall_elim",["axiom","PA5"],["var",0]]],["eq_subst",["eq",["var",0],["var",1]],["eq_sym",["forall_elim",["hyp",8],["var",0]]],["eq_refl",["var",0]]]]]]]]]]]]]]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["imp_intro",["forall_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",1]],["succ",["var",2]]],["hyp",0]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",6],["var",1]],["var",0]],["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["var",2]],["eq_trans",["eq_sym",["hyp",1]],["hyp",0]]]]],["exists_elim",["hyp",0],["exists_elim",["forall_elim",["imp_elim",["forall_elim",["hyp",4],["var",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",9],["var",0]],["var",4]],["eq_subst",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["hyp",2],["and_elim_r",["hyp",0]]]]],["var",3]],["exists_elim",["and_elim_r",["hyp",0]],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_intro",["var",4],["and_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["var",4]],["var",7]],["var",8]],["var",6]],["var",5]],["and_elim_l",["hyp",5]]],["and_elim_l",["hyp",4]]],["exists_intro",["var",2],["exists_intro",["add",["var",3],["mul",["var",6],["var",0]]],["exists_intro",["var",0],["exists_intro",["add",["var",1],["mul",["var",6],["var",2]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",7]],["var",8]],["var",6]],["var",5]],["var",4]],["var",3]],["var",2]],["var",1]],["var",0]],["and_elim_l",["hyp",5]]],["hyp",0]]]]]]]]]]]]]]],["forall_elim",["imp_elim",["forall_elim",["hyp",2],["var",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",1]],["var",2]],["hyp",0]]],["var",0]]]]]]]]],["var",0]]]]]]]]]]]],[104,["forall",["forall",["exists",["and",["and",["and",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]],["exists",["exists",["exists",["exists",["eq",["add",["mul",["var",6],["var",3]],["mul",["var",5],["var",2]]],["add",["var",4],["add",["mul",["var",6],["var",1]],["mul",["var",5],["var",0]]]]]]]]]]]]],[13,72],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",0],["var",0]],["var",0]],["forall_elim",["hyp",1],["var",0]]],["var",1]]]]]]],[736,["forall",["forall",["forall",["forall",["forall",["forall",["forall",["forall",["imp",["eq",["add",["mul",["var",7],["var",4]],["mul",["var",6],["var",3]]],["add",["var",5],["add",["mul",["var",7],["var",2]],["mul",["var",6],["var",1]]]]],["eq",["add",["mul",["var",7],["mul",["var",4],["var",0]]],["mul",["mul",["var",6],["var",0]],["var",3]]],["add",["mul",["var",5],["var",0]],["add",["mul",["var",7],["mul",["var",2],["var",0]]],["mul",["mul",["var",6],["var",0]],["var",1]]]]]]]]]]]]]],[8,6,11],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["eq_trans",["cong_add",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",7]],["var",4]],["var",0]]],["eq_refl",["mul",["mul",["var",6],["var",0]],["var",3]]]],["eq_trans",["cong_add",["eq_refl",["mul",["mul",["var",7],["var",4]],["var",0]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",6]],["var",0]],["var",3]],["eq_trans",["cong_mul",["eq_refl",["var",6]],["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",3]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",6]],["var",3]],["var",0]]]]]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",7],["var",4]]],["mul",["var",6],["var",3]]],["var",0]]],["eq_trans",["cong_mul",["hyp",0],["eq_refl",["var",0]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",5]],["add",["mul",["var",7],["var",2]],["mul",["var",6],["var",1]]]],["var",0]],["eq_trans",["cong_add",["eq_refl",["mul",["var",5],["var",0]]],["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",7],["var",2]]],["mul",["var",6],["var",1]]],["var",0]]],["eq_trans",["cong_add",["eq_refl",["mul",["var",5],["var",0]]],["cong_add",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",7]],["var",2]],["var",0]],["eq_refl",["mul",["mul",["var",6],["var",1]],["var",0]]]]],["cong_add",["eq_refl",["mul",["var",5],["var",0]]],["cong_add",["eq_refl",["mul",["var",7],["mul",["var",2],["var",0]]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",6]],["var",1]],["var",0]],["eq_trans",["cong_mul",["eq_refl",["var",6]],["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",6]],["var",0]],["var",1]]]]]]]]]]]]]]]]]]]]]]]]]]],[776,["forall",["forall",["forall",["forall",["forall",["forall",["forall",["forall",["imp",["exists",["eq",["var",7],["mul",["var",8],["var",0]]]],["imp",["exists",["eq",["var",6],["mul",["var",8],["var",0]]]],["imp",["eq",["add",["mul",["var",6],["var",3]],["mul",["var",5],["var",2]]],["add",["var",4],["add",["mul",["var",6],["var",1]],["mul",["var",5],["var",0]]]]],["exists",["eq",["var",5],["mul",["var",8],["var",0]]]]]]]]]]]]]]],[8,7,2,58],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",2],["exists_elim",["hyp",2],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["var",9]],["mul",["var",1],["var",5]]],["mul",["var",0],["var",4]]],["eq_trans",["cong_add",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",9]],["var",1]],["var",5]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",9]],["var",0]],["var",4]]]],["eq_trans",["eq_subst",["eq",["add",["mul",["mul",["var",10],["var",2]],["var",6]],["mul",["mul",["var",10],["var",1]],["var",5]]],["add",["mul",["var",0],["var",6]],["mul",["var",8],["var",5]]]],["eq_sym",["hyp",5]],["eq_subst",["eq",["add",["mul",["mul",["var",10],["var",2]],["var",6]],["mul",["mul",["var",10],["var",1]],["var",5]]],["add",["mul",["mul",["var",10],["var",2]],["var",6]],["mul",["var",0],["var",5]]]],["eq_sym",["hyp",4]],["eq_refl",["add",["mul",["mul",["var",9],["var",1]],["var",5]],["mul",["mul",["var",9],["var",0]],["var",4]]]]]],["eq_trans",["hyp",6],["eq_trans",["forall_elim",["forall_elim",["hyp",10],["var",6]],["add",["mul",["var",8],["var",3]],["mul",["var",7],["var",2]]]],["eq_trans",["eq_subst",["eq",["add",["add",["mul",["var",0],["var",4]],["mul",["var",8],["var",3]]],["var",7]],["add",["add",["mul",["mul",["var",10],["var",2]],["var",4]],["mul",["mul",["var",10],["var",1]],["var",3]]],["var",7]]],["eq_sym",["hyp",5]],["eq_subst",["eq",["add",["add",["mul",["mul",["var",10],["var",2]],["var",4]],["mul",["var",0],["var",3]]],["var",7]],["add",["add",["mul",["mul",["var",10],["var",2]],["var",4]],["mul",["mul",["var",10],["var",1]],["var",3]]],["var",7]]],["eq_sym",["hyp",4]],["eq_refl",["add",["add",["mul",["mul",["var",9],["var",1]],["var",3]],["mul",["mul",["var",9],["var",0]],["var",2]]],["var",6]]]]],["eq_trans",["cong_add",["cong_add",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",9]],["var",1]],["var",3]],["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",9]],["var",0]],["var",2]]],["eq_refl",["var",6]]],["cong_add",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["var",9]],["mul",["var",1],["var",3]]],["mul",["var",0],["var",2]]]],["eq_refl",["var",6]]]]]]]]]]]],["forall_elim",["hyp",0],["var",6]]]],["forall_elim",["hyp",0],["add",["mul",["var",1],["var",3]],["mul",["var",0],["var",2]]]]]],["forall_elim",["hyp",0],["add",["mul",["var",1],["var",5]],["mul",["var",0],["var",4]]]]]],["forall_elim",["hyp",5],["var",9]]]]]]]]]]]]]]]]]]]]],[296,["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["exists",["exists",["exists",["exists",["eq",["add",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]],["add",["succ",["zero"]],["add",["mul",["var",5],["var",1]],["mul",["var",4],["var",0]]]]]]]]]]]],[73],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",0]],["exists_elim",["and_elim_r",["hyp",0]],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_intro",["var",3],["exists_intro",["var",2],["exists_intro",["var",1],["exists_intro",["var",0],["eq_subst",["eq",["add",["mul",["var",7],["var",4]],["mul",["var",6],["var",3]]],["add",["var",0],["add",["mul",["var",7],["var",2]],["mul",["var",6],["var",1]]]]],["imp_elim",["imp_elim",["forall_elim",["hyp",5],["var",4]],["and_elim_l",["and_elim_l",["and_elim_l",["hyp",4]]]]],["and_elim_r",["and_elim_l",["and_elim_l",["hyp",4]]]]],["hyp",0]]]]]]]]]]]]]]]],[424,["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["exists",["eq",["mul",["var",2],["var",1]],["mul",["var",3],["var",0]]]],["exists",["eq",["var",1],["mul",["var",3],["var",0]]]]]]]]],[53,9,76,74,75],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["hyp",1]],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",6]],["var",6]],["mul",["var",5],["var",4]]],["var",4]],["mul",["var",3],["var",4]]],["var",2]],["mul",["var",1],["var",4]]],["var",0]],["forall_elim",["hyp",10],["var",6]]],["hyp",4]],["eq_subst",["eq",["add",["mul",["var",7],["mul",["var",4],["var",5]]],["mul",["mul",["var",6],["var",5]],["var",3]]],["add",["var",0],["add",["mul",["var",7],["mul",["var",2],["var",5]]],["mul",["mul",["var",6],["var",5]],["var",1]]]]],["forall_elim",["hyp",9],["var",4]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",6]],["var",5]],["succ",["zero"]]],["var",3]],["var",2]],["var",1]],["var",0]],["var",4]],["hyp",0]]]]]]]]]]]]]]]]]]],[400,["forall",["forall",["or",["eq",["var",1],["var",0]],["imp",["eq",["var",1],["var",0]],["bot"]]]]],[],["forall_intro",["forall_elim",["ind",["forall",["or",["eq",["var",1],["var",0]],["imp",["eq",["var",1],["var",0]],["bot"]]]],["forall_intro",["forall_elim",["ind",["or",["eq",["zero"],["var",0]],["imp",["eq",["zero"],["var",0]],["bot"]]],["or_intro_l",["eq_refl",["zero"]]],["forall_intro",["imp_intro",["or_intro_r",["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["var",0]],["eq_sym",["hyp",0]]]]]]]],["var",0]]],["forall_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["or",["eq",["succ",["var",2]],["var",0]],["imp",["eq",["succ",["var",2]],["var",0]],["bot"]]],["or_intro_r",["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["var",1]],["hyp",0]]]],["forall_intro",["imp_intro",["imp_elim",["imp_intro",["or_elim",["hyp",0],["or_intro_l",["cong_s",["hyp",0]]],["or_intro_r",["imp_intro",["imp_elim",["hyp",1],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["var",2]],["var",0]],["hyp",0]]]]]]],["forall_elim",["hyp",1],["var",0]]]]]],["var",0]]]]]],["var",0]]]],[144,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["var",2],["mul",["var",1],["var",0]]],["imp",["eq",["var",1],["zero"]],["bot"]]]]]]],[4],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["hyp",2],["eq_trans",["hyp",1],["eq_subst",["eq",["mul",["var",0],["var",1]],["zero"]],["eq_sym",["hyp",0]],["forall_elim",["hyp",3],["var",0]]]]]]]]]]]]],[208,["forall",["forall",["imp",["and",["imp",["eq",["var",1],["succ",["zero"]]],["bot"]],["forall",["forall",["imp",["eq",["var",3],["mul",["var",1],["var",0]]],["or",["eq",["var",1],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["or",["eq",["var",0],["succ",["zero"]]],["eq",["var",1],["var",0]]]]]]],[10],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",0],["or_elim",["imp_elim",["forall_elim",["forall_elim",["and_elim_r",["hyp",2]],["var",1]],["var",0]],["hyp",0]],["or_intro_l",["hyp",0]],["or_intro_r",["eq_trans",["hyp",1],["eq_subst",["eq",["mul",["var",2],["var",0]],["var",2]],["eq_sym",["hyp",0]],["forall_elim",["hyp",4],["var",1]]]]]]]]]]]]],[320,["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["mul",["var",4],["var",3]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[56,77],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",2],["var",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",0]],["var",3]],["var",2]],["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",5],["var",0]],["hyp",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",1]],["var",0]],["var",2]],["hyp",2]],["hyp",1]]]]]]],["hyp",1]]],["hyp",0]]]]]]]]]]]]],[232,["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["var",3],["var",2]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[81,68],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",1],["var",0]]],["var",2]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",1]],["var",0]],["var",2]],["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",2]],["var",1]],["hyp",1]]],["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",2]],["var",0]],["hyp",0]]]]]]]]]]]],[160,["forall",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["var",1]],["mul",["var",2],["var",1]]]]]]],[28,39,9],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["eq_subst",["exists",["eq",["add",["var",0],["var",1]],["mul",["var",3],["var",2]]]],["forall_elim",["hyp",1],["var",0]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["succ",["zero"]]],["var",1]],["var",0]],["imp_elim",["forall_elim",["hyp",3],["var",1]],["hyp",0]]]]]]]]]]],[64,["forall",["eq",["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]],["mul",["mul",["var",0],["var",0]],["mul",["var",0],["var",0]]]]],[8],["imp_intro",["forall_intro",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["mul",["var",0],["var",0]]],["var",0]],["var",0]]]]],[656,["forall",["exists",["or",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]],["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]],[],["ind",["exists",["or",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]],["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["exists_intro",["zero"],["or_intro_l",["eq_trans",["eq_refl",["zero"]],["eq_trans",["eq_refl",["zero"]],["eq_sym",["eq_trans",["cong_mul",["cong_s",["cong_s",["eq_refl",["zero"]]]],["eq_refl",["zero"]]],["forall_elim",["axiom","PA5"],["succ",["succ",["zero"]]]]]]]]]],["forall_intro",["imp_intro",["exists_elim",["hyp",0],["or_elim",["hyp",0],["exists_intro",["var",0],["or_intro_r",["eq_subst",["eq",["succ",["var",0]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_refl",["succ",["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]],["exists_intro",["add",["var",0],["succ",["zero"]]],["or_intro_l",["eq_subst",["eq",["succ",["var",0]],["mul",["succ",["succ",["zero"]]],["add",["var",1],["succ",["zero"]]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["succ",["var",0]],["mul",["succ",["succ",["zero"]]],["add",["var",1],["succ",["zero"]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["succ",["succ",["var",0]]],["mul",["succ",["succ",["zero"]]],["add",["var",1],["succ",["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["succ",["succ",["zero"]]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["zero"]]],["eq_subst",["eq",["succ",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["succ",["succ",["zero"]]],["succ",["var",0]]]],["eq_sym",["forall_elim",["axiom","PA3"],["var",0]]],["eq_subst",["eq",["succ",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["succ",["zero"]]]],["var",0]]],["eq_subst",["eq",["succ",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["succ",["zero"]]]],["eq_subst",["eq",["succ",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["succ",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["succ",["var",0]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_refl",["succ",["succ",["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]]]]]]]]]]]]]]],[608,["forall",["forall",["forall",["imp",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",1]]],["imp",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]],["bot"]]]]]],[44],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["forall_elim",["axiom","PA1"],["zero"]],["eq_sym",["and_elim_r",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["succ",["succ",["zero"]]]],["var",2]],["var",1]],["zero"]],["var",0]],["succ",["zero"]]],["eq_subst",["eq",["var",0],["add",["mul",["succ",["succ",["zero"]]],["var",2]],["zero"]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["mul",["succ",["succ",["zero"]]],["var",2]],["var",0]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_refl",["mul",["succ",["succ",["zero"]]],["var",1]]]]]],["exists_intro",["succ",["zero"]],["eq_trans",["eq_trans",["cong_add",["cong_s",["eq_refl",["zero"]]],["cong_s",["eq_refl",["zero"]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["zero"]]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["succ",["zero"]]]]]],["eq_trans",["eq_refl",["succ",["succ",["zero"]]]],["eq_sym",["eq_refl",["succ",["succ",["zero"]]]]]]]]],["hyp",0]],["exists_intro",["zero"],["eq_trans",["eq_trans",["cong_add",["eq_refl",["zero"]],["cong_s",["cong_s",["eq_refl",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["zero"]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["zero"]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["zero"]]]]]]],["eq_trans",["eq_refl",["succ",["succ",["zero"]]]],["eq_sym",["eq_refl",["succ",["succ",["zero"]]]]]]]]]]]]]]]]]]],[216,["forall",["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]],["bot"]]]],[86],["imp_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",3]],["hyp",4]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",4],["var",2]]]]]]]]]],[200,["forall",["forall",["imp",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["imp",["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["add",["var",2],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]],[7],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["var",1],["var",0]],["eq_subst",["eq",["add",["var",0],["var",3]],["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["var",0]],["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["succ",["succ",["zero"]]]],["var",1]],["var",0]]],["eq_refl",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]]]]]]]]],[392,["forall",["forall",["imp",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["add",["var",2],["var",1]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]]],[7,3],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["var",1],["var",0]],["eq_subst",["eq",["add",["var",0],["var",3]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["var",0]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["var",0]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["var",0]]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["var",0],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]],["zero"]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["succ",["succ",["zero"]]]],["var",1]],["var",0]]],["eq_refl",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]]]]]]]]]]]]]]]],[392,["forall",["forall",["imp",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["add",["var",2],["var",1]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]]],[7,1],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["var",1],["var",0]],["eq_subst",["eq",["add",["var",0],["var",3]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["add",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",0]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["var",0],["mul",["succ",["succ",["zero"]]],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",1]]],["zero"]]],["eq_subst",["eq",["add",["succ",["var",0]],["mul",["succ",["succ",["zero"]]],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_subst",["eq",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]],["zero"]]],["eq_subst",["eq",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]]],["eq_subst",["eq",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["succ",["succ",["zero"]]]],["var",1]],["var",0]]],["eq_subst",["eq",["var",0],["succ",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",4],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_refl",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]]]]]]]]]]]]]]]],[752,["forall",["forall",["imp",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["add",["var",2],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]],[7,1,3,2],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["add",["var",1],["var",0]],["succ",["zero"]]],["eq_subst",["eq",["add",["var",0],["var",3]],["mul",["succ",["succ",["zero"]]],["add",["add",["var",2],["var",1]],["succ",["zero"]]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["add",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",0]],["mul",["succ",["succ",["zero"]]],["add",["add",["var",2],["var",1]],["succ",["zero"]]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["var",0],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]]],["mul",["succ",["succ",["zero"]]],["add",["add",["var",2],["var",1]],["succ",["zero"]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",1]]],["zero"]]],["eq_subst",["eq",["add",["succ",["var",0]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]]],["mul",["succ",["succ",["zero"]]],["add",["add",["var",2],["var",1]],["succ",["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_subst",["eq",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["var",0]],["mul",["succ",["succ",["zero"]]],["add",["add",["var",2],["var",1]],["succ",["zero"]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["succ",["var",0]]],["mul",["succ",["succ",["zero"]]],["add",["add",["var",2],["var",1]],["succ",["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["var",0],["mul",["succ",["succ",["zero"]]],["add",["add",["var",2],["var",1]],["succ",["zero"]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["succ",["succ",["zero"]]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["add",["var",1],["var",0]]],["zero"]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["succ",["succ",["zero"]]],["succ",["var",0]]]],["eq_sym",["forall_elim",["axiom","PA3"],["add",["var",1],["var",0]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["succ",["zero"]]]],["add",["var",1],["var",0]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]],["succ",["zero"]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]],["zero"]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["succ",["var",0]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["succ",["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["succ",["succ",["zero"]]]],["var",1]],["var",0]]],["eq_subst",["eq",["succ",["var",0]],["succ",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",4],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["var",0]],["succ",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_subst",["eq",["succ",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",2]]]]],["succ",["succ",["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",4],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_refl",["succ",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",0]],["mul",["succ",["succ",["zero"]]],["var",1]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[128,["forall",["forall",["imp",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["mul",["var",2],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]],[8],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_intro",["mul",["var",0],["var",1]],["eq_subst",["eq",["mul",["var",0],["var",2]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",2]]]],["eq_sym",["hyp",0]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["succ",["succ",["zero"]]]],["var",0]],["var",1]]]]]]]]]],[160,["forall",["forall",["eq",["mul",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]],[8,6],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["eq_trans",["forall_elim",["forall_elim",["hyp",0],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["succ",["succ",["zero"]]]],["var",0]],["var",1]],["cong_mul",["eq_refl",["succ",["succ",["zero"]]]],["forall_elim",["forall_elim",["hyp",0],["var",0]],["var",1]]]]]]]]]],[640,["forall",["forall",["imp",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["mul",["var",2],["var",1]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]]],[7,11,3,1,93],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_intro",["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]],["var",0]],["var",1]],["eq_subst",["eq",["mul",["var",0],["var",3]],["add",["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",1]],["var",2]]],["succ",["zero"]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",0]],["add",["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",1]],["var",2]]],["succ",["zero"]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["var",0],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]]],["add",["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",1]],["var",2]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",1]]],["zero"]]],["eq_subst",["eq",["mul",["succ",["var",0]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]]],["add",["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",1]],["var",2]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["var",0]],["add",["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",1]],["var",2]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["succ",["var",0]]],["add",["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",2]],["succ",["zero"]]],["var",1]],["var",2]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]],["var",0]],["var",1]]]],["zero"]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["add",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]],["var",0]],["var",1]]]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["mul",["succ",["succ",["zero"]]],["add",["mul",["var",0],["var",1]],["var",2]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",1]]],["zero"]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["mul",["succ",["succ",["zero"]]],["add",["mul",["succ",["var",0]],["var",1]],["var",2]]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_subst",["eq",["var",0],["succ",["mul",["succ",["succ",["zero"]]],["add",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["var",1]],["var",2]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["var",0],["succ",["mul",["succ",["succ",["zero"]]],["add",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["var",1]],["var",2]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",2]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",2]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["succ",["succ",["zero"]]]],["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["var",0]]],["var",1]]],["cong_s",["cong_add",["forall_elim",["forall_elim",["hyp",4],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_refl",["mul",["succ",["succ",["zero"]]],["var",1]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[512,["forall",["eq",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["mul",["succ",["succ",["zero"]]],["mul",["succ",["succ",["zero"]]],["var",0]]]]],[8],["imp_intro",["forall_intro",["eq_trans",["cong_mul",["eq_trans",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]],["eq_trans",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]],["eq_sym",["eq_trans",["cong_mul",["cong_s",["cong_s",["eq_refl",["zero"]]]],["cong_s",["cong_s",["eq_refl",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["succ",["zero"]]]],["succ",["zero"]]],["eq_trans",["cong_add",["eq_trans",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["succ",["zero"]]]],["zero"]],["eq_trans",["cong_add",["forall_elim",["axiom","PA5"],["succ",["succ",["zero"]]]],["eq_refl",["succ",["succ",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["zero"]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["zero"]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["zero"]]]]]]]],["eq_refl",["succ",["succ",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["succ",["zero"]]]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["succ",["zero"]]]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["succ",["succ",["zero"]]]]]]]]]]]]]],["eq_refl",["var",0]]],["forall_elim",["forall_elim",["forall_elim",["hyp",0],["succ",["succ",["zero"]]]],["succ",["succ",["zero"]]]],["var",0]]]]]],[352,["forall",["forall",["imp",["and",["imp",["eq",["var",1],["succ",["zero"]]],["bot"]],["forall",["forall",["imp",["eq",["var",3],["mul",["var",1],["var",0]]],["or",["eq",["var",1],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]]]]],["or",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["exists",["eq",["var",1],["mul",["var",2],["var",0]]]]]]]],[67,80,69],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["forall_elim",["forall_elim",["hyp",3],["var",1]],["var",0]],["or_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",0]],["hyp",1]],["and_elim_l",["and_elim_l",["hyp",0]]]],["or_intro_l",["imp_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["eq_subst",["and",["and",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",3],["mul",["succ",["zero"]],["var",0]]]]],["forall",["imp",["exists",["eq",["var",5],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["succ",["zero"]],["mul",["var",1],["var",0]]]]]]]],["hyp",0],["eq_subst",["and",["and",["exists",["eq",["var",4],["mul",["var",2],["var",0]]]],["exists",["eq",["var",3],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",5],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["succ",["zero"]],["mul",["var",1],["var",0]]]]]]]],["hyp",0],["eq_subst",["and",["and",["exists",["eq",["var",4],["mul",["var",2],["var",0]]]],["exists",["eq",["var",3],["mul",["var",2],["var",0]]]]],["forall",["imp",["exists",["eq",["var",5],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]],["hyp",0],["hyp",1]]]]]],["or_intro_r",["exists_elim",["and_elim_r",["and_elim_l",["hyp",1]]],["exists_intro",["var",0],["eq_subst",["eq",["var",3],["mul",["var",0],["var",1]]],["eq_sym",["hyp",1]],["hyp",0]]]]]]]]]]]]]],[144,["forall",["forall",["imp",["and",["imp",["eq",["var",1],["succ",["zero"]]],["bot"]],["forall",["forall",["imp",["eq",["var",3],["mul",["var",1],["var",0]]],["or",["eq",["var",1],["succ",["zero"]]],["eq",["var",0],["succ",["zero"]]]]]]]],["imp",["imp",["exists",["eq",["var",1],["mul",["var",2],["var",0]]]],["bot"]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]],[96],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]],["hyp",1]],["hyp",0],["bot_elim",["imp_elim",["hyp",1],["hyp",0]]]]]]]]]],[616,["forall",["forall",["forall",["forall",["forall",["imp",["and",["and",["exists",["eq",["var",4],["mul",["var",5],["var",0]]]],["exists",["eq",["var",3],["mul",["var",5],["var",0]]]]],["forall",["imp",["exists",["eq",["var",5],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["exists",["eq",["var",6],["mul",["var",1],["var",0]]]]]]]],["imp",["imp",["eq",["var",4],["zero"]],["bot"]],["imp",["eq",["var",3],["mul",["var",4],["var",1]]],["imp",["eq",["var",2],["mul",["var",4],["var",0]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]]],[64,8,10,48,57],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",7]],["var",6]],["var",5]],["mul",["var",7],["var",2]]],["hyp",7]],["exists_intro",["var",1],["eq_trans",["hyp",5],["eq_trans",["cong_mul",["eq_refl",["var",7]],["hyp",1]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["var",7]],["var",2]],["var",1]]]]]]],["exists_intro",["var",0],["eq_trans",["hyp",4],["eq_trans",["cong_mul",["eq_refl",["var",7]],["hyp",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["var",7]],["var",2]],["var",0]]]]]]],["imp_elim",["forall_elim",["hyp",9],["var",3]],["exists_intro",["var",0],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",8]],["succ",["zero"]]],["mul",["var",3],["var",0]]],["hyp",7]],["eq_trans",["forall_elim",["hyp",11],["var",8]],["eq_trans",["hyp",0],["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",8]],["var",3]],["var",0]]]]]]]]]]]]]]]]]]]]]]]]]]]],[504,["forall",["forall",["imp",["exists",["eq",["add",["var",2],["var",1]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]],[85,88,91,87],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["forall_elim",["hyp",4],["var",1]],["exists_elim",["forall_elim",["hyp",5],["var",1]],["or_elim",["hyp",1],["or_elim",["hyp",1],["bot_elim",["imp_elim",["imp_elim",["forall_elim",["hyp",5],["add",["var",3],["var",2]]],["hyp",4]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",3]],["var",2]],["exists_intro",["var",1],["hyp",1]]],["exists_intro",["var",0],["hyp",0]]]]],["or_intro_l",["and_intro",["exists_intro",["var",1],["hyp",1]],["exists_intro",["var",0],["hyp",0]]]]],["or_elim",["hyp",1],["or_intro_r",["and_intro",["exists_intro",["var",1],["hyp",1]],["exists_intro",["var",0],["hyp",0]]]],["bot_elim",["imp_elim",["imp_elim",["forall_elim",["hyp",5],["add",["var",3],["var",2]]],["hyp",4]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",6],["var",3]],["var",2]],["exists_intro",["var",1],["hyp",1]]],["exists_intro",["var",0],["hyp",0]]]]]]]]]]]]]]]]],[112,["forall",["forall",["exists",["and",["and",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["exists",["eq",["var",2],["mul",["var",1],["var",0]]]]]]]]]]],[67],["imp_intro",["forall_intro",["forall_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]]]]],[200,["forall",["forall",["forall",["forall",["eq",["add",["add",["var",3],["var",2]],["add",["var",1],["var",0]]],["add",["add",["var",3],["var",1]],["add",["var",2],["var",0]]]]]]]],[2,70],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["eq_trans",["cong_add",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",2]],["eq_refl",["add",["var",1],["var",0]]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",2]],["var",3]],["var",1]],["var",0]],["cong_add",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",3]],["eq_refl",["add",["var",2],["var",0]]]]]]]]]]]]],[264,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["var",2],["mul",["var",1],["var",0]]],["imp",["eq",["var",0],["zero"]],["bot"]]]]]]],[79,6],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["hyp",0],["hyp",5]],["eq_trans",["hyp",4],["forall_elim",["forall_elim",["hyp",6],["var",1]],["var",0]]]],["hyp",3]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",4],["var",2]]]]]]]]]]]],[744,["forall",["forall",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["and",["and",["exists",["eq",["var",2],["mul",["var",3],["var",0]]]],["exists",["eq",["var",1],["mul",["var",3],["var",0]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["exists",["eq",["var",4],["mul",["var",1],["var",0]]]]]]]],["exists",["exists",["and",["and",["eq",["var",3],["mul",["var",4],["var",1]]],["eq",["var",2],["mul",["var",4],["var",0]]]],["and",["and",["imp",["eq",["var",4],["zero"]],["bot"]],["imp",["eq",["var",1],["zero"]],["bot"]]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]]]]],[62,63,79,102,98],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",1]],["var",0]],["hyp",0]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",3]],["var",2]],["var",1]],["hyp",1]],["exists_intro",["var",1],["exists_intro",["var",0],["and_intro",["and_intro",["hyp",1],["hyp",0]],["and_intro",["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",3]],["var",4]],["var",1]],["hyp",5]],["hyp",2]],["hyp",0]]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",3]],["var",4]],["var",1]],["hyp",5]],["hyp",2]],["hyp",0]]]],["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",4]],["var",0]],["hyp",4]],["hyp",1]],["hyp",0]]],["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",4]],["var",3]],["var",2]],["var",1]],["var",0]],["hyp",2]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",3]],["var",4]],["var",1]],["hyp",5]],["hyp",2]],["hyp",0]]]],["hyp",1]],["hyp",0]]]]]]]]]]]]]]]]]]]]],[328,["forall",["forall",["forall",["forall",["eq",["mul",["mul",["var",3],["var",2]],["mul",["var",1],["var",0]]],["mul",["mul",["var",3],["var",1]],["mul",["var",2],["var",0]]]]]]]],[8,6],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",2]],["mul",["var",1],["var",0]]],["eq_trans",["cong_mul",["eq_refl",["var",3]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",1]],["var",0]]]],["eq_trans",["cong_mul",["eq_refl",["var",3]],["cong_mul",["forall_elim",["forall_elim",["hyp",0],["var",2]],["var",1]],["eq_refl",["var",0]]]],["eq_trans",["cong_mul",["eq_refl",["var",3]],["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",2]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",1]],["mul",["var",2],["var",0]]]]]]]]]]]]]]],[176,["forall",["eq",["mul",["succ",["succ",["zero"]]],["var",0]],["add",["var",0],["var",0]]]],[6,10],["imp_intro",["imp_intro",["forall_intro",["eq_trans",["forall_elim",["forall_elim",["hyp",1],["succ",["succ",["zero"]]]],["var",0]],["eq_subst",["eq",["var",0],["add",["var",1],["var",1]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["var",0]],["succ",["zero"]]]],["imp_elim",["imp_intro",["eq_subst",["eq",["add",["var",0],["var",1]],["add",["var",1],["var",1]]],["eq_sym",["hyp",0]],["eq_refl",["add",["var",0],["var",0]]]]],["forall_elim",["hyp",0],["var",0]]]]]]]]],[240,["forall",["exists",["eq",["add",["var",0],["succ",["mul",["var",1],["var",1]]]],["mul",["succ",["var",1]],["succ",["var",1]]]]]],[26,39,12,40,34],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["mul",["var",0],["var",0]]],["mul",["succ",["var",0]],["var",0]]],["mul",["succ",["var",0]],["succ",["var",0]]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",0]],["succ",["var",0]]],["var",0]],["forall_elim",["hyp",4],["var",0]]]],["imp_elim",["forall_elim",["forall_elim",["hyp",1],["succ",["var",0]]],["var",0]],["imp_intro",["imp_elim",["forall_elim",["hyp",3],["var",0]],["hyp",0]]]]]]]]]]]],[400,["forall",["forall",["forall",["forall",["eq",["mul",["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["add",["add",["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]],["mul",["mul",["var",3],["var",0]],["mul",["var",3],["var",0]]]],["add",["mul",["mul",["var",2],["var",1]],["mul",["var",2],["var",1]]],["mul",["mul",["var",2],["var",0]],["mul",["var",2],["var",0]]]]]]]]]],[11,7,104],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["eq_subst",["eq",["var",0],["add",["add",["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]],["mul",["mul",["var",4],["var",1]],["mul",["var",4],["var",1]]]],["add",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",3],["var",3]]],["mul",["var",2],["var",2]]],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]]],["eq_subst",["eq",["add",["var",0],["mul",["mul",["var",3],["var",3]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]]],["add",["add",["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]],["mul",["mul",["var",4],["var",1]],["mul",["var",4],["var",1]]]],["add",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",3],["var",3]]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]]],["eq_subst",["eq",["add",["add",["mul",["mul",["var",4],["var",4]],["mul",["var",2],["var",2]]],["mul",["mul",["var",4],["var",4]],["mul",["var",1],["var",1]]]],["var",0]],["add",["add",["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]],["mul",["mul",["var",4],["var",1]],["mul",["var",4],["var",1]]]],["add",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",2],["var",2]]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]]],["cong_add",["cong_add",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",3]],["var",3]],["var",1]],["var",1]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",3]],["var",3]],["var",0]],["var",0]]],["cong_add",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",2]],["var",2]],["var",1]],["var",1]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",2]],["var",2]],["var",0]],["var",0]]]]]]]]]]]]]]],[208,["forall",["forall",["forall",["forall",["eq",["mul",["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["add",["add",["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]],["mul",["mul",["var",2],["var",0]],["mul",["var",2],["var",0]]]],["add",["mul",["mul",["var",3],["var",0]],["mul",["var",3],["var",0]]],["mul",["mul",["var",2],["var",1]],["mul",["var",2],["var",1]]]]]]]]]],[107,2,101],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["eq_trans",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",3]],["var",2]],["var",1]],["var",0]],["eq_trans",["cong_add",["eq_refl",["add",["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]],["mul",["mul",["var",3],["var",0]],["mul",["var",3],["var",0]]]]],["forall_elim",["forall_elim",["hyp",1],["mul",["mul",["var",2],["var",1]],["mul",["var",2],["var",1]]]],["mul",["mul",["var",2],["var",0]],["mul",["var",2],["var",0]]]]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]]],["mul",["mul",["var",3],["var",0]],["mul",["var",3],["var",0]]]],["mul",["mul",["var",2],["var",0]],["mul",["var",2],["var",0]]]],["mul",["mul",["var",2],["var",1]],["mul",["var",2],["var",1]]]]]]]]]]]]]],[184,["forall",["forall",["forall",["eq",["add",["var",2],["add",["var",1],["var",0]]],["add",["var",1],["add",["var",2],["var",0]]]]]]],[3,2],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",1]],["var",0]]],["eq_trans",["cong_add",["forall_elim",["forall_elim",["hyp",0],["var",2]],["var",1]],["eq_refl",["var",0]]],["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",2]],["var",0]]]]]]]]]],[1776,["forall",["forall",["forall",["imp",["or",["eq",["var",2],["add",["var",1],["var",0]]],["eq",["var",1],["add",["var",2],["var",0]]]],["eq",["add",["mul",["var",0],["var",0]],["add",["mul",["var",2],["var",1]],["mul",["var",2],["var",1]]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]]]]]],[7,11,8,6,3,2,109],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["hyp",0],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",2]],["mul",["var",3],["var",2]]]],["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["add",["var",2],["var",1]],["var",2]],["mul",["var",0],["var",2]]]],["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["add",["var",2],["var",1]],["var",2]],["mul",["add",["var",2],["var",1]],["var",2]]]],["add",["mul",["var",0],["var",3]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["add",["var",2],["var",1]],["var",2]],["mul",["add",["var",2],["var",1]],["var",2]]]],["add",["mul",["add",["var",2],["var",1]],["var",0]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["add",["var",2],["var",1]],["var",2]],["mul",["add",["var",2],["var",1]],["var",2]]]],["add",["var",0],["mul",["var",2],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["add",["var",1],["var",0]]],["var",1]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["var",0],["mul",["add",["var",2],["var",1]],["var",2]]]],["add",["add",["mul",["add",["var",2],["var",1]],["var",2]],["mul",["add",["var",2],["var",1]],["var",1]]],["mul",["var",2],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",1]],["var",0]],["var",1]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["var",0]]],["add",["add",["mul",["add",["var",2],["var",1]],["var",2]],["mul",["add",["var",2],["var",1]],["var",1]]],["mul",["var",2],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",1]],["var",0]],["var",1]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]]]],["add",["add",["var",0],["mul",["add",["var",2],["var",1]],["var",1]]],["mul",["var",2],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",1]],["var",0]],["var",1]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]]]],["add",["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["var",0]],["mul",["var",2],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",1]],["var",0]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]]]],["add",["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["add",["var",0],["mul",["var",1],["var",1]]]],["mul",["var",2],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["var",1]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["var",0]],["add",["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["add",["mul",["var",1],["var",2]],["mul",["var",1],["var",1]]]],["mul",["var",2],["var",2]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",1],["var",1]]],["mul",["var",0],["var",1]]],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",1]]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",1]]]],["add",["mul",["var",0],["var",1]],["mul",["var",0],["var",0]]]],["mul",["var",1],["var",1]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",1],["var",1]]],["mul",["var",0],["var",1]]],["add",["add",["mul",["var",0],["var",1]],["mul",["var",0],["var",0]]],["mul",["var",1],["var",1]]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]]]]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",0],["var",1]]],["mul",["var",0],["var",0]]],["mul",["var",1],["var",1]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["var",0]]]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["mul",["var",1],["var",2]],["add",["mul",["var",1],["var",1]],["mul",["var",2],["var",2]]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",3],["mul",["var",1],["var",1]]],["mul",["var",0],["var",1]]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",1],["var",1]]],["add",["mul",["var",0],["var",1]],["add",["mul",["var",0],["var",1]],["mul",["var",1],["var",1]]]]],["eq_trans",["cong_add",["eq_refl",["mul",["var",1],["var",1]]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",0],["var",1]]],["add",["mul",["var",0],["var",1]],["mul",["var",1],["var",1]]]]],["cong_add",["eq_refl",["mul",["var",1],["var",1]]],["cong_add",["eq_refl",["mul",["var",0],["var",1]]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",0],["var",1]]],["mul",["var",1],["var",1]]]]]]]]]]]]]]]]]]]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["var",0]],["mul",["var",3],["var",2]]]],["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["add",["var",3],["var",1]]],["mul",["var",3],["var",0]]]],["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["add",["var",3],["var",1]]],["mul",["var",3],["add",["var",3],["var",1]]]]],["add",["mul",["var",3],["var",3]],["mul",["var",0],["var",2]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["add",["var",3],["var",1]]],["mul",["var",3],["add",["var",3],["var",1]]]]],["add",["mul",["var",3],["var",3]],["mul",["add",["var",3],["var",1]],["var",0]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["var",0],["mul",["var",3],["add",["var",3],["var",1]]]]],["add",["mul",["var",3],["var",3]],["mul",["add",["var",3],["var",1]],["add",["var",3],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",2]],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]],["var",0]]],["add",["mul",["var",3],["var",3]],["mul",["add",["var",3],["var",1]],["add",["var",3],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",2]],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]],["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]]]],["add",["mul",["var",3],["var",3]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["add",["var",2],["var",0]]],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]],["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]]]],["add",["mul",["var",3],["var",3]],["add",["var",0],["mul",["add",["var",3],["var",1]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",0]],["var",2]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]],["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]]]],["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",0]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",3],["var",3]],["var",0]],["add",["mul",["var",3],["var",3]],["mul",["var",3],["var",1]]]]],["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["mul",["var",3],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["mul",["var",3],["var",3]],["var",0]]]],["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["mul",["var",3],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]]]],["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["var",0],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["var",0]],["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",2],["var",2]]],["mul",["var",0],["var",2]]],["add",["mul",["var",2],["var",2]],["mul",["var",0],["var",2]]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]]]]],["add",["mul",["var",3],["var",3]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",2],["var",2]]],["mul",["var",0],["var",2]]],["add",["mul",["var",0],["var",2]],["mul",["var",0],["var",0]]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["var",0]]]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",3],["mul",["var",2],["var",2]]],["mul",["var",0],["var",2]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",3]],["mul",["var",3],["var",3]]]]]],["add",["mul",["var",3],["var",3]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["hyp",3],["mul",["var",2],["var",2]]],["add",["mul",["var",0],["var",2]],["add",["mul",["var",0],["var",2]],["mul",["var",0],["var",0]]]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",3]],["mul",["var",3],["var",3]]]]]],["add",["mul",["var",3],["var",3]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",0],["var",2]]],["add",["mul",["var",0],["var",2]],["mul",["var",0],["var",0]]]],["mul",["var",2],["var",2]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",3]],["mul",["var",3],["var",3]]]]]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",0],["var",2]]],["mul",["var",0],["var",0]]],["mul",["var",2],["var",2]]]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",2],["var",2]]],["add",["mul",["var",0],["var",2]],["add",["mul",["var",0],["var",2]],["mul",["var",2],["var",2]]]]],["eq_trans",["cong_add",["eq_refl",["mul",["var",2],["var",2]]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",0],["var",2]]],["add",["mul",["var",0],["var",2]],["mul",["var",2],["var",2]]]]],["cong_add",["eq_refl",["mul",["var",2],["var",2]]],["cong_add",["eq_refl",["mul",["var",0],["var",2]]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",0],["var",2]]],["mul",["var",2],["var",2]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[384,["forall",["forall",["eq",["mul",["add",["var",1],["var",0]],["add",["var",1],["var",0]]],["add",["mul",["var",1],["var",1]],["add",["mul",["var",1],["var",0]],["add",["mul",["var",1],["var",0]],["mul",["var",0],["var",0]]]]]]]],[7,11,3,6],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["eq_subst",["eq",["var",0],["add",["mul",["var",2],["var",2]],["add",["mul",["var",2],["var",1]],["add",["mul",["var",2],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["add",["var",1],["var",0]]],["var",1]],["var",0]]],["eq_subst",["eq",["add",["var",0],["mul",["add",["var",2],["var",1]],["var",1]]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",2],["var",1]],["add",["mul",["var",2],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]],["var",1]]],["eq_subst",["eq",["add",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",2]]],["var",0]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",2],["var",1]],["add",["mul",["var",2],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]],["var",0]]],["eq_subst",["eq",["var",0],["add",["mul",["var",2],["var",2]],["add",["mul",["var",2],["var",1]],["add",["mul",["var",2],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",1],["var",1]]],["mul",["var",0],["var",1]]],["add",["mul",["var",1],["var",0]],["mul",["var",0],["var",0]]]]],["eq_subst",["eq",["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["var",0],["mul",["var",1],["var",1]]]]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",2],["var",1]],["add",["mul",["var",2],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["mul",["var",1],["var",2]],["mul",["var",1],["var",1]]]]],["add",["mul",["var",2],["var",2]],["add",["var",0],["add",["mul",["var",2],["var",1]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["mul",["var",1],["var",2]],["mul",["var",1],["var",1]]]]],["add",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",2]],["add",["var",0],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]]],["eq_refl",["add",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",1]],["add",["mul",["var",0],["var",1]],["mul",["var",0],["var",0]]]]]]]]]]]]]]]]]]]],[240,["forall",["forall",["eq",["mul",["add",["var",1],["var",0]],["add",["var",1],["var",0]]],["add",["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]],["add",["mul",["var",1],["var",0]],["mul",["var",1],["var",0]]]]]]],[111,2,109,3],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["eq_trans",["forall_elim",["forall_elim",["hyp",3],["var",1]],["var",0]],["eq_trans",["cong_add",["eq_refl",["mul",["var",1],["var",1]]],["eq_trans",["cong_add",["eq_refl",["mul",["var",1],["var",0]]],["forall_elim",["forall_elim",["hyp",2],["mul",["var",1],["var",0]]],["mul",["var",0],["var",0]]]],["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",1],["var",0]]],["mul",["var",0],["var",0]]],["mul",["var",1],["var",0]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["add",["mul",["var",1],["var",0]],["mul",["var",1],["var",0]]]]]]]]]]]]]],[208,["forall",["forall",["forall",["forall",["eq",["mul",["mul",["var",3],["var",1]],["mul",["var",2],["var",0]]],["mul",["mul",["var",3],["var",0]],["mul",["var",2],["var",1]]]]]]]],[104,6],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",2]],["var",1]],["var",0]]],["eq_trans",["cong_mul",["eq_refl",["mul",["var",3],["var",2]]],["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",2]],["var",0]],["var",1]]]]]]]]]]],[440,["forall",["forall",["forall",["forall",["forall",["imp",["or",["eq",["mul",["var",4],["var",1]],["add",["mul",["var",3],["var",2]],["var",0]]],["eq",["mul",["var",3],["var",2]],["add",["mul",["var",4],["var",1]],["var",0]]]],["eq",["mul",["add",["mul",["var",4],["var",4]],["mul",["var",3],["var",3]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]],["add",["mul",["add",["mul",["var",4],["var",2]],["mul",["var",3],["var",1]]],["add",["mul",["var",4],["var",2]],["mul",["var",3],["var",1]]]],["mul",["var",0],["var",0]]]]]]]]]],[108,110,112,113,2,3],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["eq_trans",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",4]],["var",3]],["var",2]],["var",1]],["eq_subst",["eq",["add",["add",["mul",["mul",["var",5],["var",3]],["mul",["var",5],["var",3]]],["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]]],["var",0]],["add",["mul",["add",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]],["add",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]]],["mul",["var",1],["var",1]]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",4],["var",1]]],["mul",["var",3],["var",2]]],["var",0]],["hyp",0]],["eq_subst",["eq",["add",["add",["mul",["mul",["var",5],["var",3]],["mul",["var",5],["var",3]]],["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]]],["add",["mul",["var",1],["var",1]],["add",["mul",["mul",["var",5],["var",2]],["mul",["var",4],["var",3]]],["mul",["mul",["var",5],["var",2]],["mul",["var",4],["var",3]]]]]],["add",["var",0],["mul",["var",1],["var",1]]]],["eq_sym",["eq_subst",["eq",["mul",["add",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]],["add",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]]],["add",["add",["mul",["mul",["var",5],["var",3]],["mul",["var",5],["var",3]]],["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]]],["add",["mul",["mul",["var",5],["var",2]],["mul",["var",4],["var",3]]],["var",0]]]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",4]],["var",3]],["var",2]],["var",1]],["eq_subst",["eq",["mul",["add",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]],["add",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]]],["add",["add",["mul",["mul",["var",5],["var",3]],["mul",["var",5],["var",3]]],["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]]],["add",["var",0],["mul",["mul",["var",5],["var",3]],["mul",["var",4],["var",2]]]]]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",4]],["var",3]],["var",2]],["var",1]],["forall_elim",["forall_elim",["hyp",4],["mul",["var",4],["var",2]]],["mul",["var",3],["var",1]]]]]],["eq_trans",["cong_add",["eq_refl",["add",["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]],["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]]]],["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["add",["mul",["mul",["var",4],["var",1]],["mul",["var",3],["var",2]]],["mul",["mul",["var",4],["var",1]],["mul",["var",3],["var",2]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["add",["mul",["mul",["var",4],["var",2]],["mul",["var",4],["var",2]]],["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]]]],["add",["mul",["mul",["var",4],["var",1]],["mul",["var",3],["var",2]]],["mul",["mul",["var",4],["var",1]],["mul",["var",3],["var",2]]]]],["mul",["var",0],["var",0]]]]]]]]]]]]]]]]]]]]],[328,["forall",["forall",["forall",["forall",["eq",["mul",["mul",["var",3],["var",2]],["mul",["var",1],["var",0]]],["mul",["mul",["var",3],["var",1]],["mul",["var",2],["var",0]]]]]]]],[8,6],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",2]],["mul",["var",1],["var",0]]],["eq_trans",["cong_mul",["eq_refl",["var",3]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",1]],["var",0]]]],["eq_trans",["cong_mul",["eq_refl",["var",3]],["cong_mul",["forall_elim",["forall_elim",["hyp",0],["var",2]],["var",1]],["eq_refl",["var",0]]]],["eq_trans",["cong_mul",["eq_refl",["var",3]],["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",2]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",3]],["var",1]],["mul",["var",2],["var",0]]]]]]]]]]]]]]],[80,["forall",["forall",["eq",["mul",["mul",["var",1],["var",0]],["mul",["var",1],["var",0]]],["mul",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]]]],[115],["imp_intro",["forall_intro",["forall_intro",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]],["var",1]],["var",0]]]]]],[144,["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["mul",["var",0],["var",0]],["zero"]],["bot"]]]],[21],["imp_intro",["forall_intro",["imp_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",0]],["hyp",0]],["imp_elim",["hyp",2],["hyp",0]],["imp_elim",["hyp",2],["hyp",0]]]]]]]],[280,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["mul",["mul",["var",2],["var",2]],["var",1]],["mul",["mul",["var",2],["var",2]],["var",0]]],["eq",["var",1],["var",0]]]]]]],[117,48],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["hyp",1],["hyp",6]],["hyp",0]]]],["forall_elim",["hyp",6],["var",2]]]],["hyp",3]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",2],["mul",["var",2],["var",2]]]]]]]]]]]],[952,["forall",["forall",["forall",["imp",["eq",["var",1],["add",["var",2],["var",0]]],["eq",["add",["mul",["var",2],["var",2]],["mul",["var",0],["add",["var",1],["var",2]]]],["mul",["var",1],["var",1]]]]]]],[7,11,6,3,2],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["mul",["var",1],["add",["var",0],["var",3]]]],["mul",["var",2],["var",2]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["mul",["var",1],["add",["add",["var",3],["var",1]],["var",3]]]],["mul",["var",0],["var",2]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["mul",["var",1],["add",["add",["var",3],["var",1]],["var",3]]]],["mul",["add",["var",3],["var",1]],["var",0]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["var",0]],["mul",["add",["var",3],["var",1]],["add",["var",3],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",0]],["add",["var",2],["var",0]]],["var",2]]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["add",["var",0],["mul",["var",1],["var",3]]]],["mul",["add",["var",3],["var",1]],["add",["var",3],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",0]],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]],["mul",["var",1],["var",3]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["add",["var",2],["var",0]]],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]],["mul",["var",1],["var",3]]]],["add",["var",0],["mul",["add",["var",3],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",0]],["var",2]]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]],["mul",["var",1],["var",3]]]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",0]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["add",["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]],["mul",["var",1],["var",3]]]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["var",0],["mul",["var",1],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",0]]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["var",0]],["add",["add",["mul",["var",3],["var",3]],["mul",["var",1],["var",3]]],["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",2]]],["mul",["var",0],["var",0]]],["mul",["var",0],["var",2]]]],["eq_subst",["eq",["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",1]],["mul",["var",1],["var",3]]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",2],["var",2]]],["mul",["var",0],["var",2]]],["add",["mul",["var",0],["var",2]],["mul",["var",0],["var",0]]]]],["eq_subst",["eq",["var",0],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",1],["mul",["var",2],["var",2]]],["add",["mul",["var",0],["var",2]],["add",["mul",["var",0],["var",0]],["mul",["var",0],["var",2]]]]]],["eq_subst",["eq",["var",0],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",2]]],["add",["mul",["var",0],["var",0]],["mul",["var",0],["var",2]]]],["mul",["var",2],["var",2]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",3]],["var",0]],["add",["mul",["var",3],["var",3]],["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",3]],["mul",["var",1],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",0],["var",2]]],["mul",["var",2],["var",2]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",1]],["add",["mul",["var",1],["var",3]],["mul",["var",3],["var",3]]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["hyp",1],["mul",["var",2],["var",2]]],["add",["mul",["var",0],["var",2]],["add",["mul",["var",0],["var",2]],["mul",["var",0],["var",0]]]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",1]],["add",["mul",["var",1],["var",3]],["mul",["var",3],["var",3]]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",2]]],["add",["mul",["var",0],["var",2]],["mul",["var",0],["var",0]]]],["mul",["var",2],["var",2]]]],["eq_subst",["eq",["add",["mul",["var",1],["var",3]],["add",["mul",["var",1],["var",1]],["add",["mul",["var",1],["var",3]],["mul",["var",3],["var",3]]]]],["add",["mul",["var",1],["var",3]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",2]]],["mul",["var",0],["var",0]]],["mul",["var",2],["var",2]]]],["cong_add",["eq_refl",["mul",["var",0],["var",2]]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["var",0],["var",2]]],["mul",["var",2],["var",2]]]],["eq_trans",["cong_add",["forall_elim",["forall_elim",["hyp",1],["mul",["var",0],["var",0]]],["mul",["var",0],["var",2]]],["eq_refl",["mul",["var",2],["var",2]]]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",2]]],["mul",["var",0],["var",0]]],["mul",["var",2],["var",2]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[208,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["exists",["eq",["add",["var",0],["mul",["var",2],["var",2]]],["mul",["var",1],["var",1]]]]]]],[39,38,14],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["mul",["var",1],["var",1]]],["mul",["var",0],["var",1]]],["mul",["var",0],["var",0]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",1]],["var",0]],["var",1]],["hyp",0]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",0]],["var",0]],["hyp",0]]]]]]]]]],[864,["forall",["forall",["imp",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]],["exists",["eq",["mul",["var",2],["var",2]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["zero"]]]]]]],[8,6],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_intro",["mul",["var",0],["var",0]],["eq_subst",["eq",["mul",["var",0],["var",2]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["zero"]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["mul",["succ",["succ",["zero"]]],["var",1]],["var",0]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["zero"]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]],["var",0]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",0],["var",0]]]]],["eq_subst",["eq",["var",0],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["succ",["succ",["zero"]]]],["var",0]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_trans",["cong_mul",["eq_refl",["succ",["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",0]],["succ",["succ",["zero"]]]],["var",0]]]],["eq_trans",["cong_mul",["eq_refl",["succ",["succ",["zero"]]]],["cong_mul",["forall_elim",["forall_elim",["hyp",1],["var",0]],["succ",["succ",["zero"]]]],["eq_refl",["var",0]]]],["eq_trans",["cong_mul",["eq_refl",["succ",["succ",["zero"]]]],["forall_elim",["forall_elim",["forall_elim",["hyp",2],["succ",["succ",["zero"]]]],["var",0]],["var",0]]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["succ",["succ",["zero"]]]],["succ",["succ",["zero"]]]],["mul",["var",0],["var",0]]]],["cong_mul",["eq_trans",["eq_trans",["cong_mul",["cong_s",["cong_s",["eq_refl",["zero"]]]],["cong_s",["cong_s",["eq_refl",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["succ",["zero"]]]],["succ",["zero"]]],["eq_trans",["cong_add",["eq_trans",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["succ",["zero"]]]],["zero"]],["eq_trans",["cong_add",["forall_elim",["axiom","PA5"],["succ",["succ",["zero"]]]],["eq_refl",["succ",["succ",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["zero"]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["zero"]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["zero"]]]]]]]],["eq_refl",["succ",["succ",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["succ",["zero"]]]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["succ",["zero"]]]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["succ",["succ",["zero"]]]]]]]]]]],["eq_trans",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]],["eq_sym",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]]]]],["eq_refl",["mul",["var",0],["var",0]]]]]]]]]]]]]]]]]]],[1872,["forall",["forall",["imp",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]],["exists",["eq",["mul",["var",2],["var",2]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["succ",["zero"]]]]]]]],[7,11,8,6,3,2,0,5,9,4,93,95],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_intro",["add",["mul",["var",0],["var",0]],["var",0]],["eq_subst",["eq",["mul",["var",0],["var",2]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]],["succ",["zero"]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]],["var",0]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]],["succ",["zero"]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["var",0],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["mul",["succ",["var",0]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["succ",["zero"]]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["var",0]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["zero"]]],["var",0]]],["zero"]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["succ",["var",0]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]],["succ",["zero"]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",0],["var",0]],["var",0]]]],["zero"]]],["eq_subst",["eq",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["succ",["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",0],["var",0]],["var",0]]]]],["eq_subst",["eq",["var",0],["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["mul",["succ",["succ",["zero"]]],["var",0]]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["var",0],["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["mul",["var",1],["var",1]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["succ",["succ",["succ",["succ",["zero"]]]]]],["mul",["var",0],["var",0]]],["var",0]]],["eq_subst",["eq",["succ",["add",["var",0],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",9],["succ",["mul",["succ",["succ",["zero"]]],["var",0]]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["add",["var",0],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["add",["add",["var",0],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["succ",["succ",["zero"]]]],["var",0]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["var",0]],["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["mul",["succ",["succ",["zero"]]],["mul",["var",0],["mul",["succ",["succ",["zero"]]],["var",0]]]]],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["succ",["var",0]],["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",7],["mul",["succ",["succ",["zero"]]],["mul",["var",0],["mul",["succ",["succ",["zero"]]],["var",0]]]]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["mul",["succ",["succ",["zero"]]],["var",0]]]]],["eq_subst",["eq",["succ",["var",0]],["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["mul",["var",0],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["eq_subst",["eq",["succ",["add",["mul",["succ",["succ",["zero"]]],["var",1]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["mul",["succ",["succ",["zero"]]],["var",1]]]]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["hyp",7],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["mul",["var",0],["var",0]]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]]],["cong_s",["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["mul",["var",0],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["eq_trans",["cong_add",["eq_sym",["eq_subst",["eq",["var",0],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["add",["var",0],["mul",["succ",["succ",["zero"]]],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["zero"]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["var",0]],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["var",0],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["zero"]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["var",0]],["mul",["succ",["succ",["zero"]]],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["var",0],["var",1]]],["mul",["succ",["succ",["zero"]]],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["zero"]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["var",0]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["var",0],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["zero"]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["var",0],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["var",0],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["zero"]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]],["var",0]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["var",0]]],["eq_subst",["eq",["add",["add",["mul",["zero"],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]],["add",["var",0],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",5],["zero"]],["var",0]]],["eq_subst",["eq",["add",["add",["var",0],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",3],["add",["add",["mul",["zero"],["var",0]],["var",0]],["var",0]]]],["eq_subst",["eq",["add",["add",["zero"],["add",["add",["var",0],["var",1]],["var",1]]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]],["add",["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",3],["var",0]]],["eq_subst",["eq",["add",["add",["zero"],["add",["add",["zero"],["var",1]],["var",1]]],["add",["add",["var",0],["var",1]],["var",1]]],["add",["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",3],["var",0]]],["eq_subst",["eq",["add",["add",["zero"],["add",["add",["zero"],["var",1]],["var",1]]],["add",["add",["zero"],["var",1]],["var",1]]],["add",["add",["add",["var",0],["var",1]],["var",1]],["add",["add",["mul",["zero"],["var",1]],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",3],["var",0]]],["eq_subst",["eq",["add",["add",["zero"],["add",["add",["zero"],["var",1]],["var",1]]],["add",["add",["zero"],["var",1]],["var",1]]],["add",["add",["add",["zero"],["var",1]],["var",1]],["add",["add",["var",0],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",3],["var",0]]],["eq_subst",["eq",["add",["var",0],["add",["add",["zero"],["var",1]],["var",1]]],["add",["add",["add",["zero"],["var",1]],["var",1]],["add",["add",["zero"],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",6],["add",["add",["zero"],["var",0]],["var",0]]]],["eq_subst",["eq",["add",["add",["var",0],["var",1]],["add",["add",["zero"],["var",1]],["var",1]]],["add",["add",["add",["zero"],["var",1]],["var",1]],["add",["add",["zero"],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",6],["var",0]]],["eq_subst",["eq",["add",["add",["var",1],["var",1]],["add",["var",0],["var",1]]],["add",["add",["add",["zero"],["var",1]],["var",1]],["add",["add",["zero"],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",6],["var",0]]],["eq_subst",["eq",["add",["add",["var",1],["var",1]],["add",["var",1],["var",1]]],["add",["add",["var",0],["var",1]],["add",["add",["zero"],["var",1]],["var",1]]]],["eq_sym",["forall_elim",["hyp",6],["var",0]]],["eq_subst",["eq",["add",["add",["var",1],["var",1]],["add",["var",1],["var",1]]],["add",["add",["var",1],["var",1]],["add",["var",0],["var",1]]]],["eq_sym",["forall_elim",["hyp",6],["var",0]]],["eq_refl",["add",["add",["var",0],["var",0]],["add",["var",0],["var",0]]]]]]]]]]]]]]]]]]]]]]]]]]],["eq_refl",["mul",["succ",["succ",["zero"]]],["mul",["var",0],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["eq_trans",["cong_add",["eq_sym",["forall_elim",["hyp",1],["var",0]]],["eq_refl",["mul",["succ",["succ",["zero"]]],["mul",["var",0],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["cong_add",["eq_refl",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]],["eq_trans",["cong_mul",["eq_refl",["succ",["succ",["zero"]]]],["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",0]]],["eq_sym",["forall_elim",["hyp",1],["mul",["var",0],["var",0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[304,["forall",["or",["exists",["eq",["mul",["var",1],["var",1]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["zero"]]]],["exists",["eq",["mul",["var",1],["var",1]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["succ",["zero"]]]]]]],[85,121,122],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_elim",["imp_intro",["exists_elim",["hyp",0],["or_elim",["hyp",0],["or_intro_l",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",2]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",4],["var",1]]]],["or_intro_r",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",2]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",3],["var",1]]]]]]],["forall_elim",["hyp",2],["var",0]]]]]]]],[176,["forall",["forall",["eq",["add",["mul",["var",1],["var",0]],["mul",["var",0],["var",1]]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]],[6,5,9],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["eq_trans",["cong_add",["eq_refl",["mul",["var",1],["var",0]]],["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",1]]],["eq_sym",["eq_trans",["forall_elim",["forall_elim",["hyp",1],["succ",["zero"]]],["mul",["var",1],["var",0]]],["cong_add",["forall_elim",["hyp",0],["mul",["var",1],["var",0]]],["eq_refl",["mul",["var",1],["var",0]]]]]]]]]]]]],[280,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["eq",["add",["mul",["var",0],["var",0]],["mul",["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]],["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]]]],["mul",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]]]]]]],[114,2,124],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["eq_trans",["forall_elim",["forall_elim",["hyp",2],["mul",["var",0],["var",0]]],["mul",["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]],["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]]]],["eq_sym",["eq_subst",["eq",["mul",["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]],["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]]],["add",["mul",["mul",["succ",["succ",["zero"]]],["mul",["var",3],["var",2]]],["var",0]],["mul",["var",1],["var",1]]]],["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",1]],["eq_subst",["eq",["mul",["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]],["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]]],["add",["mul",["var",0],["add",["mul",["var",3],["var",2]],["mul",["var",2],["var",3]]]],["mul",["var",1],["var",1]]]],["forall_elim",["forall_elim",["hyp",1],["var",2]],["var",1]],["eq_subst",["eq",["mul",["add",["mul",["var",3],["var",3]],["mul",["var",2],["var",2]]],["var",0]],["add",["mul",["add",["mul",["var",3],["var",2]],["mul",["var",2],["var",3]]],["add",["mul",["var",3],["var",2]],["mul",["var",2],["var",3]]]],["mul",["var",1],["var",1]]]],["forall_elim",["forall_elim",["hyp",2],["mul",["var",1],["var",1]]],["mul",["var",2],["var",2]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",1]],["var",2]],["var",0]],["or_intro_l",["hyp",0]]]]]]]]]]]]]]]],[96,["forall",["forall",["forall",["imp",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["eq",["add",["mul",["var",1],["var",1]],["mul",["var",2],["var",2]]],["mul",["var",0],["var",0]]]]]]],[2],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["eq_trans",["forall_elim",["forall_elim",["hyp",1],["mul",["var",1],["var",1]]],["mul",["var",2],["var",2]]],["hyp",0]]]]]]]],[192,["forall",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["eq",["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]],["zero"]],["bot"]]]]],[45,21],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["hyp",0]]],["imp_elim",["hyp",2],["hyp",0]],["imp_elim",["hyp",2],["hyp",0]]]]]]]]]],[136,["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]],[],["forall_intro",["forall_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",1]],["hyp",2]]],["forall_elim",["hyp",2],["var",0]]]]]]]]]],[224,["forall",["forall",["forall",["imp",["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["and",["eq",["add",["mul",["var",1],["var",1]],["mul",["var",2],["var",2]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[126,128],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["and_intro",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["var",0]],["hyp",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["hyp",0]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]]]],[112,["forall",["forall",["forall",["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]],["eq",["add",["mul",["mul",["var",2],["var",2]],["mul",["var",2],["var",2]]],["mul",["mul",["var",1],["var",1]],["mul",["var",1],["var",1]]]],["mul",["var",0],["var",0]]]]]]],[84],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["eq_subst",["eq",["add",["mul",["mul",["var",3],["var",3]],["mul",["var",3],["var",3]]],["var",0]],["mul",["var",1],["var",1]]],["forall_elim",["hyp",1],["var",1]],["eq_subst",["eq",["add",["var",0],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["var",1],["var",1]]],["forall_elim",["hyp",1],["var",2]],["hyp",0]]]]]]]]],[520,["forall",["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",3]],["var",4]]],["imp",["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["exists",["exists",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]]]],["imp",["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["mul",["var",2],["var",2]]]]]],["bot"]]]]]]]],[27,14,29],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_elim",["ind",["forall",["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",3]],["var",4]]],["imp",["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["exists",["exists",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]]]],["imp",["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["mul",["var",2],["var",2]]]]]],["bot"]]]]]]],["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]],["imp_elim",["forall_elim",["hyp",5],["var",2]],["hyp",2]]]]]]]]],["forall_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",0]],["var",2]],["hyp",0]],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",0]],["var",2]],["var",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",0]],["var",6]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["succ",["var",0]]],["var",5]],["succ",["var",6]]],["and_elim_r",["hyp",0]]],["hyp",5]]]],["hyp",4]],["and_elim_l",["hyp",0]]]]]]]]]]]]]]],["var",0]]]]]]],[264,["imp",["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["exists",["exists",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]]]],["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["bot"]]]]]],[131,13],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["hyp",0],["forall_elim",["hyp",6],["var",0]]],["hyp",5]],["hyp",4]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",3],["var",0]]]]]]]]]]],[336,["imp",["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["exists",["exists",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]]]],["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["bot"]]]]]]]]],[132,84,21],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["imp_elim",["hyp",7],["hyp",4]],["var",2]],["var",1]],["mul",["var",0],["var",0]]],["and_intro",["hyp",3],["and_intro",["hyp",2],["and_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",6],["var",0]],["var",0]],["hyp",0]],["imp_elim",["hyp",3],["hyp",0]],["imp_elim",["hyp",3],["hyp",0]]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["var",0]],["forall_elim",["hyp",6],["var",0]],["hyp",0]]]]]]]]]]]]]]]]]],[128,["forall",["imp",["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["mul",["var",1],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],[92],["imp_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",2]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",1],["var",0]]]]]]],[144,["forall",["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["mul",["var",1],["var",1]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]],[94],["imp_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",2]],["hyp",2]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",1],["var",0]]]]]]],[384,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["exists",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",0]]]],["imp",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]]]]],[134,135,85,90,86],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["forall_elim",["hyp",5],["var",0]],["or_elim",["hyp",0],["bot_elim",["exists_elim",["imp_elim",["forall_elim",["hyp",9],["var",3]],["hyp",3]],["exists_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",7],["mul",["var",3],["var",3]]],["var",2]],["imp_elim",["forall_elim",["hyp",9],["var",3]],["hyp",3]]],["exists_intro",["var",1],["hyp",1]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["mul",["var",5],["var",5]]],["var",1]],["var",0]],["hyp",1]],["eq_trans",["hyp",6],["hyp",0]]]]]],["exists_intro",["var",0],["hyp",0]]]]]]]]]]]]]]]],[384,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]]]]],[135,134,85,88,86],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["forall_elim",["hyp",5],["var",0]],["or_elim",["hyp",0],["bot_elim",["exists_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",6],["mul",["var",2],["var",2]]],["var",1]],["imp_elim",["forall_elim",["hyp",8],["var",2]],["hyp",2]]],["exists_intro",["var",0],["hyp",0]]],["exists_elim",["imp_elim",["forall_elim",["hyp",10],["var",4]],["hyp",4]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["mul",["var",5],["var",5]]],["var",1]],["var",0]],["eq_trans",["hyp",6],["hyp",1]]],["hyp",0]]]]],["exists_intro",["var",0],["hyp",0]]]]]]]]]]]]]]]],[376,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["or",["and",["exists",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]]]],[136,137],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["or_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",2]],["var",1]],["var",0]],["hyp",4]],["hyp",1]],["hyp",0]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",1]],["var",0]],["hyp",4]],["hyp",1]],["hyp",0]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]]]]]],[520,["forall",["forall",["imp",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["exists",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]],[134,135,89,90],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["forall_elim",["hyp",9],["var",1]],["hyp",3]]],["imp_elim",["forall_elim",["hyp",8],["var",0]],["hyp",2]]]],["forall_elim",["hyp",0],["mul",["var",0],["var",0]]]]],["forall_elim",["hyp",5],["mul",["var",1],["var",1]]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["forall_elim",["hyp",8],["var",1]],["hyp",3]]],["imp_elim",["forall_elim",["hyp",9],["var",0]],["hyp",2]]]],["forall_elim",["hyp",0],["mul",["var",0],["var",0]]]]],["forall_elim",["hyp",4],["mul",["var",1],["var",1]]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]]]]]],[248,["forall",["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]],[86,51,97,68],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",1],["succ",["succ",["zero"]]]],["var",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["succ",["succ",["zero"]]]],["var",0]],["hyp",3]],["imp_intro",["exists_elim",["hyp",0],["exists_elim",["hyp",2],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",1]],["var",0]],["hyp",1]],["hyp",0]]]]]]]]]]]]]],[168,["forall",["forall",["imp",["exists",["eq",["var",1],["mul",["var",2],["var",0]]]],["exists",["eq",["mul",["var",1],["var",1]],["mul",["var",2],["var",0]]]]]]],[55],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",3]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",1],["var",1]]]]]]]],[408,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[141,59,10,2,82],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",3]],["var",2]],["var",2]],["hyp",2]],["hyp",2]],["var",0]],["hyp",0]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",0]],["mul",["var",3],["var",3]]],["var",1]],["succ",["zero"]]],["mul",["var",2],["var",2]]],["imp_elim",["forall_elim",["forall_elim",["hyp",8],["var",0]],["var",3]],["hyp",0]]],["hyp",1]],["eq_subst",["eq",["mul",["var",4],["var",4]],["add",["var",0],["mul",["var",3],["var",3]]]],["eq_sym",["forall_elim",["hyp",6],["var",1]]],["eq_trans",["hyp",3],["forall_elim",["forall_elim",["hyp",5],["mul",["var",2],["var",2]]],["var",1]]]]]]]]]]]]]]]]]]]],[392,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[141,54,68,82],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",3]],["var",3]],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",3]],["var",2]],["hyp",2]]],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",3]],["var",2]],["hyp",2]]],["var",0]],["hyp",0]],["eq_subst",["exists",["eq",["var",1],["mul",["var",2],["var",0]]]],["eq_sym",["hyp",3]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",0]],["mul",["var",2],["var",2]]],["var",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",0]],["var",2]],["hyp",0]]],["hyp",1]]]]]]]]]]]]]]]]],[472,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["var",4],["var",3]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[142,143,82],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",7]],["hyp",6]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",7],["var",2]]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",7]],["hyp",6]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",6],["var",2]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",2]]]],["forall_elim",["hyp",2],["var",0]]]]]]]]]]]],[296,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["or",["and",["exists",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["succ",["succ",["zero"]]],["mul",["var",4],["var",3]]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]],[138,140,144,82],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",0]],["succ",["succ",["zero"]]]],["mul",["var",2],["var",1]]],["imp_elim",["forall_elim",["hyp",5],["var",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",2]],["var",1]],["var",0]],["hyp",2]],["hyp",0]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["var",0]],["hyp",2]],["hyp",1]]]]]]]]]]]]]],[360,["forall",["forall",["forall",["imp",["eq",["mul",["var",2],["var",2]],["add",["mul",["var",1],["var",1]],["var",0]]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["or",["and",["exists",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["add",["mul",["var",0],["var",0]],["mul",["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]],["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]]]],["mul",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]]],["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["succ",["succ",["zero"]]],["mul",["var",4],["var",3]]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]],[125,145],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["and_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",5]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",4],["var",2]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["hyp",0],["hyp",5]],["hyp",4]],["hyp",3]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",3],["var",2]]]]]]]]]]]]],[424,["forall",["forall",["forall",["imp",["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]],[141,59,10,82],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",3]],["var",2]],["var",2]],["and_elim_r",["hyp",2]]],["and_elim_r",["hyp",2]]],["var",0]],["hyp",1]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",0]],["mul",["var",1],["var",1]]],["mul",["var",3],["var",3]]],["succ",["zero"]]],["mul",["var",2],["var",2]]],["imp_elim",["forall_elim",["forall_elim",["hyp",6],["var",0]],["var",1]],["hyp",0]]],["imp_elim",["forall_elim",["forall_elim",["hyp",6],["var",0]],["var",3]],["hyp",1]]],["eq_subst",["eq",["mul",["var",2],["var",2]],["add",["var",0],["mul",["var",3],["var",3]]]],["eq_sym",["forall_elim",["hyp",4],["mul",["var",3],["var",3]]]],["eq_sym",["and_elim_l",["hyp",2]]]]]]]]]]]]]]]]]],[288,["forall",["forall",["forall",["imp",["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["imp",["exists",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["bot"]]]]]]],[51],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",1],["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",6]],["hyp",5]]],["forall_elim",["hyp",2],["succ",["succ",["zero"]]]]]]],["and_elim_r",["hyp",6]]]],["and_elim_l",["hyp",5]]]],["and_elim_r",["hyp",3]]]],["and_elim_l",["hyp",2]]]]]]]]]]],[736,["forall",["forall",["imp",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["succ",["succ",["zero"]]]]]]]]]],[122,7,3,2],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",1],["exists_elim",["hyp",1],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",3]],["var",1]],["hyp",1]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",8],["var",3]],["var",1]],["hyp",1]],["exists_intro",["add",["var",1],["var",0]],["eq_subst",["eq",["add",["var",0],["mul",["var",5],["var",5]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",2],["var",1]]],["succ",["succ",["zero"]]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["add",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]],["succ",["zero"]]],["var",0]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",2],["var",1]]],["succ",["succ",["zero"]]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["var",0],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]],["succ",["zero"]]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",2],["var",1]]],["succ",["succ",["zero"]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]],["zero"]]],["eq_subst",["eq",["add",["succ",["var",0]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]],["succ",["zero"]]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",2],["var",1]]],["succ",["succ",["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["eq_subst",["eq",["add",["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]]],["var",0]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",2],["var",1]]],["succ",["succ",["zero"]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]],["zero"]]],["eq_subst",["eq",["add",["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]]],["succ",["var",0]]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",2],["var",1]]],["succ",["succ",["zero"]]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]]],["eq_subst",["eq",["var",0],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",2],["var",1]]],["succ",["succ",["zero"]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",1],["var",0]]]],["succ",["zero"]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",1],["var",0]]]],["zero"]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["succ",["succ",["var",0]]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["add",["var",1],["var",0]]]]],["eq_subst",["eq",["succ",["add",["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["succ",["succ",["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["succ",["succ",["succ",["succ",["zero"]]]]]],["var",1]],["var",0]]],["eq_subst",["eq",["succ",["var",0]],["succ",["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",6],["succ",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]]],["eq_subst",["eq",["succ",["var",0]],["succ",["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]],["eq_subst",["eq",["succ",["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",2]]]]],["succ",["succ",["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",6],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]]]],["eq_refl",["succ",["succ",["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",1]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[1288,["forall",["forall",["imp",["eq",["mul",["var",1],["var",1]],["add",["mul",["succ",["succ",["succ",["succ",["zero"]]]]],["var",0]],["succ",["succ",["zero"]]]]],["bot"]]]],[123,44],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["forall_elim",["hyp",2],["var",1]],["exists_elim",["hyp",0],["imp_elim",["forall_elim",["axiom","PA1"],["succ",["zero"]]],["and_elim_r",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["succ",["succ",["succ",["succ",["zero"]]]]]],["mul",["var",2],["var",2]]],["var",1]],["succ",["succ",["zero"]]]],["var",0]],["zero"]],["hyp",2]],["exists_intro",["succ",["zero"]],["eq_trans",["eq_trans",["cong_add",["cong_s",["eq_refl",["zero"]]],["cong_s",["cong_s",["cong_s",["eq_refl",["zero"]]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["zero"]]],["succ",["succ",["zero"]]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["zero"]]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["zero"]]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["succ",["zero"]]]]]]]]]],["eq_trans",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]],["eq_sym",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]]]]]]],["hyp",0]],["exists_intro",["succ",["succ",["succ",["zero"]]]],["eq_trans",["eq_trans",["cong_add",["cong_s",["cong_s",["cong_s",["eq_refl",["zero"]]]]],["cong_s",["eq_refl",["zero"]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["succ",["succ",["zero"]]]]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["succ",["succ",["succ",["zero"]]]]]]]],["eq_trans",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]],["eq_sym",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]]]]]]]]]],["exists_elim",["hyp",0],["imp_elim",["forall_elim",["axiom","PA1"],["zero"]],["imp_elim",["forall_elim",["forall_elim",["axiom","PA2"],["succ",["zero"]]],["zero"]],["and_elim_r",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["succ",["succ",["succ",["succ",["zero"]]]]]],["mul",["var",2],["var",2]]],["var",1]],["succ",["succ",["zero"]]]],["var",0]],["succ",["zero"]]],["hyp",2]],["exists_intro",["succ",["zero"]],["eq_trans",["eq_trans",["cong_add",["cong_s",["eq_refl",["zero"]]],["cong_s",["cong_s",["cong_s",["eq_refl",["zero"]]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["zero"]]],["succ",["succ",["zero"]]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["zero"]]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["zero"]]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["succ",["zero"]]]]]]]]]],["eq_trans",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]],["eq_sym",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]]]]]]],["hyp",0]],["exists_intro",["succ",["succ",["zero"]]],["eq_trans",["eq_trans",["cong_add",["cong_s",["cong_s",["eq_refl",["zero"]]]],["cong_s",["cong_s",["eq_refl",["zero"]]]]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["succ",["zero"]]]],["succ",["zero"]]],["cong_s",["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["succ",["succ",["zero"]]]],["zero"]],["cong_s",["forall_elim",["axiom","PA3"],["succ",["succ",["zero"]]]]]]]]],["eq_trans",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]],["eq_sym",["eq_refl",["succ",["succ",["succ",["succ",["zero"]]]]]]]]]]]]]]]]]]]]]],[208,["forall",["forall",["forall",["imp",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["imp",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["bot"]]]]]]],[149,150],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["hyp",1]],["hyp",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",0]],["eq_trans",["eq_sym",["hyp",3]],["hyp",0]]]]]]]]]]]]],[144,["forall",["or",["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]],[85],["imp_intro",["forall_intro",["imp_elim",["imp_intro",["exists_elim",["hyp",0],["or_elim",["hyp",0],["or_intro_l",["exists_intro",["var",0],["hyp",0]]],["or_intro_r",["exists_intro",["var",0],["hyp",0]]]]]],["forall_elim",["hyp",0],["var",0]]]]]],[392,["forall",["forall",["forall",["imp",["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["or",["and",["exists",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]]],[152,148,151],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["forall_elim",["hyp",3],["var",2]],["or_elim",["forall_elim",["hyp",4],["var",1]],["bot_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["var",0]],["hyp",2]],["hyp",1]],["hyp",0]]],["or_intro_l",["and_intro",["hyp",1],["hyp",0]]]],["or_elim",["forall_elim",["hyp",4],["var",1]],["or_intro_r",["and_intro",["hyp",1],["hyp",0]]],["bot_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",0]],["and_elim_l",["hyp",2]]],["hyp",1]],["hyp",0]]]]]]]]]]]]],[248,["forall",["imp",["exists",["eq",["mul",["var",1],["var",1]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]],[85,134,86],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["exists_elim",["forall_elim",["hyp",3],["var",0]],["or_elim",["hyp",0],["bot_elim",["exists_elim",["imp_elim",["forall_elim",["hyp",4],["var",1]],["exists_intro",["var",0],["hyp",0]]],["exists_elim",["hyp",3],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",3],["var",3]]],["var",1]],["var",0]],["hyp",1]],["hyp",0]]]]],["exists_intro",["var",0],["hyp",0]]]]]]]]]],[232,["forall",["forall",["forall",["imp",["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]]]]],[153,139,154],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["forall_elim",["hyp",1],["var",0]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",2]],["var",1]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",0]],["hyp",0]]],["exists_intro",["var",0],["eq_trans",["eq_sym",["and_elim_l",["hyp",1]]],["hyp",0]]]]]]]]]]]]],[432,["forall",["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",5],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["exists",["eq",["var",4],["mul",["var",2],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]],[56],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",7]],["hyp",5]]],["forall_elim",["hyp",0],["var",4]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",6],["var",2]]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",6]],["hyp",4]]],["forall_elim",["hyp",0],["var",3]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",6],["var",1]]]]],["forall_elim",["hyp",4],["var",0]]]]]]]]]]]]]]],[312,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["exists",["eq",["add",["var",0],["succ",["mul",["var",2],["var",2]]]],["mul",["var",1],["var",1]]]]]]],[106,120,33],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",6],["var",1]]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",5]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",5],["succ",["var",1]]]]]],["forall_elim",["hyp",0],["mul",["var",0],["var",0]]]]],["forall_elim",["hyp",0],["mul",["succ",["var",1]],["succ",["var",1]]]]]],["forall_elim",["hyp",1],["mul",["var",1],["var",1]]]]]]]]]]],[344,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["mul",["var",2],["var",2]]],["mul",["var",1],["var",1]]]],["exists",["eq",["add",["var",0],["var",2]],["var",1]]]]]],[35,157,36],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["or_elim",["hyp",0],["hyp",0],["bot_elim",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",4]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",7],["var",0]]]],["hyp",5]]],["forall_elim",["hyp",0],["mul",["var",1],["var",1]]]]],["forall_elim",["hyp",4],["mul",["var",0],["var",0]]]]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",3],["var",1]]]]]]]]]],[344,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["succ",["mul",["var",2],["var",2]]]],["mul",["var",1],["var",1]]]],["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]]]]],[35,120,36],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["or_elim",["hyp",0],["bot_elim",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["hyp",5]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",4]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",7],["var",0]]]]],["forall_elim",["hyp",0],["mul",["var",0],["var",0]]]]],["forall_elim",["hyp",4],["mul",["var",1],["var",1]]]]],["hyp",0]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",3],["var",0]]]]]]]]]],[440,["forall",["forall",["imp",["eq",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]],["eq",["var",1],["var",0]]]]],[158,13,18],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_subst",["exists",["eq",["add",["var",0],["var",1]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",4]],["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",6],["mul",["var",0],["var",0]]]]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",5],["var",1]]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_subst",["exists",["eq",["add",["var",0],["mul",["var",2],["var",2]]],["var",1]]],["eq_sym",["hyp",4]],["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",6],["mul",["var",0],["var",0]]]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",5],["var",0]]]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",1],["var",1]]]]]]]]]],[104,["forall",["imp",["eq",["mul",["var",0],["var",0]],["zero"]],["eq",["var",0],["zero"]]]],[21],["imp_intro",["forall_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",1],["var",0]],["var",0]],["hyp",0]],["hyp",0],["hyp",0]]]]]],[2672,["forall",["forall",["forall",["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["var",5],["mul",["var",2],["var",1]]],["imp",["eq",["var",3],["mul",["var",2],["var",0]]],["imp",["eq",["mul",["var",5],["var",4]],["mul",["var",3],["var",3]]],["imp",["forall",["imp",["exists",["eq",["var",7],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",6],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["eq",["var",5],["mul",["var",1],["var",1]]],["eq",["var",4],["mul",["var",0],["var",0]]]]]]]]]]]]]]]]],[48,49,8,6,116,117,68,81,77,10,9],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["mul",["var",0],["var",0]]],["var",1]],["var",4]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",0]],["var",0]],["var",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",11],["var",1]],["var",0]],["hyp",0]]],["imp_elim",["forall_elim",["forall_elim",["hyp",11],["var",1]],["var",0]],["hyp",0]]]],["exists_intro",["var",2],["eq_trans",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",17],["var",2]],["mul",["var",1],["var",4]]],["mul",["var",2],["mul",["var",0],["var",0]]]],["hyp",6]],["eq_trans",["eq_subst",["eq",["mul",["var",3],["mul",["var",2],["var",5]]],["mul",["var",0],["var",5]]],["eq_sym",["hyp",4]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",15],["var",2]],["var",1]],["var",4]]]],["eq_trans",["hyp",2],["eq_subst",["eq",["mul",["var",0],["var",4]],["mul",["var",3],["mul",["var",3],["mul",["var",1],["var",1]]]]],["eq_sym",["hyp",3]],["eq_subst",["eq",["mul",["mul",["var",3],["var",1]],["var",0]],["mul",["var",3],["mul",["var",3],["mul",["var",1],["var",1]]]]],["eq_sym",["hyp",3]],["eq_trans",["forall_elim",["forall_elim",["hyp",13],["var",2]],["var",0]],["forall_elim",["forall_elim",["forall_elim",["hyp",15],["var",2]],["var",2]],["mul",["var",0],["var",0]]]]]]]]],["forall_elim",["forall_elim",["hyp",14],["var",2]],["mul",["var",0],["var",0]]]]]],["and_intro",["eq_trans",["eq_subst",["eq",["var",7],["mul",["var",0],["mul",["var",3],["var",3]]]],["imp_elim",["imp_elim",["forall_elim",["hyp",2],["var",0]],["exists_intro",["mul",["var",2],["var",2]],["eq_trans",["hyp",5],["eq_subst",["eq",["mul",["var",0],["var",3]],["mul",["var",1],["mul",["var",3],["var",3]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",17],["mul",["var",2],["var",0]]],["var",3]],["mul",["var",1],["var",1]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",14],["var",1]],["hyp",7]],["hyp",0]]]],["eq_trans",["eq_subst",["eq",["mul",["mul",["var",3],["var",1]],["mul",["var",2],["var",2]]],["mul",["var",3],["var",0]]],["eq_sym",["hyp",0]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",2]],["var",0]],["mul",["var",1],["var",1]]],["cong_mul",["eq_refl",["var",2]],["forall_elim",["forall_elim",["hyp",15],["var",0]],["mul",["var",1],["var",1]]]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",18],["var",3]],["mul",["var",2],["var",5]]],["mul",["var",3],["mul",["var",1],["var",1]]]],["hyp",7]],["eq_trans",["eq_subst",["eq",["mul",["var",4],["mul",["var",3],["var",6]]],["mul",["var",0],["var",6]]],["eq_sym",["hyp",5]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",3]],["var",2]],["var",5]]]],["eq_trans",["hyp",3],["eq_subst",["eq",["mul",["var",0],["var",5]],["mul",["var",4],["mul",["var",4],["mul",["var",2],["var",2]]]]],["eq_sym",["hyp",4]],["eq_subst",["eq",["mul",["mul",["var",4],["var",2]],["var",0]],["mul",["var",4],["mul",["var",4],["mul",["var",2],["var",2]]]]],["eq_sym",["hyp",4]],["eq_trans",["forall_elim",["forall_elim",["hyp",14],["var",3]],["var",1]],["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",3]],["var",3]],["mul",["var",1],["var",1]]]]]]]]]]],["eq_trans",["cong_mul",["forall_elim",["forall_elim",["hyp",15],["var",2]],["var",0]],["eq_refl",["var",2]]],["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",0]],["var",2]],["var",2]]]]]]],["exists_intro",["mul",["var",1],["var",1]],["eq_trans",["hyp",0],["forall_elim",["forall_elim",["hyp",15],["mul",["var",1],["var",1]]],["var",0]]]]],["eq_trans",["hyp",5],["eq_subst",["eq",["mul",["var",0],["var",3]],["mul",["var",1],["mul",["var",3],["var",3]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",17],["mul",["var",2],["var",0]]],["var",3]],["mul",["var",1],["var",1]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",14],["var",1]],["hyp",7]],["hyp",0]]]],["eq_trans",["eq_subst",["eq",["mul",["mul",["var",3],["var",1]],["mul",["var",2],["var",2]]],["mul",["var",3],["var",0]]],["eq_sym",["hyp",0]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",2]],["var",0]],["mul",["var",1],["var",1]]],["cong_mul",["eq_refl",["var",2]],["forall_elim",["forall_elim",["hyp",15],["var",0]],["mul",["var",1],["var",1]]]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",18],["var",3]],["mul",["var",2],["var",5]]],["mul",["var",3],["mul",["var",1],["var",1]]]],["hyp",7]],["eq_trans",["eq_subst",["eq",["mul",["var",4],["mul",["var",3],["var",6]]],["mul",["var",0],["var",6]]],["eq_sym",["hyp",5]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",3]],["var",2]],["var",5]]]],["eq_trans",["hyp",3],["eq_subst",["eq",["mul",["var",0],["var",5]],["mul",["var",4],["mul",["var",4],["mul",["var",2],["var",2]]]]],["eq_sym",["hyp",4]],["eq_subst",["eq",["mul",["mul",["var",4],["var",2]],["var",0]],["mul",["var",4],["mul",["var",4],["mul",["var",2],["var",2]]]]],["eq_sym",["hyp",4]],["eq_trans",["forall_elim",["forall_elim",["hyp",14],["var",3]],["var",1]],["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",3]],["var",3]],["mul",["var",1],["var",1]]]]]]]]]]],["eq_trans",["cong_mul",["forall_elim",["forall_elim",["hyp",15],["var",2]],["var",0]],["eq_refl",["var",2]]],["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",0]],["var",2]],["var",2]]]]]],["forall_elim",["hyp",8],["mul",["var",2],["var",2]]]],["eq_trans",["eq_subst",["eq",["var",6],["mul",["mul",["var",2],["var",2]],["var",0]]],["imp_elim",["imp_elim",["forall_elim",["hyp",2],["var",0]],["exists_intro",["mul",["var",2],["var",2]],["eq_trans",["hyp",5],["eq_subst",["eq",["mul",["var",0],["var",3]],["mul",["var",1],["mul",["var",3],["var",3]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",17],["mul",["var",2],["var",0]]],["var",3]],["mul",["var",1],["var",1]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",14],["var",1]],["hyp",7]],["hyp",0]]]],["eq_trans",["eq_subst",["eq",["mul",["mul",["var",3],["var",1]],["mul",["var",2],["var",2]]],["mul",["var",3],["var",0]]],["eq_sym",["hyp",0]],["eq_trans",["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",2]],["var",0]],["mul",["var",1],["var",1]]],["cong_mul",["eq_refl",["var",2]],["forall_elim",["forall_elim",["hyp",15],["var",0]],["mul",["var",1],["var",1]]]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",18],["var",3]],["mul",["var",2],["var",5]]],["mul",["var",3],["mul",["var",1],["var",1]]]],["hyp",7]],["eq_trans",["eq_subst",["eq",["mul",["var",4],["mul",["var",3],["var",6]]],["mul",["var",0],["var",6]]],["eq_sym",["hyp",5]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",3]],["var",2]],["var",5]]]],["eq_trans",["hyp",3],["eq_subst",["eq",["mul",["var",0],["var",5]],["mul",["var",4],["mul",["var",4],["mul",["var",2],["var",2]]]]],["eq_sym",["hyp",4]],["eq_subst",["eq",["mul",["mul",["var",4],["var",2]],["var",0]],["mul",["var",4],["mul",["var",4],["mul",["var",2],["var",2]]]]],["eq_sym",["hyp",4]],["eq_trans",["forall_elim",["forall_elim",["hyp",14],["var",3]],["var",1]],["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",3]],["var",3]],["mul",["var",1],["var",1]]]]]]]]]]],["eq_trans",["cong_mul",["forall_elim",["forall_elim",["hyp",15],["var",2]],["var",0]],["eq_refl",["var",2]]],["forall_elim",["forall_elim",["forall_elim",["hyp",16],["var",0]],["var",2]],["var",2]]]]]]],["exists_intro",["mul",["var",1],["var",1]],["eq_trans",["hyp",0],["forall_elim",["forall_elim",["hyp",15],["mul",["var",1],["var",1]]],["var",0]]]]],["hyp",0]],["forall_elim",["hyp",9],["mul",["var",1],["var",1]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[1768,["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["eq",["mul",["var",2],["var",1]],["mul",["var",0],["var",0]]],["exists",["exists",["and",["eq",["var",4],["mul",["var",1],["var",1]]],["eq",["var",3],["mul",["var",0],["var",0]]]]]]]]]]],[78,21,10,100,62,63,79,102,98,162],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["or_elim",["forall_elim",["forall_elim",["hyp",11],["var",0]],["zero"]],["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",11],["var",2]],["var",1]],["eq_trans",["hyp",1],["eq_subst",["eq",["mul",["var",0],["var",1]],["zero"]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["zero"],["var",0]],["zero"]],["eq_sym",["hyp",0]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_refl",["zero"]]]]]]],["exists_intro",["zero"],["exists_intro",["succ",["zero"]],["and_intro",["eq_subst",["eq",["var",0],["mul",["zero"],["zero"]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_refl",["zero"]]]],["eq_subst",["eq",["var",0],["mul",["succ",["zero"]],["succ",["zero"]]]],["eq_sym",["imp_elim",["imp_elim",["forall_elim",["hyp",3],["var",1]],["exists_intro",["zero"],["eq_subst",["eq",["var",0],["mul",["var",2],["zero"]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",1]]],["eq_refl",["zero"]]]]]],["exists_intro",["succ",["zero"]],["eq_sym",["forall_elim",["hyp",11],["var",1]]]]]],["eq_subst",["eq",["succ",["zero"]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["zero"]]],["zero"]]],["eq_subst",["eq",["succ",["zero"]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["zero"]],["zero"]]],["zero"]]],["eq_subst",["eq",["succ",["zero"]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["zero"]],["zero"]]]],["eq_subst",["eq",["succ",["zero"]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA5"],["succ",["zero"]]]],["eq_refl",["succ",["zero"]]]]]]]]]]],["exists_intro",["succ",["zero"]],["exists_intro",["zero"],["and_intro",["eq_subst",["eq",["var",0],["mul",["succ",["zero"]],["succ",["zero"]]]],["eq_sym",["imp_elim",["imp_elim",["forall_elim",["hyp",3],["var",2]],["exists_intro",["succ",["zero"]],["eq_sym",["forall_elim",["hyp",11],["var",2]]]]],["exists_intro",["zero"],["eq_subst",["eq",["var",0],["mul",["var",3],["zero"]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",2]]],["eq_refl",["zero"]]]]]]],["eq_subst",["eq",["succ",["zero"]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA6"],["succ",["zero"]]],["zero"]]],["eq_subst",["eq",["succ",["zero"]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["mul",["succ",["zero"]],["zero"]]],["zero"]]],["eq_subst",["eq",["succ",["zero"]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["succ",["zero"]],["zero"]]]],["eq_subst",["eq",["succ",["zero"]],["succ",["var",0]]],["eq_sym",["forall_elim",["axiom","PA5"],["succ",["zero"]]]],["eq_refl",["succ",["zero"]]]]]]]],["eq_subst",["eq",["var",0],["mul",["zero"],["zero"]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_refl",["zero"]]]]]]]],["exists_elim",["forall_elim",["forall_elim",["hyp",9],["var",2]],["var",0]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",0]],["var",3]],["var",1]],["hyp",0]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",1]],["var",4]],["var",2]],["hyp",1]],["exists_intro",["var",1],["exists_intro",["var",0],["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",5]],["var",4]],["var",3]],["var",2]],["var",1]],["var",0]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",3]],["var",2]],["var",0]],["hyp",4]],["hyp",1]],["hyp",0]]]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",3]],["var",2]],["var",0]],["hyp",4]],["hyp",1]],["hyp",0]]]],["hyp",1]],["hyp",0]],["hyp",4]],["hyp",5]],["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",5]],["var",3]],["var",1]],["var",0]],["hyp",2]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",3]],["var",2]],["var",0]],["hyp",4]],["hyp",1]],["hyp",0]]]],["hyp",1]],["hyp",0]]]]]]]]]]]]]]]]]]]]]]]]],[704,["forall",["forall",["forall",["forall",["forall",["forall",["imp",["imp",["eq",["var",3],["zero"]],["bot"]],["imp",["eq",["var",5],["mul",["var",3],["var",2]]],["imp",["eq",["var",4],["mul",["var",3],["var",1]]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["eq",["mul",["var",4],["var",4]],["mul",["mul",["var",5],["var",5]],["var",0]]],["exists",["eq",["var",5],["mul",["var",6],["var",0]]]]]]]]]]]]]]],[118,116,8,82,10],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["exists_intro",["var",1],["eq_trans",["hyp",2],["eq_subst",["eq",["mul",["var",4],["var",2]],["mul",["var",0],["var",2]]],["eq_sym",["hyp",3]],["eq_subst",["eq",["mul",["var",4],["var",2]],["mul",["mul",["var",4],["var",0]],["var",2]]],["eq_sym",["imp_elim",["imp_elim",["forall_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",2]],["var",1]],["var",1]],["hyp",1]],["hyp",1]],["var",2]],["exists_intro",["succ",["zero"]],["eq_sym",["forall_elim",["hyp",5],["var",2]]]]],["exists_intro",["mul",["var",2],["var",0]],["eq_trans",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",3]],["mul",["var",1],["var",1]]],["mul",["mul",["var",2],["var",2]],["var",0]]],["hyp",4]],["eq_trans",["eq_subst",["eq",["mul",["mul",["var",4],["var",4]],["mul",["var",2],["var",2]]],["mul",["var",0],["var",5]]],["eq_sym",["hyp",2]],["eq_subst",["eq",["mul",["mul",["var",4],["var",4]],["mul",["var",2],["var",2]]],["mul",["mul",["var",4],["var",2]],["var",0]]],["eq_sym",["hyp",2]],["eq_sym",["forall_elim",["forall_elim",["hyp",8],["var",3]],["var",1]]]]],["eq_trans",["hyp",0],["eq_subst",["eq",["mul",["mul",["var",0],["var",6]],["var",1]],["mul",["mul",["var",4],["var",4]],["mul",["mul",["var",3],["var",3]],["var",1]]]],["eq_sym",["hyp",3]],["eq_subst",["eq",["mul",["mul",["mul",["var",4],["var",3]],["var",0]],["var",1]],["mul",["mul",["var",4],["var",4]],["mul",["mul",["var",3],["var",3]],["var",1]]]],["eq_sym",["hyp",3]],["eq_trans",["cong_mul",["forall_elim",["forall_elim",["hyp",8],["var",3]],["var",2]],["eq_refl",["var",0]]],["forall_elim",["forall_elim",["forall_elim",["hyp",7],["mul",["var",3],["var",3]]],["mul",["var",2],["var",2]]],["var",0]]]]]]]],["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",2]],["var",0]]]]]],["cong_mul",["eq_sym",["forall_elim",["hyp",5],["var",3]]],["eq_refl",["var",1]]]]]]]]]]]]]]]]]]]]]]]],[904,["forall",["forall",["imp",["exists",["eq",["mul",["var",1],["var",1]],["mul",["mul",["var",2],["var",2]],["var",0]]]],["exists",["eq",["var",1],["mul",["var",2],["var",0]]]]]]],[78,4,161,100,62,63,79,98,164],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["or_elim",["forall_elim",["forall_elim",["hyp",10],["var",2]],["zero"]],["exists_intro",["zero"],["eq_subst",["eq",["var",0],["mul",["var",3],["zero"]]],["eq_sym",["imp_elim",["forall_elim",["hyp",9],["var",1]],["eq_trans",["hyp",1],["eq_subst",["eq",["mul",["mul",["var",0],["var",3]],["var",1]],["zero"]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["var",0],["var",1]],["zero"]],["eq_sym",["forall_elim",["hyp",10],["var",2]]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["hyp",10],["var",0]]],["eq_refl",["zero"]]]]]]]],["eq_subst",["eq",["zero"],["var",0]],["eq_sym",["forall_elim",["axiom","PA5"],["var",2]]],["eq_refl",["zero"]]]]],["exists_elim",["forall_elim",["forall_elim",["hyp",8],["var",2]],["var",1]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",0]],["var",3]],["var",2]],["hyp",0]],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",1]],["var",4]],["var",3]],["hyp",1]],["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",5]],["var",4]],["var",2]],["var",1]],["var",0]],["var",3]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",5]],["var",2]],["var",1]],["hyp",4]],["hyp",2]],["hyp",0]]]],["hyp",1]],["hyp",0]],["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",5]],["var",4]],["var",1]],["var",0]],["hyp",2]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",5]],["var",2]],["var",1]],["hyp",4]],["hyp",2]],["hyp",0]]]],["hyp",1]],["hyp",0]]],["hyp",4]]]]]]]]]]]]]]]]]]]],[264,["forall",["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["succ",["var",2]]],["add",["var",2],["var",1]]]]]]],[23,2,1],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["imp_elim",["forall_elim",["hyp",3],["var",0]],["hyp",0]],["exists_intro",["var",0],["eq_subst",["eq",["add",["var",1],["succ",["var",3]]],["add",["var",3],["var",0]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["var",0],["add",["var",3],["succ",["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",2]]],["eq_subst",["eq",["succ",["add",["var",1],["var",3]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",2]],["var",0]]],["eq_subst",["eq",["succ",["add",["var",1],["var",3]]],["succ",["var",0]]],["eq_sym",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",0]]],["eq_refl",["succ",["add",["var",0],["var",2]]]]]]]]]]]]]]]]],[240,["forall",["forall",["forall",["imp",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["succ",["var",3]]],["var",1]]]]]]]],[166,47,159],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",2]],["var",0]],["eq_subst",["exists",["eq",["add",["var",0],["succ",["mul",["var",4],["var",4]]]],["var",1]]],["hyp",1],["imp_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",2],["var",2]]],["mul",["var",1],["var",1]]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",1]],["hyp",1]],["hyp",1]],["hyp",0]]]]]]]]]]]]]]],[440,["forall",["forall",["imp",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["imp",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["add",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]]],[99,86,2],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["hyp",2],["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",6],["var",2]],["var",0]],["exists_elim",["hyp",1],["exists_intro",["var",0],["eq_trans",["eq_trans",["forall_elim",["forall_elim",["hyp",5],["var",3]],["var",1]],["hyp",1]],["hyp",0]]]]],["bot_elim",["exists_elim",["and_elim_l",["hyp",0]],["exists_elim",["hyp",4],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",4]],["var",1]],["var",0]],["hyp",1]],["hyp",0]]]]],["exists_elim",["and_elim_r",["hyp",0]],["exists_intro",["var",0],["eq_trans",["eq_trans",["eq_sym",["hyp",2]],["forall_elim",["forall_elim",["hyp",6],["var",1]],["var",3]]],["cong_add",["eq_refl",["var",3]],["hyp",0]]]]]]]]]]]]]]]],[312,["forall",["forall",["eq",["add",["add",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]],["var",1]],["mul",["succ",["succ",["zero"]]],["add",["var",1],["var",0]]]]]],[105,3,2],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["eq_subst",["eq",["add",["add",["var",2],["var",0]],["var",2]],["mul",["succ",["succ",["zero"]]],["add",["var",2],["var",1]]]],["eq_sym",["forall_elim",["hyp",2],["var",0]]],["eq_subst",["eq",["add",["add",["var",2],["add",["var",1],["var",1]]],["var",2]],["var",0]],["eq_sym",["forall_elim",["hyp",2],["add",["var",1],["var",0]]]],["eq_subst",["eq",["var",0],["add",["add",["var",2],["var",1]],["add",["var",2],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["add",["var",0],["var",0]]],["var",1]]],["eq_subst",["eq",["add",["var",2],["var",0]],["add",["add",["var",2],["var",1]],["add",["var",2],["var",1]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",0]],["var",0]],["var",1]]],["eq_subst",["eq",["add",["var",2],["add",["var",1],["add",["var",1],["var",2]]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",0]],["add",["var",1],["var",0]]]],["eq_subst",["eq",["add",["var",2],["add",["var",1],["add",["var",1],["var",2]]]],["add",["var",2],["add",["var",1],["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]]],["eq_refl",["add",["var",1],["add",["var",0],["add",["var",0],["var",1]]]]]]]]]]]]]]]]],[136,["forall",["forall",["eq",["add",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]],["add",["add",["var",1],["var",0]],["var",0]]]]],[105,3],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["eq_subst",["eq",["add",["var",2],["var",0]],["add",["add",["var",2],["var",1]],["var",1]]],["eq_sym",["forall_elim",["hyp",1],["var",0]]],["eq_subst",["eq",["add",["var",2],["add",["var",1],["var",1]]],["var",0]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]],["var",0]]],["eq_refl",["add",["var",1],["add",["var",0],["var",0]]]]]]]]]]],[624,["forall",["forall",["forall",["forall",["imp",["eq",["add",["mul",["var",3],["var",3]],["mul",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["var",2],["var",2]]],["imp",["eq",["var",2],["add",["var",3],["mul",["succ",["succ",["zero"]]],["var",0]]]],["eq",["mul",["add",["var",3],["var",0]],["var",0]],["mul",["var",1],["var",1]]]]]]]]],[119,169,115,116,25,48,47,12,6],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["eq_trans",["forall_elim",["forall_elim",["hyp",2],["add",["var",3],["var",0]]],["var",0]],["eq_sym",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["succ",["succ",["zero"]]],["succ",["succ",["zero"]]]]],["mul",["var",1],["var",1]]],["mul",["var",0],["add",["var",3],["var",0]]]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",5],["succ",["succ",["zero"]]]],["succ",["succ",["zero"]]]],["imp_intro",["imp_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["hyp",0]]]],["imp_intro",["imp_elim",["forall_elim",["hyp",5],["succ",["zero"]]],["hyp",0]]]],["hyp",0]]]],["eq_trans",["eq_sym",["forall_elim",["forall_elim",["hyp",7],["succ",["succ",["zero"]]]],["var",1]]],["eq_trans",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["mul",["var",3],["var",3]]],["mul",["mul",["succ",["succ",["zero"]]],["var",1]],["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["mul",["succ",["succ",["zero"]]],["var",0]],["mul",["succ",["succ",["zero"]]],["add",["var",3],["var",0]]]]],["eq_trans",["hyp",1],["eq_sym",["eq_subst",["eq",["add",["mul",["var",4],["var",4]],["mul",["mul",["succ",["succ",["zero"]]],["var",1]],["var",0]]],["mul",["var",3],["var",3]]],["eq_subst",["eq",["add",["var",0],["var",4]],["mul",["succ",["succ",["zero"]]],["add",["var",4],["var",1]]]],["eq_sym",["hyp",0]],["forall_elim",["forall_elim",["hyp",9],["var",3]],["var",0]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",3]],["var",2]],["mul",["succ",["succ",["zero"]]],["var",0]]],["hyp",0]]]]]],["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["succ",["succ",["zero"]]]],["var",0]],["succ",["succ",["zero"]]]],["add",["var",3],["var",0]]]]]]]]]]]]]]]]]]]]]]]],[440,["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["eq",["var",1],["add",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["forall",["imp",["exists",["eq",["add",["var",4],["var",2]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[59,54,170,10,2],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",3],["var",0]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",0]],["add",["var",3],["var",1]]],["var",1]],["succ",["zero"]]],["var",3]],["hyp",1]],["hyp",0]],["eq_trans",["forall_elim",["forall_elim",["hyp",4],["var",3]],["var",1]],["cong_add",["eq_sym",["forall_elim",["hyp",5],["var",1]]],["eq_refl",["var",3]]]]]],["eq_subst",["exists",["eq",["var",1],["mul",["var",2],["var",0]]]],["eq_sym",["hyp",2]],["eq_subst",["exists",["eq",["var",1],["mul",["var",2],["var",0]]]],["eq_sym",["forall_elim",["forall_elim",["hyp",6],["var",3]],["var",1]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",0]],["add",["var",3],["var",1]]],["var",1]],["hyp",1]],["hyp",0]]]]]]]]]]]]]]]]]]],[208,["forall",["forall",["imp",["forall",["imp",["exists",["eq",["mul",["var",3],["var",3]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["var",2],["var",2]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]],[141],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["hyp",0],["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",0]],["var",2]],["hyp",2]]],["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",0]],["var",1]],["hyp",1]]]],["forall_elim",["hyp",2],["var",0]]]]]]]]]]],[216,["forall",["imp",["exists",["eq",["mul",["var",1],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],[152,135,86],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["or_elim",["forall_elim",["hyp",3],["var",0]],["hyp",0],["bot_elim",["exists_elim",["hyp",1],["exists_elim",["imp_elim",["forall_elim",["hyp",4],["var",1]],["hyp",1]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",2],["var",2]]],["var",1]],["var",0]],["hyp",1]],["hyp",0]]]]]]]]]]]],[304,["forall",["forall",["imp",["exists",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]]]]],[99,174,154],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["hyp",0]],["or_intro_l",["and_intro",["imp_elim",["forall_elim",["hyp",3],["var",1]],["and_elim_l",["hyp",0]]],["imp_elim",["forall_elim",["hyp",2],["var",0]],["and_elim_r",["hyp",0]]]]],["or_intro_r",["and_intro",["imp_elim",["forall_elim",["hyp",2],["var",1]],["and_elim_l",["hyp",0]]],["imp_elim",["forall_elim",["hyp",3],["var",0]],["and_elim_r",["hyp",0]]]]]]]]]]]]],[256,["forall",["forall",["forall",["forall",["forall",["imp",["eq",["var",3],["add",["var",4],["mul",["succ",["succ",["zero"]]],["var",2]]]],["imp",["eq",["add",["var",4],["var",2]],["mul",["var",1],["var",1]]],["imp",["eq",["var",2],["mul",["var",0],["var",0]]],["and",["eq",["var",3],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]]]]]]]]]]],[170,2],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["and_intro",["eq_trans",["eq_trans",["hyp",2],["forall_elim",["forall_elim",["hyp",4],["var",4]],["var",2]]],["cong_add",["hyp",1],["hyp",0]]],["eq_trans",["eq_sym",["hyp",1]],["eq_trans",["forall_elim",["forall_elim",["hyp",3],["var",4]],["var",2]],["cong_add",["hyp",0],["eq_refl",["var",4]]]]]]]]]]]]]]]]],[232,["forall",["forall",["forall",["forall",["forall",["imp",["eq",["mul",["add",["var",4],["var",3]],["var",3]],["mul",["var",2],["var",2]]],["imp",["eq",["add",["var",4],["var",3]],["mul",["var",1],["var",1]]],["imp",["eq",["var",3],["mul",["var",0],["var",0]]],["eq",["mul",["succ",["succ",["zero"]]],["var",2]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]],[116,160],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["cong_mul",["eq_refl",["succ",["succ",["zero"]]]],["imp_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["mul",["var",1],["var",0]]],["eq_trans",["eq_sym",["hyp",2]],["eq_trans",["cong_mul",["hyp",1],["hyp",0]],["eq_sym",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",0]]]]]]]]]]]]]]]]]],[168,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",2]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]]]]]]],[166,159],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",2],["var",0]],["var",1]],["eq_subst",["exists",["eq",["add",["var",0],["succ",["mul",["var",2],["var",2]]]],["var",1]]],["eq_sym",["hyp",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",0],["var",0]]],["var",2]],["hyp",1]]]]]]]]]]]],[344,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["eq",["var",0],["zero"]],["bot"]]]]]]]],[102,4],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["succ",["succ",["zero"]]]],["mul",["var",1],["var",0]]],["hyp",2]],["hyp",1]],["eq_subst",["eq",["mul",["var",0],["var",1]],["zero"]],["eq_sym",["hyp",0]],["forall_elim",["hyp",3],["var",0]]]]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["succ",["succ",["zero"]]]],["mul",["var",1],["var",0]]],["hyp",2]],["hyp",1]],["eq_subst",["eq",["mul",["var",2],["var",0]],["zero"]],["eq_sym",["hyp",0]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["axiom","PA5"],["var",1]]],["eq_refl",["zero"]]]]]]]]]]]]]]],[632,["forall",["forall",["forall",["imp",["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["exists",["and",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",1]]],["and",["eq",["var",2],["add",["var",4],["mul",["succ",["succ",["zero"]]],["var",0]]]],["and",["eq",["mul",["add",["var",4],["var",0]],["var",0]],["mul",["var",1],["var",1]]],["forall",["imp",["exists",["eq",["add",["var",6],["var",2]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]]]]]],[147,155,167,30,168,171,172],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",6],["var",2]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",2]],["var",1]],["var",0]],["and_elim_l",["hyp",3]]],["hyp",2]]]],["hyp",1]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",2]],["var",1]],["var",0]],["hyp",3]]],["exists_elim",["hyp",1],["exists_intro",["var",0],["exists_intro",["var",1],["and_intro",["hyp",0],["and_intro",["hyp",1],["and_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",4]],["var",2]],["var",0]],["var",1]],["eq_subst",["eq",["add",["mul",["var",5],["var",5]],["mul",["var",0],["mul",["succ",["succ",["zero"]]],["var",1]]]],["mul",["var",3],["var",3]]],["hyp",0],["eq_subst",["eq",["add",["mul",["var",5],["var",5]],["mul",["var",4],["var",0]]],["mul",["var",3],["var",3]]],["hyp",0],["and_elim_l",["hyp",5]]]]],["hyp",1]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",4]],["var",2]],["var",1]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",4]],["var",3]],["var",2]],["hyp",5]]],["hyp",1]]]]]]]]]]]]]]]]]]]]]]]],[1312,["forall",["forall",["forall",["forall",["forall",["imp",["imp",["eq",["var",4],["zero"]],["bot"]],["imp",["imp",["eq",["var",3],["zero"]],["bot"]],["imp",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["eq",["var",3],["mul",["succ",["succ",["zero"]]],["var",1]]],["imp",["eq",["var",2],["add",["var",4],["mul",["succ",["succ",["zero"]]],["var",0]]]],["imp",["eq",["mul",["add",["var",4],["var",0]],["var",0]],["mul",["var",1],["var",1]]],["imp",["forall",["imp",["exists",["eq",["add",["var",6],["var",2]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",4],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",6]]],["eq",["var",5],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]]]]]]]]]]]]],[163,176,177,173,179,178,175],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",13],["add",["var",4],["var",0]]],["var",0]],["var",1]],["hyp",0]],["hyp",1]],["exists_elim",["hyp",0],["exists_intro",["var",1],["exists_intro",["var",0],["and_intro",["and_elim_r",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["var",5]],["var",1]],["var",0]],["hyp",7]],["eq_trans",["hyp",5],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",13],["var",6]],["var",2]],["var",3]],["var",1]],["var",0]],["hyp",3]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]]]]],["and_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",6]],["var",1]],["var",0]],["hyp",8]],["and_elim_r",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["var",6]],["var",4]],["var",2]],["var",1]],["var",0]],["hyp",4]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]]]],["and_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",12],["var",1]],["var",0]],["eq_subst",["forall",["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["var",3],["var",3]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and_elim_l",["hyp",0]],["eq_subst",["forall",["imp",["exists",["eq",["add",["var",9],["var",5]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and_elim_r",["hyp",0]],["hyp",2]]]],["and_intro",["imp_elim",["forall_elim",["forall_elim",["hyp",9],["var",1]],["var",0]],["eq_subst",["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["and_elim_l",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["var",6]],["var",4]],["var",2]],["var",1]],["var",0]],["hyp",4]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]]],["hyp",6]]],["and_intro",["and_elim_l",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["var",6]],["var",4]],["var",2]],["var",1]],["var",0]],["hyp",4]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]]],["and_intro",["and_elim_r",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["var",6]],["var",4]],["var",2]],["var",1]],["var",0]],["hyp",4]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]]],["eq_trans",["hyp",5],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",13],["var",6]],["var",2]],["var",3]],["var",1]],["var",0]],["hyp",3]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[496,["forall",["forall",["forall",["imp",["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]]]]]]]]],[180,155,181],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",2]],["var",1]],["var",0]],["hyp",4]],["hyp",2]],["hyp",1]],["hyp",0]],["exists_elim",["hyp",0],["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",4]],["var",3]],["var",2]],["var",1]],["var",0]],["hyp",5]],["hyp",4]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",4]],["var",3]],["var",2]],["hyp",6]]],["and_elim_l",["hyp",0]]],["and_elim_l",["and_elim_r",["hyp",0]]]],["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]]],["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]]]]]]]]]]]]],[416,["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",1],["var",1]],["mul",["var",2],["var",2]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]],[129],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["and_intro",["hyp",3],["and_intro",["hyp",5],["and_intro",["hyp",1],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["hyp",3]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",7],["var",2]]]]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]]],[1136,["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]],["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]]]]]],[153,129,182],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",0]],["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]],["exists_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",2]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",0]],["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]]],["and_elim_l",["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",1]]],["and_elim_r",["hyp",0]]],["and_elim_l",["hyp",0]]],["exists_elim",["hyp",0],["exists_intro",["var",1],["exists_intro",["var",0],["and_intro",["and_elim_l",["hyp",0]],["and_intro",["and_elim_l",["and_elim_r",["hyp",0]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]],["or_intro_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]]]]]]]]]],["exists_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",2]],["var",1]],["var",0]],["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]],["and_elim_l",["hyp",1]]],["and_elim_l",["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]],["exists_elim",["hyp",0],["exists_intro",["var",1],["exists_intro",["var",0],["and_intro",["and_elim_l",["hyp",0]],["and_intro",["and_elim_l",["and_elim_r",["hyp",0]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]],["or_intro_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]]]]]]]]]]]]]]]]]]],[368,["forall",["forall",["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["imp",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",2]]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["eq",["var",2],["zero"]],["bot"]]]]]]]]],[30,27,160,31],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["and_intro",["imp_intro",["imp_elim",["hyp",3],["imp_elim",["forall_elim",["hyp",6],["var",0]],["eq_subst",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["hyp",0],["imp_elim",["forall_elim",["forall_elim",["hyp",7],["var",0]],["var",1]],["hyp",2]]]]]],["imp_intro",["imp_elim",["forall_elim",["hyp",4],["var",0]],["eq_subst",["exists",["eq",["add",["var",0],["succ",["var",2]]],["var",1]]],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",1]],["var",0]],["eq_trans",["hyp",1],["eq_subst",["eq",["add",["mul",["var",1],["var",1]],["var",0]],["mul",["var",1],["var",1]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["var",0],["mul",["var",1],["var",1]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["var",0],["var",0]]]],["eq_refl",["mul",["var",0],["var",0]]]]]]],["hyp",2]]]]]]]]]]]]]]]],[1512,["forall",["forall",["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]],["and",["imp",["eq",["var",4],["zero"]],["bot"]],["and",["imp",["eq",["var",3],["zero"]],["bot"]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",4],["var",4]],["mul",["var",3],["var",3]]],["mul",["var",2],["var",2]]],["forall",["imp",["exists",["eq",["var",6],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",5],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]]]],[185,146,127,47,12],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["and_intro",["and_elim_r",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",4]],["var",1]],["var",0]],["and_elim_l",["hyp",0]]],["and_elim_l",["and_elim_r",["hyp",0]]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]]],["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",3],["succ",["succ",["zero"]]]],["mul",["var",1],["var",0]]],["imp_intro",["imp_elim",["forall_elim",["hyp",3],["succ",["zero"]]],["hyp",0]]]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",0]],["and_elim_l",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",4]],["var",1]],["var",0]],["and_elim_l",["hyp",2]]],["and_elim_l",["and_elim_r",["hyp",2]]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",2]]]]]]]]]],["and_elim_l",["hyp",2]]],["hyp",0]]]],["eq_trans",["eq_sym",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]]]]],["hyp",0]]]],["and_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",0]],["and_elim_l",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",6],["var",4]],["var",1]],["var",0]],["and_elim_l",["hyp",1]]],["and_elim_l",["and_elim_r",["hyp",1]]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]]]]]]],["eq_trans",["eq_sym",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]]]],["hyp",0]]]],["eq_subst",["and",["eq",["add",["mul",["var",5],["var",5]],["mul",["var",4],["var",4]]],["mul",["var",0],["var",3]]],["forall",["imp",["exists",["eq",["var",7],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",6],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["eq_sym",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]],["eq_subst",["and",["eq",["add",["mul",["var",5],["var",5]],["mul",["var",4],["var",4]]],["mul",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["var",0]]],["forall",["imp",["exists",["eq",["var",7],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",6],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["eq_sym",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]],["eq_subst",["and",["eq",["add",["mul",["var",5],["var",5]],["mul",["var",0],["var",4]]],["mul",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]]],["forall",["imp",["exists",["eq",["var",7],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",6],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["eq_sym",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]],["eq_subst",["and",["eq",["add",["mul",["var",5],["var",5]],["mul",["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]],["var",0]]],["mul",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]]],["forall",["imp",["exists",["eq",["var",7],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",6],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["eq_sym",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]],["eq_subst",["and",["eq",["add",["mul",["var",5],["var",5]],["mul",["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]],["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]]]],["mul",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]]],["forall",["imp",["exists",["eq",["var",7],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["eq_sym",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["var",0]],["var",4]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]],["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]]]]]]]]]]]]]]]]]]]],[720,["forall",["forall",["forall",["imp",["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]],[186,183],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["hyp",0],["exists_elim",["hyp",0],["or_elim",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",4]],["var",3]],["var",2]],["var",1]],["var",0]],["and_intro",["and_elim_l",["hyp",1]],["and_intro",["and_elim_l",["and_elim_r",["hyp",1]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["hyp",1]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]]],["hyp",0]]]]]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",3]],["var",4]],["var",2]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",3]],["var",4]],["var",2]],["var",1]],["var",0]],["and_intro",["and_elim_l",["hyp",1]],["and_intro",["and_elim_l",["and_elim_r",["hyp",1]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["hyp",1]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]]],["hyp",0]]]]]]]]]]]]]]]]]],[272,["forall",["forall",["forall",["and",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]],["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]]],["imp",["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]]],[184,187],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["and_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",1],["var",2]]],["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["hyp",0]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",0],["var",2]]]]]]]]]],[160,["forall",["forall",["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["mul",["var",1],["var",0]],["zero"]],["bot"]]]]]],[21],["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",3],["var",1]],["var",0]],["hyp",0]],["imp_elim",["hyp",3],["hyp",0]],["imp_elim",["hyp",2],["hyp",0]]]]]]]]]],[168,["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["mul",["var",0],["var",0]],["zero"]],["bot"]]]],[189],["imp_intro",["forall_intro",["imp_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["hyp",0],["hyp",3]],["hyp",3]],["hyp",2]]],["forall_elim",["hyp",0],["var",0]]]],["forall_elim",["hyp",2],["var",0]]]]]]]],[232,["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["forall",["imp",["exists",["eq",["mul",["var",3],["var",3]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["var",2],["var",2]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]],[81,82],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",1]],["mul",["var",0],["var",0]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",0]],["var",0]],["hyp",0]],["hyp",0]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["var",0]],["var",0]],["hyp",0]],["hyp",0]]]]]]]]],[128,["forall",["forall",["eq",["mul",["mul",["mul",["var",1],["var",0]],["mul",["var",1],["var",0]]],["mul",["mul",["var",1],["var",0]],["mul",["var",1],["var",0]]]],["mul",["mul",["mul",["var",1],["var",1]],["mul",["var",1],["var",1]]],["mul",["mul",["var",0],["var",0]],["mul",["var",0],["var",0]]]]]]],[116],["imp_intro",["forall_intro",["forall_intro",["eq_trans",["cong_mul",["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]],["forall_elim",["forall_elim",["hyp",0],["var",1]],["var",0]]],["forall_elim",["forall_elim",["hyp",0],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]]]]]]],[408,["forall",["forall",["forall",["forall",["forall",["forall",["imp",["eq",["var",5],["mul",["var",2],["var",1]]],["imp",["eq",["var",4],["mul",["var",2],["var",0]]],["imp",["eq",["add",["mul",["mul",["mul",["var",5],["var",5]],["var",5]],["var",5]],["mul",["mul",["mul",["var",4],["var",4]],["var",4]],["var",4]]],["mul",["var",3],["var",3]]],["eq",["mul",["var",3],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["mul",["var",2],["var",2]]],["add",["mul",["mul",["var",1],["var",1]],["mul",["var",1],["var",1]]],["mul",["mul",["var",0],["var",0]],["mul",["var",0],["var",0]]]]]]]]]]]]]]],[130,192,7],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["eq_trans",["eq_sym",["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["mul",["var",3],["var",3]]],["mul",["mul",["var",2],["var",2]],["mul",["var",2],["var",2]]]],["var",0]],["mul",["var",4],["var",4]]],["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",0]],["eq_subst",["eq",["add",["var",0],["mul",["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]],["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]]]],["mul",["var",4],["var",4]]],["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]]],["mul",["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]],["mul",["mul",["var",3],["var",1]],["var",0]]]],["mul",["var",4],["var",4]]],["hyp",1],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]]],["mul",["mul",["mul",["var",3],["var",1]],["mul",["var",3],["var",1]]],["mul",["var",0],["var",5]]]],["mul",["var",4],["var",4]]],["hyp",1],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]]],["mul",["mul",["mul",["var",3],["var",1]],["var",0]],["mul",["var",5],["var",5]]]],["mul",["var",4],["var",4]]],["hyp",1],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]]],["mul",["mul",["var",0],["var",5]],["mul",["var",5],["var",5]]]],["mul",["var",4],["var",4]]],["hyp",1],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["mul",["var",3],["var",2]],["var",0]]],["mul",["mul",["var",5],["var",5]],["mul",["var",5],["var",5]]]],["mul",["var",4],["var",4]]],["hyp",2],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",2]],["mul",["var",3],["var",2]]],["mul",["var",0],["var",6]]],["mul",["mul",["var",5],["var",5]],["mul",["var",5],["var",5]]]],["mul",["var",4],["var",4]]],["hyp",2],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",2]],["var",0]],["mul",["var",6],["var",6]]],["mul",["mul",["var",5],["var",5]],["mul",["var",5],["var",5]]]],["mul",["var",4],["var",4]]],["hyp",2],["eq_subst",["eq",["add",["mul",["mul",["var",0],["var",6]],["mul",["var",6],["var",6]]],["mul",["mul",["var",5],["var",5]],["mul",["var",5],["var",5]]]],["mul",["var",4],["var",4]]],["hyp",2],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",5]],["var",4]],["var",3]],["hyp",0]]]]]]]]]]]]],["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["mul",["var",2],["var",2]],["mul",["var",2],["var",2]]]],["mul",["mul",["var",1],["var",1]],["mul",["var",1],["var",1]]]],["mul",["mul",["var",0],["var",0]],["mul",["var",0],["var",0]]]]]]]]]]]]]]]]]]],[408,["forall",["forall",["forall",["forall",["forall",["imp",["imp",["eq",["var",4],["zero"]],["bot"]],["imp",["eq",["var",0],["mul",["mul",["var",4],["var",4]],["var",1]]],["imp",["eq",["mul",["var",0],["var",0]],["mul",["mul",["mul",["var",4],["var",4]],["mul",["var",4],["var",4]]],["add",["mul",["mul",["var",3],["var",3]],["mul",["var",3],["var",3]]],["mul",["mul",["var",2],["var",2]],["mul",["var",2],["var",2]]]]]],["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["var",1],["var",1]]]]]]]]]]],[190,116,48,84],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["eq_trans",["cong_add",["forall_elim",["hyp",3],["var",3]],["forall_elim",["hyp",3],["var",2]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["mul",["var",4],["var",4]],["mul",["var",4],["var",4]]]],["add",["mul",["mul",["var",3],["var",3]],["mul",["var",3],["var",3]]],["mul",["mul",["var",2],["var",2]],["mul",["var",2],["var",2]]]]],["mul",["var",1],["var",1]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",7],["mul",["var",4],["var",4]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",8],["var",4]],["hyp",4]],["hyp",0]]]],["hyp",0]]]],["eq_trans",["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["var",0],["var",1]],["mul",["mul",["mul",["var",5],["var",5]],["mul",["var",5],["var",5]]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",1]],["eq_subst",["eq",["mul",["mul",["mul",["var",5],["var",5]],["var",2]],["var",0]],["mul",["mul",["mul",["var",5],["var",5]],["mul",["var",5],["var",5]]],["mul",["var",2],["var",2]]]],["eq_sym",["hyp",1]],["forall_elim",["forall_elim",["hyp",5],["mul",["var",4],["var",4]]],["var",1]]]]]]]]]]]]]]]]]]]],[240,["forall",["forall",["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["exists",["eq",["add",["var",0],["succ",["var",2]]],["add",["var",2],["var",1]]]]]]],[23,2,1],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["imp_elim",["forall_elim",["hyp",3],["var",0]],["hyp",0]],["exists_intro",["var",0],["eq_subst",["eq",["add",["var",1],["succ",["var",3]]],["add",["var",3],["var",0]]],["eq_sym",["hyp",0]],["eq_trans",["forall_elim",["forall_elim",["axiom","PA4"],["var",0]],["var",2]],["eq_trans",["cong_s",["forall_elim",["forall_elim",["hyp",3],["var",0]],["var",2]]],["eq_sym",["forall_elim",["forall_elim",["axiom","PA4"],["var",2]],["var",0]]]]]]]]]]]]]]],[496,["forall",["forall",["forall",["forall",["imp",["imp",["eq",["var",3],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["eq",["var",2],["mul",["var",3],["var",3]]],["imp",["eq",["var",0],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]],["exists",["eq",["add",["var",0],["succ",["var",4]]],["var",1]]]]]]]]]]],[83,190,195,34,14],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["eq_subst",["exists",["eq",["add",["var",0],["succ",["var",5]]],["var",1]]],["eq_sym",["hyp",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",3]],["mul",["var",2],["var",2]]],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",3]],["var",2]],["mul",["var",2],["var",2]]],["eq_subst",["exists",["eq",["add",["var",0],["var",5]],["var",1]]],["eq_sym",["hyp",1]],["imp_elim",["forall_elim",["forall_elim",["hyp",8],["var",3]],["var",3]],["hyp",3]]]],["imp_elim",["forall_elim",["forall_elim",["hyp",8],["var",2]],["var",2]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",8],["var",3]],["hyp",4]],["eq_subst",["eq",["var",0],["zero"]],["hyp",2],["hyp",0]]]]]]],["imp_elim",["forall_elim",["forall_elim",["hyp",6],["mul",["var",2],["var",2]]],["mul",["var",1],["var",1]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",8],["var",1]],["hyp",3]],["hyp",0]]]]]]]]]]]]]]]]]]]],[248,["forall",["imp",["exists",["eq",["mul",["var",1],["var",1]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],[85,135,86],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["imp_intro",["exists_elim",["forall_elim",["hyp",3],["var",0]],["or_elim",["hyp",0],["exists_intro",["var",0],["hyp",0]],["bot_elim",["exists_elim",["hyp",2],["exists_elim",["imp_elim",["forall_elim",["hyp",5],["var",2]],["exists_intro",["var",1],["hyp",1]]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",3],["var",3]]],["var",1]],["var",0]],["hyp",1]],["hyp",0]]]]]]]]]]]]],[176,["forall",["forall",["eq",["mul",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]],[8,6],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["eq_trans",["eq_sym",["forall_elim",["forall_elim",["forall_elim",["hyp",1],["var",1]],["succ",["succ",["zero"]]]],["var",0]]],["eq_trans",["cong_mul",["forall_elim",["forall_elim",["hyp",0],["var",1]],["succ",["succ",["zero"]]]],["eq_refl",["var",0]]],["forall_elim",["forall_elim",["forall_elim",["hyp",1],["succ",["succ",["zero"]]]],["var",1]],["var",0]]]]]]]]],[672,["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["exists",["eq",["var",3],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["eq",["mul",["var",0],["var",0]],["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]]],["exists",["exists",["and",["eq",["var",4],["mul",["var",1],["var",1]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",0],["var",0]]]]]]]]]]]]],[140,82,163,198,197,48,116,8,12],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",2]],["mul",["succ",["succ",["zero"]]],["var",1]]],["var",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",2]],["succ",["succ",["zero"]]]],["var",1]],["imp_elim",["forall_elim",["hyp",11],["var",2]],["hyp",1]]],["hyp",2]]],["eq_trans",["forall_elim",["forall_elim",["hyp",8],["var",2]],["var",1]],["eq_sym",["hyp",0]]]],["exists_elim",["hyp",0],["exists_elim",["imp_elim",["forall_elim",["hyp",9],["var",0]],["exists_intro",["var",3],["eq_sym",["and_elim_r",["hyp",0]]]]],["exists_intro",["var",2],["exists_intro",["var",0],["and_intro",["and_elim_l",["hyp",1]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["succ",["succ",["zero"]]]],["var",4]],["mul",["succ",["succ",["zero"]]],["mul",["var",0],["var",0]]]],["imp_intro",["imp_elim",["forall_elim",["hyp",7],["succ",["zero"]]],["hyp",0]]]],["eq_trans",["and_elim_r",["hyp",1]],["eq_subst",["eq",["mul",["var",0],["var",2]],["mul",["succ",["succ",["zero"]]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",1]]]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["mul",["succ",["succ",["zero"]]],["var",1]],["var",0]],["mul",["succ",["succ",["zero"]]],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",1]]]]],["eq_sym",["hyp",0]],["eq_trans",["forall_elim",["forall_elim",["hyp",8],["succ",["succ",["zero"]]]],["var",0]],["forall_elim",["forall_elim",["forall_elim",["hyp",7],["succ",["succ",["zero"]]]],["succ",["succ",["zero"]]]],["mul",["var",0],["var",0]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[1160,["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["exists",["exists",["and",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["exists",["eq",["add",["var",0],["var",1]],["var",4]]]]]]]]]]],[67,103,193,165,102,194,190,83],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["forall_elim",["forall_elim",["hyp",8],["var",2]],["var",1]],["exists_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",8],["var",0]],["var",3]],["var",2]],["and_elim_l",["hyp",1]]],["and_elim_l",["and_elim_r",["hyp",1]]]],["hyp",0]],["exists_elim",["hyp",0],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["hyp",8],["mul",["var",2],["var",2]]],["var",3]],["exists_intro",["add",["mul",["mul",["var",1],["var",1]],["mul",["var",1],["var",1]]],["mul",["mul",["var",0],["var",0]],["mul",["var",0],["var",0]]]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",5]],["var",4]],["var",3]],["var",2]],["var",1]],["var",0]],["and_elim_l",["and_elim_l",["hyp",0]]]],["and_elim_r",["and_elim_l",["hyp",0]]]],["and_elim_r",["and_elim_r",["and_elim_r",["hyp",3]]]]]]],["exists_intro",["var",2],["exists_intro",["var",1],["exists_intro",["var",0],["and_intro",["and_intro",["and_intro",["and_elim_r",["and_elim_l",["and_elim_r",["hyp",1]]]],["and_intro",["and_elim_l",["and_elim_r",["and_elim_r",["hyp",1]]]],["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",4]],["mul",["var",3],["var",3]]],["var",0]],["and_elim_l",["and_elim_r",["and_elim_r",["hyp",5]]]]],["hyp",1]],["hyp",0]]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",3]],["var",2]],["var",1]],["var",0]],["var",4]],["and_elim_l",["and_elim_l",["and_elim_r",["hyp",1]]]]],["hyp",0]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",6]],["var",5]],["var",4]],["var",3]],["var",2]],["var",1]],["and_elim_l",["and_elim_l",["hyp",1]]]],["and_elim_r",["and_elim_l",["hyp",1]]]],["and_elim_r",["and_elim_r",["and_elim_r",["hyp",4]]]]]]]]],["and_elim_r",["and_elim_r",["and_elim_r",["hyp",1]]]]],["eq_subst",["exists",["eq",["add",["var",0],["var",2]],["var",1]]],["eq_sym",["hyp",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",3],["var",3]]],["var",0]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",7],["var",3]],["and_elim_l",["and_elim_l",["and_elim_r",["hyp",2]]]]],["hyp",0]]]]]]]]]]]]]]]]]]]]]]]]]],[320,["forall",["forall",["forall",["imp",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["mul",["var",2],["var",2]]]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",0],["var",0]]],["mul",["var",1],["var",1]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]],[143,156,10,2],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["and_intro",["eq_trans",["forall_elim",["forall_elim",["hyp",2],["mul",["var",2],["var",2]]],["mul",["var",0],["var",0]]],["eq_sym",["hyp",0]]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["var",2],["var",2]]],["var",0]],["var",2]],["var",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",1]],["var",0]],["mul",["var",2],["var",2]]],["hyp",0]],["hyp",1]]],["exists_intro",["var",2],["eq_refl",["mul",["var",2],["var",2]]]]],["exists_intro",["succ",["zero"]],["eq_sym",["forall_elim",["hyp",3],["var",0]]]]]]]]]]]]]]]],[1272,["forall",["forall",["forall",["forall",["forall",["forall",["forall",["imp",["imp",["eq",["var",6],["zero"]],["bot"]],["imp",["imp",["eq",["var",4],["zero"]],["bot"]],["imp",["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["imp",["eq",["var",5],["mul",["var",6],["var",6]]],["imp",["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",3],["var",3]]]],["imp",["eq",["var",5],["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]]],["imp",["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",2],["var",1]]]],["imp",["eq",["var",0],["add",["mul",["var",5],["var",5]],["mul",["var",4],["var",4]]]],["exists",["exists",["and",["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["imp",["eq",["var",8],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["mul",["var",8],["var",8]]]]]],["exists",["eq",["add",["var",0],["succ",["var",9]]],["var",3]]]]]]]]]]]]]]]]]]]]],[48,12,163,79,102,84,196],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",2]],["var",1]],["var",3]],["hyp",5]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["succ",["succ",["zero"]]]],["mul",["var",2],["var",1]]],["mul",["var",3],["var",3]]],["imp_intro",["imp_elim",["forall_elim",["hyp",14],["succ",["zero"]]],["hyp",0]]]],["eq_trans",["eq_sym",["hyp",1]],["hyp",3]]]],["exists_elim",["hyp",0],["exists_intro",["var",1],["exists_intro",["var",0],["and_intro",["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["var",4]],["var",1]],["var",1]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",15],["mul",["var",4],["var",3]]],["var",4]],["var",3]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",15],["var",6]],["succ",["succ",["zero"]]]],["mul",["var",4],["var",3]]],["hyp",11]],["hyp",6]],["hyp",0]]]],["eq_refl",["mul",["var",4],["var",3]]]],["hyp",0]]]],["and_elim_l",["hyp",1]]],["hyp",0]]],["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["var",3]],["var",0]],["var",0]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",14],["mul",["var",4],["var",3]]],["var",4]],["var",3]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",15],["var",6]],["succ",["succ",["zero"]]]],["mul",["var",4],["var",3]]],["hyp",11]],["hyp",6]],["hyp",0]]]],["eq_refl",["mul",["var",4],["var",3]]]],["hyp",0]]]],["and_elim_r",["hyp",1]]],["hyp",0]]],["and_intro",["hyp",9],["eq_trans",["cong_add",["forall_elim",["hyp",11],["var",1]],["forall_elim",["hyp",11],["var",0]]],["eq_trans",["cong_add",["cong_mul",["eq_sym",["and_elim_l",["hyp",0]]],["eq_sym",["and_elim_l",["hyp",0]]]],["cong_mul",["eq_sym",["and_elim_r",["hyp",0]]],["eq_sym",["and_elim_r",["hyp",0]]]]],["eq_trans",["eq_sym",["hyp",4]],["hyp",6]]]]]]],["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",8]],["var",7]],["var",6]],["var",2]],["hyp",9]],["hyp",8]],["hyp",6]],["hyp",2]]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[416,["forall",["forall",["forall",["imp",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["and",["eq",["add",["mul",["mul",["var",2],["var",2]],["mul",["var",2],["var",2]]],["mul",["mul",["var",1],["var",1]],["mul",["var",1],["var",1]]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["mul",["var",4],["var",4]],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["mul",["var",3],["var",3]],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[130,191],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["and_intro",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",2]],["var",1]],["var",0]],["hyp",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",9],["var",2]],["var",1]],["hyp",6]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]],["and_elim_r",["hyp",2]]]],["and_elim_l",["hyp",1]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]]]],[456,["forall",["forall",["forall",["imp",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["and",["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["var",0],["var",0]]]]]],["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]],[2,68],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["and_intro",["and_intro",["hyp",3],["and_intro",["hyp",5],["and_intro",["hyp",1],["eq_trans",["forall_elim",["forall_elim",["hyp",10],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["hyp",0]]]]],["imp_elim",["forall_elim",["forall_elim",["hyp",9],["var",2]],["var",1]],["hyp",6]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]],["and_elim_r",["hyp",2]]]],["and_elim_l",["hyp",1]]]],["and_elim_r",["hyp",1]]]],["and_elim_l",["hyp",0]]]]]]]]]],[1824,["forall",["forall",["forall",["imp",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["imp",["exists",["eq",["mul",["var",3],["var",3]],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["imp",["exists",["eq",["mul",["var",2],["var",2]],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["exists",["exists",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]]]]]],[203,182,190,201,155,199,179,79,154,202],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["exists_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["mul",["var",2],["var",2]]],["mul",["var",1],["var",1]]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",2]],["var",1]],["var",0]],["hyp",2]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",11],["var",2]],["and_elim_l",["and_elim_l",["hyp",3]]]],["hyp",0]]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",11],["var",1]],["and_elim_l",["and_elim_r",["and_elim_l",["hyp",3]]]]],["hyp",0]]]],["hyp",1]],["hyp",0]],["exists_elim",["hyp",0],["exists_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",1]],["var",0]],["var",3]],["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",10],["var",4]],["var",0]],["var",1]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",11],["var",4]],["var",1]],["var",0]],["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]]]],["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]],["exists_elim",["hyp",0],["exists_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",15],["var",6]],["var",2]],["var",3]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",13],["var",6]],["var",3]],["var",2]],["and_elim_l",["and_elim_r",["and_elim_r",["hyp",2]]]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",2]]]]]]]]],["and_elim_l",["and_elim_l",["hyp",6]]]],["and_elim_l",["hyp",2]]],["imp_elim",["forall_elim",["hyp",8],["var",6]],["hyp",5]]],["exists_intro",["mul",["var",0],["var",0]],["and_elim_r",["hyp",0]]]],["exists_elim",["hyp",0],["exists_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",9],["var",3]],["var",5]],["var",4]],["var",2]],["var",1]],["var",0]],["var",6]],["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",12],["var",5]],["var",3]],["var",3]],["and_elim_l",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",13],["mul",["var",7],["var",7]]],["var",5]],["var",4]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",18],["var",7]],["and_elim_l",["and_elim_r",["and_elim_l",["hyp",10]]]]],["hyp",0]]]],["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",5]]]]]]]]]],["and_elim_l",["hyp",3]]],["hyp",0]]]],["and_elim_l",["hyp",4]]],["and_elim_l",["and_elim_r",["and_elim_r",["hyp",0]]]]],["and_elim_l",["hyp",2]]],["and_elim_r",["hyp",2]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]],["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",0]]]]]]]],["and_elim_l",["and_elim_r",["and_elim_r",["and_elim_r",["and_elim_r",["hyp",4]]]]]]],["exists_elim",["hyp",0],["exists_intro",["var",1],["exists_intro",["var",0],["exists_intro",["var",5],["hyp",0]]]]]]]]]]]]]]]]]]]]]]]]]]]]],[408,["forall",["forall",["forall",["imp",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]],["exists",["exists",["exists",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]]]],[203,153,204,205],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["mul",["var",2],["var",2]]],["mul",["var",1],["var",1]]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["var",0]],["hyp",0]]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",1]],["var",2]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",0]],["hyp",1]]],["and_elim_r",["hyp",0]]],["and_elim_l",["hyp",0]]],["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",2],["var",2]],["var",1]],["var",0]],["hyp",1]],["and_elim_l",["hyp",0]]],["and_elim_r",["hyp",0]]]]]]]]]]]]],[384,["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["exists",["exists",["and",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",4]]]]]]]]]]],[200,206,33],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",0]],["hyp",0]],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",1]],["var",0]],["and_elim_l",["hyp",0]]],["exists_elim",["hyp",0],["exists_elim",["hyp",0],["exists_intro",["var",2],["exists_intro",["var",1],["exists_intro",["var",0],["and_intro",["and_elim_l",["hyp",0]],["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",7],["var",0]],["var",3]],["var",6]],["and_elim_r",["hyp",0]]],["and_elim_r",["hyp",3]]]]]]]]]]]]]]]]]]]]],[56,["forall",["forall",["forall",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]]]]],["bot"]]]]],[132,207],["imp_intro",["imp_intro",["imp_elim",["hyp",1],["hyp",0]]]]],[56,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["bot"]]]]]]]],[133,207],["imp_intro",["imp_intro",["imp_elim",["hyp",1],["hyp",0]]]]],[352,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]],["imp",["eq",["var",0],["zero"]],["bot"]]]]]]],[190,84,45,4],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",6],["mul",["var",2],["var",2]]],["imp_intro",["imp_elim",["imp_elim",["forall_elim",["hyp",7],["var",2]],["hyp",3]],["hyp",0]]]],["eq_trans",["eq_sym",["forall_elim",["hyp",5],["var",2]]],["imp_elim",["forall_elim",["forall_elim",["hyp",4],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_trans",["hyp",1],["eq_subst",["eq",["mul",["var",0],["var",1]],["zero"]],["eq_sym",["hyp",0]],["eq_subst",["eq",["mul",["zero"],["var",0]],["zero"]],["eq_sym",["hyp",0]],["eq_subst",["eq",["var",0],["zero"]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_refl",["zero"]]]]]]]]]]]]]]]]]]]],[352,["forall",["forall",["forall",["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]],["or",["eq",["var",2],["zero"]],["eq",["var",1],["zero"]]]]]]],[78,208,210],["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["or_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["zero"]],["or_intro_l",["hyp",0]],["or_elim",["forall_elim",["forall_elim",["hyp",4],["var",1]],["zero"]],["or_intro_r",["hyp",0]],["bot_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["var",0]],["and_intro",["hyp",1],["and_intro",["hyp",0],["and_intro",["imp_intro",["imp_elim",["imp_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",4],["var",2]],["var",1]],["var",0]],["hyp",2]],["hyp",3]],["hyp",0]]],["hyp",2]]]]]]]]]]]]]]]],[208,["forall",["forall",["forall",["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["or",["eq",["var",2],["zero"]],["eq",["var",1],["zero"]]]]]]],[211,84],["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["imp_intro",["imp_elim",["hyp",0],["eq_trans",["hyp",3],["forall_elim",["hyp",4],["var",0]]]]],["forall_elim",["hyp",0],["mul",["var",0],["var",0]]]]],["forall_elim",["hyp",0],["var",1]]]],["forall_elim",["hyp",2],["var",2]]]]]]]]]],[1760,["forall",["forall",["forall",["and",["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["or",["and",["eq",["var",2],["zero"]],["eq",["var",1],["var",0]]],["and",["eq",["var",1],["zero"]],["eq",["var",2],["var",0]]]]],["imp",["or",["and",["eq",["var",2],["zero"]],["eq",["var",1],["var",0]]],["and",["eq",["var",1],["zero"]],["eq",["var",2],["var",0]]]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]]]]]]],[212,160,84,4,0],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["imp_intro",["forall_intro",["forall_intro",["forall_intro",["and_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",1]],["var",0]],["hyp",0]],["or_intro_l",["and_intro",["hyp",0],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",1]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["eq_trans",["eq_sym",["forall_elim",["hyp",4],["var",1]]],["eq_trans",["eq_trans",["eq_sym",["eq_subst",["eq",["add",["mul",["mul",["mul",["var",0],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["mul",["mul",["zero"],["var",0]],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["mul",["var",0],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["mul",["zero"],["var",0]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["var",0],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["zero"],["var",0]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["var",0],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["var",0],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["eq_sym",["forall_elim",["hyp",2],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]]],["eq_refl",["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]]]]]]]]]]],["hyp",1]],["forall_elim",["hyp",4],["var",0]]]]]]]],["or_intro_r",["and_intro",["hyp",0],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["var",2]],["var",0]],["imp_elim",["forall_elim",["forall_elim",["hyp",5],["mul",["var",2],["var",2]]],["mul",["var",0],["var",0]]],["eq_trans",["eq_sym",["forall_elim",["hyp",4],["var",2]]],["eq_trans",["eq_trans",["eq_sym",["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["var",0],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["zero"],["var",0]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["var",0],["var",2]],["var",2]]],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["zero"],["var",0]],["var",2]]],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["var",0],["var",2]]],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["zero"],["var",0]]],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["hyp",0]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["var",0]],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["var",0],["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]]],["eq_refl",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]]]]]]]]]]],["hyp",1]],["forall_elim",["hyp",4],["var",0]]]]]]]]]],["imp_intro",["or_elim",["hyp",0],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",0],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["zero"],["var",0]],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["mul",["mul",["var",0],["var",3]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["mul",["zero"],["var",0]],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["mul",["var",0],["var",3]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["zero"],["var",0]],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["var",0],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["zero"],["mul",["mul",["mul",["var",0],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_subst",["eq",["add",["zero"],["mul",["mul",["mul",["var",1],["var",0]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_subst",["eq",["add",["zero"],["mul",["mul",["mul",["var",1],["var",1]],["var",0]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_subst",["eq",["add",["zero"],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",0]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_subst",["eq",["var",0],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["hyp",2],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]]],["eq_refl",["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]]]]]]]]]]]]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["var",0],["var",2]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["zero"],["var",0]],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["var",0],["var",2]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["mul",["zero"],["var",0]],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["var",0],["var",2]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["mul",["zero"],["var",0]]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_l",["hyp",0]]],["eq_subst",["eq",["add",["mul",["mul",["mul",["var",3],["var",3]],["var",3]],["var",3]],["var",0]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["axiom","PA5"],["zero"]]],["eq_subst",["eq",["var",0],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["forall_elim",["axiom","PA3"],["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]]]],["eq_subst",["eq",["mul",["mul",["mul",["var",0],["var",3]],["var",3]],["var",3]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_subst",["eq",["mul",["mul",["mul",["var",1],["var",0]],["var",3]],["var",3]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_subst",["eq",["mul",["mul",["mul",["var",1],["var",1]],["var",0]],["var",3]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_subst",["eq",["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",0]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["eq_sym",["and_elim_r",["hyp",0]]],["eq_refl",["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]]]]]]]]]]]]]]]]]]]]]]]]]],[176,["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]],["bot"]]]]]]],[211],["imp_intro",["forall_intro",["forall_intro",["forall_intro",["imp_intro",["imp_intro",["imp_intro",["or_elim",["imp_elim",["forall_elim",["forall_elim",["forall_elim",["hyp",3],["var",2]],["var",1]],["var",0]],["hyp",0]],["imp_elim",["hyp",3],["hyp",0]],["imp_elim",["hyp",2],["hyp",0]]]]]]]]]]],[104,["and",["and",["forall",["forall",["forall",["and",["imp",["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]],["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]]],["imp",["exists",["exists",["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["exists",["eq",["add",["var",0],["succ",["var",1]]],["var",2]]],["and",["forall",["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",2],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]],["and",["or",["and",["exists",["eq",["var",2],["mul",["succ",["succ",["zero"]]],["var",0]]]],["exists",["eq",["var",1],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]]],["and",["exists",["eq",["var",2],["add",["mul",["succ",["succ",["zero"]]],["var",0]],["succ",["zero"]]]]],["exists",["eq",["var",1],["mul",["succ",["succ",["zero"]]],["var",0]]]]]],["and",["eq",["var",2],["add",["mul",["var",1],["var",1]],["mul",["var",0],["var",0]]]],["or",["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",4]]],["eq",["var",3],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]],["and",["eq",["mul",["var",1],["var",1]],["add",["mul",["var",0],["var",0]],["var",3]]],["eq",["var",4],["mul",["succ",["succ",["zero"]]],["mul",["var",1],["var",0]]]]]]]]]]]]],["and",["imp",["eq",["var",2],["zero"]],["bot"]],["and",["imp",["eq",["var",1],["zero"]],["bot"]],["and",["imp",["eq",["var",0],["zero"]],["bot"]],["and",["eq",["add",["mul",["var",2],["var",2]],["mul",["var",1],["var",1]]],["mul",["var",0],["var",0]]],["forall",["imp",["exists",["eq",["var",4],["mul",["var",1],["var",0]]]],["imp",["exists",["eq",["var",3],["mul",["var",1],["var",0]]]],["eq",["var",0],["succ",["zero"]]]]]]]]]]]]]]],["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["imp",["eq",["var",0],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["bot"]]]]]]]]],["and",["forall",["forall",["forall",["and",["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]],["or",["and",["eq",["var",2],["zero"]],["eq",["var",1],["var",0]]],["and",["eq",["var",1],["zero"]],["eq",["var",2],["var",0]]]]],["imp",["or",["and",["eq",["var",2],["zero"]],["eq",["var",1],["var",0]]],["and",["eq",["var",1],["zero"]],["eq",["var",2],["var",0]]]],["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["mul",["mul",["var",0],["var",0]],["var",0]],["var",0]]]]]]]],["forall",["forall",["forall",["imp",["imp",["eq",["var",2],["zero"]],["bot"]],["imp",["imp",["eq",["var",1],["zero"]],["bot"]],["imp",["eq",["add",["mul",["mul",["mul",["var",2],["var",2]],["var",2]],["var",2]],["mul",["mul",["mul",["var",1],["var",1]],["var",1]],["var",1]]],["mul",["var",0],["var",0]]],["bot"]]]]]]]]],[188,209,213,214],["imp_intro",["imp_intro",["imp_intro",["imp_intro",["and_intro",["and_intro",["hyp",3],["hyp",2]],["and_intro",["hyp",1],["hyp",0]]]]]]]]]]