{ "authority": "same_existing_definitions_reused_not_new_admission", "edges": [ { "kind": "definition_uses_definition", "source": "PD0002", "target": "PD0015" }, { "kind": "definition_uses_definition", "source": "PD0013", "target": "PD0015" }, { "kind": "definition_uses_definition", "source": "PD0002", "target": "PD0019" }, { "kind": "definition_uses_definition", "source": "PD0013", "target": "PD0019" }, { "kind": "definition_uses_definition", "source": "PD0002", "target": "ND0291" }, { "kind": "definition_uses_definition", "source": "PD0001", "target": "ND0291" }, { "kind": "definition_uses_definition", "source": "PD0013", "target": "ND0291" }, { "kind": "definition_uses_definition", "source": "ND0291", "target": "ND0292" }, { "kind": "definition_uses_definition", "source": "PD0002", "target": "ND0293" }, { "kind": "definition_uses_definition", "source": "PD0013", "target": "ND0293" }, { "kind": "definition_uses_definition", "source": "ND0292", "target": "ND0293" } ], "nodes": [ { "ast": [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 1 ] ], [ "var", 2 ] ] ], "id": "PD0001", "name": "Le", "parameters": [ "a", "b" ], "source": "exists h. h + a = b" }, { "ast": [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 2 ] ] ], "id": "PD0002", "name": "Lt", "parameters": [ "a", "b" ], "source": "exists h. h + S a = b" }, { "ast": [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 2 ] ] ] ], [ "var", 4 ] ] ] ] ], "id": "PD0013", "name": "BetaAt", "parameters": [ "b", "c", "i", "x" ], "source": "((exists ff_h_defined_beta_at. ff_h_defined_beta_at + S (x) = S ((S (i)) * c)) /\\ exists ff_q_defined_beta_at. b = ff_q_defined_beta_at * S ((S (i)) * c) + (x))" }, { "ast": [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 6 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "id": "PD0015", "name": "Sum", "parameters": [ "b", "c", "l", "z" ], "source": "exists ff_u_defined_sum ff_v_defined_sum. ((((exists ff_h_defined_sum_start. ff_h_defined_sum_start + S (0) = S ((S (0)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_start. ff_u_defined_sum = ff_q_defined_sum_start * S ((S (0)) * ff_v_defined_sum) + (0))) /\\ ((((exists ff_h_defined_sum_terminal. ff_h_defined_sum_terminal + S (z) = S ((S (l)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_terminal. ff_u_defined_sum = ff_q_defined_sum_terminal * S ((S (l)) * ff_v_defined_sum) + (z))) /\\ forall ff_i_defined_sum. (exists ff_lt_defined_sum_bound. ff_lt_defined_sum_bound + S ff_i_defined_sum = l) -> exists ff_a_defined_sum ff_r_defined_sum ff_s_defined_sum. ((((exists ff_h_defined_sum_summand. ff_h_defined_sum_summand + S (ff_a_defined_sum) = S ((S (ff_i_defined_sum)) * c)) /\\ exists ff_q_defined_sum_summand. b = ff_q_defined_sum_summand * S ((S (ff_i_defined_sum)) * c) + (ff_a_defined_sum))) /\\ ((((exists ff_h_defined_sum_partial. ff_h_defined_sum_partial + S (ff_r_defined_sum) = S ((S (ff_i_defined_sum)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_partial. ff_u_defined_sum = ff_q_defined_sum_partial * S ((S (ff_i_defined_sum)) * ff_v_defined_sum) + (ff_r_defined_sum))) /\\ ((((exists ff_h_defined_sum_successor. ff_h_defined_sum_successor + S (ff_s_defined_sum) = S ((S (S ff_i_defined_sum)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_successor. ff_u_defined_sum = ff_q_defined_sum_successor * S ((S (S ff_i_defined_sum)) * ff_v_defined_sum) + (ff_s_defined_sum))) /\\ ff_s_defined_sum = ff_r_defined_sum + ff_a_defined_sum)))))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 5 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ], [ "var", 4 ] ] ] ] ] ] ], "id": "PD0019", "name": "Repeat", "parameters": [ "b", "c", "a", "l" ], "source": "forall ff_i_defined_repeat. (exists ff_lt_defined_repeat_bound. ff_lt_defined_repeat_bound + S ff_i_defined_repeat = l) -> (((exists ff_h_defined_repeat_decoded. ff_h_defined_repeat_decoded + S (a) = S ((S (ff_i_defined_repeat)) * c)) /\\ exists ff_q_defined_repeat_decoded. b = ff_q_defined_repeat_decoded * S ((S (ff_i_defined_repeat)) * c) + (a)))" }, { "ast": [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 3 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 2 ] ] ] ], [ "var", 5 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 3 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 4 ], [ "zero" ] ] ] ], "id": "ND0291", "name": "BetaZeroExtend", "parameters": [ "b", "c", "L", "i", "a" ], "source": "(((exists pfa_gap_lowercontinuationinside. pfa_gap_lowercontinuationinside + S ((i)) = ((L))) /\\ ((((exists ff_h_pfp_lowercontinuationentry. ff_h_pfp_lowercontinuationentry + S ((a)) = S ((S ((i))) * (c))) /\\ exists ff_q_pfp_lowercontinuationentry. (b) = ff_q_pfp_lowercontinuationentry * S ((S ((i))) * (c)) + ((a))))))) \\/ (((exists pfc_gap_lowercontinuationoutside. pfc_gap_lowercontinuationoutside+((L))=((i))) /\\ ((((a))=0))))" }, { "ast": [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 10 ], [ "var", 2 ] ], [ "var", 9 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "var", 6 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 11 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 11 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 6 ] ], [ "var", 11 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 9 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 9 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 11 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], "id": "ND0292", "name": "PolynomialDiagonalTerm", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "i", "j", "t" ], "source": "exists pfc_complement_lowercontinuation pfc_left_lowercontinuation pfc_right_lowercontinuation. ((((j))+pfc_complement_lowercontinuation=((i))) /\\ ((((((exists pfa_gap_lowercontinuationleftinside. pfa_gap_lowercontinuationleftinside + S ((j)) = ((L))) /\\ ((((exists ff_h_pfp_lowercontinuationleftentry. ff_h_pfp_lowercontinuationleftentry + S (pfc_left_lowercontinuation) = S ((S ((j))) * (ac))) /\\ exists ff_q_pfp_lowercontinuationleftentry. (ab) = ff_q_pfp_lowercontinuationleftentry * S ((S ((j))) * (ac)) + (pfc_left_lowercontinuation)))))) \\/ (((exists pfc_gap_lowercontinuationleftoutside. pfc_gap_lowercontinuationleftoutside+((L))=((j))) /\\ (((pfc_left_lowercontinuation)=0))))) /\\ ((((((exists pfa_gap_lowercontinuationrightinside. pfa_gap_lowercontinuationrightinside + S (pfc_complement_lowercontinuation) = ((M))) /\\ ((((exists ff_h_pfp_lowercontinuationrightentry. ff_h_pfp_lowercontinuationrightentry + S (pfc_right_lowercontinuation) = S ((S (pfc_complement_lowercontinuation)) * (bc))) /\\ exists ff_q_pfp_lowercontinuationrightentry. (bb) = ff_q_pfp_lowercontinuationrightentry * S ((S (pfc_complement_lowercontinuation)) * (bc)) + (pfc_right_lowercontinuation)))))) \\/ (((exists pfc_gap_lowercontinuationrightoutside. pfc_gap_lowercontinuationrightoutside+((M))=(pfc_complement_lowercontinuation)) /\\ (((pfc_right_lowercontinuation)=0))))) /\\ ((((t))=pfc_left_lowercontinuation*pfc_right_lowercontinuation)))))))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 11 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 11 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 8 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 8 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 11 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 11 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ], "id": "ND0293", "name": "PolynomialDiagonalPrefix", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "i", "db", "dc", "l" ], "source": "forall pfc_index_lowercontinuation. (exists pfa_gap_lowercontinuationbound. pfa_gap_lowercontinuationbound + S (pfc_index_lowercontinuation) = ((l))) -> exists pfc_value_lowercontinuation. ((((exists ff_h_pfp_lowercontinuationentry. ff_h_pfp_lowercontinuationentry + S (pfc_value_lowercontinuation) = S ((S (pfc_index_lowercontinuation)) * (dc))) /\\ exists ff_q_pfp_lowercontinuationentry. (db) = ff_q_pfp_lowercontinuationentry * S ((S (pfc_index_lowercontinuation)) * (dc)) + (pfc_value_lowercontinuation))) /\\ ((exists pfc_complement_lowercontinuationterm pfc_left_lowercontinuationterm pfc_right_lowercontinuationterm. (((pfc_index_lowercontinuation)+pfc_complement_lowercontinuationterm=((i))) /\\ ((((((exists pfa_gap_lowercontinuationtermleftinside. pfa_gap_lowercontinuationtermleftinside + S (pfc_index_lowercontinuation) = ((L))) /\\ ((((exists ff_h_pfp_lowercontinuationtermleftentry. ff_h_pfp_lowercontinuationtermleftentry + S (pfc_left_lowercontinuationterm) = S ((S (pfc_index_lowercontinuation)) * (ac))) /\\ exists ff_q_pfp_lowercontinuationtermleftentry. (ab) = ff_q_pfp_lowercontinuationtermleftentry * S ((S (pfc_index_lowercontinuation)) * (ac)) + (pfc_left_lowercontinuationterm)))))) \\/ (((exists pfc_gap_lowercontinuationtermleftoutside. pfc_gap_lowercontinuationtermleftoutside+((L))=(pfc_index_lowercontinuation)) /\\ (((pfc_left_lowercontinuationterm)=0))))) /\\ ((((((exists pfa_gap_lowercontinuationtermrightinside. pfa_gap_lowercontinuationtermrightinside + S (pfc_complement_lowercontinuationterm) = ((M))) /\\ ((((exists ff_h_pfp_lowercontinuationtermrightentry. ff_h_pfp_lowercontinuationtermrightentry + S (pfc_right_lowercontinuationterm) = S ((S (pfc_complement_lowercontinuationterm)) * (bc))) /\\ exists ff_q_pfp_lowercontinuationtermrightentry. (bb) = ff_q_pfp_lowercontinuationtermrightentry * S ((S (pfc_complement_lowercontinuationterm)) * (bc)) + (pfc_right_lowercontinuationterm)))))) \\/ (((exists pfc_gap_lowercontinuationtermrightoutside. pfc_gap_lowercontinuationtermrightoutside+((M))=(pfc_complement_lowercontinuationterm)) /\\ (((pfc_right_lowercontinuationterm)=0))))) /\\ (((pfc_value_lowercontinuation)=pfc_left_lowercontinuationterm*pfc_right_lowercontinuationterm))))))))))" } ], "theorem_uses": [ { "definition": "PD0002", "kind": "theorem_uses_definition", "theorem": "CV001" }, { "definition": "PD0015", "kind": "theorem_uses_definition", "theorem": "CV001" }, { "definition": "ND0291", "kind": "theorem_uses_definition", "theorem": "CV001" }, { "definition": "ND0293", "kind": "theorem_uses_definition", "theorem": "CV001" } ] }