Actual pointwise additive congruences lift by finite induction to the three actual Sum endpoints, for every modulus and also for the empty prefix.
Alpha v34 checked-use · first admitted v33 · independently kernel and Lean verified; not Stable
Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.
forall p ab ac bb bc cb cc L A B C. (exists fs_u_pfc_distribution_sum_a fs_v_pfc_distribution_sum_a. ((((exists fs_h_pfc_distribution_sum_a_body_start. fs_h_pfc_distribution_sum_a_body_start + S (0) = S ((S (0)) * fs_v_pfc_distribution_sum_a)) /\ exists fs_q_pfc_distribution_sum_a_body_start. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_start * S ((S (0)) * fs_v_pfc_distribution_sum_a) + (0))) /\ ((((exists fs_h_pfc_distribution_sum_a_body_terminal. fs_h_pfc_distribution_sum_a_body_terminal + S (A) = S ((S (L)) * fs_v_pfc_distribution_sum_a)) /\ exists fs_q_pfc_distribution_sum_a_body_terminal. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_terminal * S ((S (L)) * fs_v_pfc_distribution_sum_a) + (A))) /\ forall fs_i_pfc_distribution_sum_a_body_steps. (exists fs_lt_pfc_distribution_sum_a_body_steps_bound. fs_lt_pfc_distribution_sum_a_body_steps_bound + S fs_i_pfc_distribution_sum_a_body_steps = L) -> exists fs_a_pfc_distribution_sum_a_body_steps fs_r_pfc_distribution_sum_a_body_steps fs_s_pfc_distribution_sum_a_body_steps. ((((exists fs_h_pfc_distribution_sum_a_body_steps_summand. fs_h_pfc_distribution_sum_a_body_steps_summand + S (fs_a_pfc_distribution_sum_a_body_steps) = S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * ac)) /\ exists fs_q_pfc_distribution_sum_a_body_steps_summand. ab = fs_q_pfc_distribution_sum_a_body_steps_summand * S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * ac) + (fs_a_pfc_distribution_sum_a_body_steps))) /\ ((((exists fs_h_pfc_distribution_sum_a_body_steps_partial. fs_h_pfc_distribution_sum_a_body_steps_partial + S (fs_r_pfc_distribution_sum_a_body_steps) = S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a)) /\ exists fs_q_pfc_distribution_sum_a_body_steps_partial. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_steps_partial * S ((S (fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a) + (fs_r_pfc_distribution_sum_a_body_steps))) /\ ((((exists fs_h_pfc_distribution_sum_a_body_steps_successor. fs_h_pfc_distribution_sum_a_body_steps_successor + S (fs_s_pfc_distribution_sum_a_body_steps) = S ((S (S fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a)) /\ exists fs_q_pfc_distribution_sum_a_body_steps_successor. fs_u_pfc_distribution_sum_a = fs_q_pfc_distribution_sum_a_body_steps_successor * S ((S (S fs_i_pfc_distribution_sum_a_body_steps)) * fs_v_pfc_distribution_sum_a) + (fs_s_pfc_distribution_sum_a_body_steps))) /\ fs_s_pfc_distribution_sum_a_body_steps = fs_r_pfc_distribution_sum_a_body_steps + fs_a_pfc_distribution_sum_a_body_steps)))))) -> (exists fs_u_pfc_distribution_sum_b fs_v_pfc_distribution_sum_b. ((((exists fs_h_pfc_distribution_sum_b_body_start. fs_h_pfc_distribution_sum_b_body_start + S (0) = S ((S (0)) * fs_v_pfc_distribution_sum_b)) /\ exists fs_q_pfc_distribution_sum_b_body_start. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_start * S ((S (0)) * fs_v_pfc_distribution_sum_b) + (0))) /\ ((((exists fs_h_pfc_distribution_sum_b_body_terminal. fs_h_pfc_distribution_sum_b_body_terminal + S (B) = S ((S (L)) * fs_v_pfc_distribution_sum_b)) /\ exists fs_q_pfc_distribution_sum_b_body_terminal. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_terminal * S ((S (L)) * fs_v_pfc_distribution_sum_b) + (B))) /\ forall fs_i_pfc_distribution_sum_b_body_steps. (exists fs_lt_pfc_distribution_sum_b_body_steps_bound. fs_lt_pfc_distribution_sum_b_body_steps_bound + S fs_i_pfc_distribution_sum_b_body_steps = L) -> exists fs_a_pfc_distribution_sum_b_body_steps fs_r_pfc_distribution_sum_b_body_steps fs_s_pfc_distribution_sum_b_body_steps. ((((exists fs_h_pfc_distribution_sum_b_body_steps_summand. fs_h_pfc_distribution_sum_b_body_steps_summand + S (fs_a_pfc_distribution_sum_b_body_steps) = S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * bc)) /\ exists fs_q_pfc_distribution_sum_b_body_steps_summand. bb = fs_q_pfc_distribution_sum_b_body_steps_summand * S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * bc) + (fs_a_pfc_distribution_sum_b_body_steps))) /\ ((((exists fs_h_pfc_distribution_sum_b_body_steps_partial. fs_h_pfc_distribution_sum_b_body_steps_partial + S (fs_r_pfc_distribution_sum_b_body_steps) = S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b)) /\ exists fs_q_pfc_distribution_sum_b_body_steps_partial. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_steps_partial * S ((S (fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b) + (fs_r_pfc_distribution_sum_b_body_steps))) /\ ((((exists fs_h_pfc_distribution_sum_b_body_steps_successor. fs_h_pfc_distribution_sum_b_body_steps_successor + S (fs_s_pfc_distribution_sum_b_body_steps) = S ((S (S fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b)) /\ exists fs_q_pfc_distribution_sum_b_body_steps_successor. fs_u_pfc_distribution_sum_b = fs_q_pfc_distribution_sum_b_body_steps_successor * S ((S (S fs_i_pfc_distribution_sum_b_body_steps)) * fs_v_pfc_distribution_sum_b) + (fs_s_pfc_distribution_sum_b_body_steps))) /\ fs_s_pfc_distribution_sum_b_body_steps = fs_r_pfc_distribution_sum_b_body_steps + fs_a_pfc_distribution_sum_b_body_steps)))))) -> (exists fs_u_pfc_distribution_sum_c fs_v_pfc_distribution_sum_c. ((((exists fs_h_pfc_distribution_sum_c_body_start. fs_h_pfc_distribution_sum_c_body_start + S (0) = S ((S (0)) * fs_v_pfc_distribution_sum_c)) /\ exists fs_q_pfc_distribution_sum_c_body_start. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_start * S ((S (0)) * fs_v_pfc_distribution_sum_c) + (0))) /\ ((((exists fs_h_pfc_distribution_sum_c_body_terminal. fs_h_pfc_distribution_sum_c_body_terminal + S (C) = S ((S (L)) * fs_v_pfc_distribution_sum_c)) /\ exists fs_q_pfc_distribution_sum_c_body_terminal. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_terminal * S ((S (L)) * fs_v_pfc_distribution_sum_c) + (C))) /\ forall fs_i_pfc_distribution_sum_c_body_steps. (exists fs_lt_pfc_distribution_sum_c_body_steps_bound. fs_lt_pfc_distribution_sum_c_body_steps_bound + S fs_i_pfc_distribution_sum_c_body_steps = L) -> exists fs_a_pfc_distribution_sum_c_body_steps fs_r_pfc_distribution_sum_c_body_steps fs_s_pfc_distribution_sum_c_body_steps. ((((exists fs_h_pfc_distribution_sum_c_body_steps_summand. fs_h_pfc_distribution_sum_c_body_steps_summand + S (fs_a_pfc_distribution_sum_c_body_steps) = S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * cc)) /\ exists fs_q_pfc_distribution_sum_c_body_steps_summand. cb = fs_q_pfc_distribution_sum_c_body_steps_summand * S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * cc) + (fs_a_pfc_distribution_sum_c_body_steps))) /\ ((((exists fs_h_pfc_distribution_sum_c_body_steps_partial. fs_h_pfc_distribution_sum_c_body_steps_partial + S (fs_r_pfc_distribution_sum_c_body_steps) = S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c)) /\ exists fs_q_pfc_distribution_sum_c_body_steps_partial. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_steps_partial * S ((S (fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c) + (fs_r_pfc_distribution_sum_c_body_steps))) /\ ((((exists fs_h_pfc_distribution_sum_c_body_steps_successor. fs_h_pfc_distribution_sum_c_body_steps_successor + S (fs_s_pfc_distribution_sum_c_body_steps) = S ((S (S fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c)) /\ exists fs_q_pfc_distribution_sum_c_body_steps_successor. fs_u_pfc_distribution_sum_c = fs_q_pfc_distribution_sum_c_body_steps_successor * S ((S (S fs_i_pfc_distribution_sum_c_body_steps)) * fs_v_pfc_distribution_sum_c) + (fs_s_pfc_distribution_sum_c_body_steps))) /\ fs_s_pfc_distribution_sum_c_body_steps = fs_r_pfc_distribution_sum_c_body_steps + fs_a_pfc_distribution_sum_c_body_steps)))))) -> (forall i a b c. (exists pfa_gap_distribution_sum_pointwise_bound. pfa_gap_distribution_sum_pointwise_bound + S (i) = (L)) -> (((exists ff_h_pfp_distribution_sum_pointwise_a. ff_h_pfp_distribution_sum_pointwise_a + S (a) = S ((S (i)) * ac)) /\ exists ff_q_pfp_distribution_sum_pointwise_a. ab = ff_q_pfp_distribution_sum_pointwise_a * S ((S (i)) * ac) + (a))) -> (((exists ff_h_pfp_distribution_sum_pointwise_b. ff_h_pfp_distribution_sum_pointwise_b + S (b) = S ((S (i)) * bc)) /\ exists ff_q_pfp_distribution_sum_pointwise_b. bb = ff_q_pfp_distribution_sum_pointwise_b * S ((S (i)) * bc) + (b))) -> (((exists ff_h_pfp_distribution_sum_pointwise_c. ff_h_pfp_distribution_sum_pointwise_c + S (c) = S ((S (i)) * cc)) /\ exists ff_q_pfp_distribution_sum_pointwise_c. cb = ff_q_pfp_distribution_sum_pointwise_c * S ((S (i)) * cc) + (c))) -> (exists pfa_offset_left_distribution_sum_pointwise_value pfa_offset_right_distribution_sum_pointwise_value. (a+b) + (p) * pfa_offset_left_distribution_sum_pointwise_value = (c) + (p) * pfa_offset_right_distribution_sum_pointwise_value)) -> (exists pfa_offset_left_distribution_sum_result pfa_offset_right_distribution_sum_result. (A+B) + (p) * pfa_offset_left_distribution_sum_result = (C) + (p) * pfa_offset_right_distribution_sum_result)
Complete tactic proof in conservative notation
All 152 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.
01Fix variables and assumptionsL1–7
Work with arbitrary variables or the premises of the current implication.