SS0016 — divisor_signed_sum_successor_intro

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

These are actual signed-table and finite-sum foundations. Equality compares represented signed values, not arbitrary encodings. MatrixMinorFourCode is reused solely as generic nested pairing, without a matrix hypothesis. Full finite signed G007 is established separately in the Möbius-inversion family.

Exact theorem in conservative defined notation

∀ F. ∀ l. ∀ a. ∀ b. ∀ c. SignedPrefixSum(F,l,a) → ArithAt(F,l,b) → (∃ x. ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. SignedDecode(a,x,y) ∧ (SignedDecode(b,z,n) ∧ (SignedDecode(c,m,k) ∧ x + z + k = y + n + m))) → SignedPrefixSum(F,S l,c)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

SignedDecode(z,p,n) · 3 SignedBalance(z,p,n) · 1 ArithAt(F,i,z) · 1 SignedPrefixSum(F,l,z) · 2 BetaAt(b,c,i,x) · 2

Actual proof prerequisites

divisor_signed_table_at_to_components divisor_signed_sum_from_components divisor_natural_sum_successor_intro gaussian_signed_add_to_balance · checked external prerequisite

Original expanded first-order statement

forall F l a b c. (exists dst_positive_code_step_sum dst_positive_scale_step_sum dst_negative_code_step_sum dst_negative_scale_step_sum dst_positive_sum_step_sum dst_negative_sum_step_sum. (((F) = (((((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) * S ((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) + ((dst_positive_scale_step_sum) + (dst_positive_scale_step_sum))) + (((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum)))) * S ((((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) * S ((dst_positive_code_step_sum) + (dst_positive_scale_step_sum)) + ((dst_positive_scale_step_sum) + (dst_positive_scale_step_sum))) + (((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum)))) + ((((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum))) + (((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) * S ((dst_negative_code_step_sum) + (dst_negative_scale_step_sum)) + ((dst_negative_scale_step_sum) + (dst_negative_scale_step_sum)))))) /\ (((exists fs_u_dst_step_sumpositive fs_v_dst_step_sumpositive. ((((exists fs_h_dst_step_sumpositive_body_start. fs_h_dst_step_sumpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_start. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_start * S ((S (0)) * fs_v_dst_step_sumpositive) + (0))) /\ ((((exists fs_h_dst_step_sumpositive_body_terminal. fs_h_dst_step_sumpositive_body_terminal + S (dst_positive_sum_step_sum) = S ((S (l)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_terminal. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_terminal * S ((S (l)) * fs_v_dst_step_sumpositive) + (dst_positive_sum_step_sum))) /\ forall fs_i_dst_step_sumpositive_body_steps. (exists fs_lt_dst_step_sumpositive_body_steps_bound. fs_lt_dst_step_sumpositive_body_steps_bound + S fs_i_dst_step_sumpositive_body_steps = l) -> exists fs_a_dst_step_sumpositive_body_steps fs_r_dst_step_sumpositive_body_steps fs_s_dst_step_sumpositive_body_steps. ((((exists fs_h_dst_step_sumpositive_body_steps_summand. fs_h_dst_step_sumpositive_body_steps_summand + S (fs_a_dst_step_sumpositive_body_steps) = S ((S (fs_i_dst_step_sumpositive_body_steps)) * dst_positive_scale_step_sum)) /\ exists fs_q_dst_step_sumpositive_body_steps_summand. dst_positive_code_step_sum = fs_q_dst_step_sumpositive_body_steps_summand * S ((S (fs_i_dst_step_sumpositive_body_steps)) * dst_positive_scale_step_sum) + (fs_a_dst_step_sumpositive_body_steps))) /\ ((((exists fs_h_dst_step_sumpositive_body_steps_partial. fs_h_dst_step_sumpositive_body_steps_partial + S (fs_r_dst_step_sumpositive_body_steps) = S ((S (fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_steps_partial. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_steps_partial * S ((S (fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive) + (fs_r_dst_step_sumpositive_body_steps))) /\ ((((exists fs_h_dst_step_sumpositive_body_steps_successor. fs_h_dst_step_sumpositive_body_steps_successor + S (fs_s_dst_step_sumpositive_body_steps) = S ((S (S fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive)) /\ exists fs_q_dst_step_sumpositive_body_steps_successor. fs_u_dst_step_sumpositive = fs_q_dst_step_sumpositive_body_steps_successor * S ((S (S fs_i_dst_step_sumpositive_body_steps)) * fs_v_dst_step_sumpositive) + (fs_s_dst_step_sumpositive_body_steps))) /\ fs_s_dst_step_sumpositive_body_steps = fs_r_dst_step_sumpositive_body_steps + fs_a_dst_step_sumpositive_body_steps)))))) /\ (((exists fs_u_dst_step_sumnegative fs_v_dst_step_sumnegative. ((((exists fs_h_dst_step_sumnegative_body_start. fs_h_dst_step_sumnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_start. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_start * S ((S (0)) * fs_v_dst_step_sumnegative) + (0))) /\ ((((exists fs_h_dst_step_sumnegative_body_terminal. fs_h_dst_step_sumnegative_body_terminal + S (dst_negative_sum_step_sum) = S ((S (l)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_terminal. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_terminal * S ((S (l)) * fs_v_dst_step_sumnegative) + (dst_negative_sum_step_sum))) /\ forall fs_i_dst_step_sumnegative_body_steps. (exists fs_lt_dst_step_sumnegative_body_steps_bound. fs_lt_dst_step_sumnegative_body_steps_bound + S fs_i_dst_step_sumnegative_body_steps = l) -> exists fs_a_dst_step_sumnegative_body_steps fs_r_dst_step_sumnegative_body_steps fs_s_dst_step_sumnegative_body_steps. ((((exists fs_h_dst_step_sumnegative_body_steps_summand. fs_h_dst_step_sumnegative_body_steps_summand + S (fs_a_dst_step_sumnegative_body_steps) = S ((S (fs_i_dst_step_sumnegative_body_steps)) * dst_negative_scale_step_sum)) /\ exists fs_q_dst_step_sumnegative_body_steps_summand. dst_negative_code_step_sum = fs_q_dst_step_sumnegative_body_steps_summand * S ((S (fs_i_dst_step_sumnegative_body_steps)) * dst_negative_scale_step_sum) + (fs_a_dst_step_sumnegative_body_steps))) /\ ((((exists fs_h_dst_step_sumnegative_body_steps_partial. fs_h_dst_step_sumnegative_body_steps_partial + S (fs_r_dst_step_sumnegative_body_steps) = S ((S (fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_steps_partial. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_steps_partial * S ((S (fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative) + (fs_r_dst_step_sumnegative_body_steps))) /\ ((((exists fs_h_dst_step_sumnegative_body_steps_successor. fs_h_dst_step_sumnegative_body_steps_successor + S (fs_s_dst_step_sumnegative_body_steps) = S ((S (S fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative)) /\ exists fs_q_dst_step_sumnegative_body_steps_successor. fs_u_dst_step_sumnegative = fs_q_dst_step_sumnegative_body_steps_successor * S ((S (S fs_i_dst_step_sumnegative_body_steps)) * fs_v_dst_step_sumnegative) + (fs_s_dst_step_sumnegative_body_steps))) /\ fs_s_dst_step_sumnegative_body_steps = fs_r_dst_step_sumnegative_body_steps + fs_a_dst_step_sumnegative_body_steps)))))) /\ (exists ge_balance_positive_step_sumresult ge_balance_negative_step_sumresult. (((((a) = 2 * (ge_balance_positive_step_sumresult) /\ (ge_balance_negative_step_sumresult) = 0) \/ exists ge_signed_half_step_sumresultdecode. (((a) = 2 * ge_signed_half_step_sumresultdecode + 1 /\ (ge_balance_positive_step_sumresult) = 0) /\ (ge_balance_negative_step_sumresult) = S ge_signed_half_step_sumresultdecode))) /\ ((dst_positive_sum_step_sum) + ge_balance_negative_step_sumresult = (dst_negative_sum_step_sum) + ge_balance_positive_step_sumresult))))))))) -> (exists dst_positive_code_step_entry dst_positive_scale_step_entry dst_negative_code_step_entry dst_negative_scale_step_entry dst_positive_step_entry dst_negative_step_entry. (((F) = (((((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) * S ((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) + ((dst_positive_scale_step_entry) + (dst_positive_scale_step_entry))) + (((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry)))) * S ((((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) * S ((dst_positive_code_step_entry) + (dst_positive_scale_step_entry)) + ((dst_positive_scale_step_entry) + (dst_positive_scale_step_entry))) + (((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry)))) + ((((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry))) + (((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) * S ((dst_negative_code_step_entry) + (dst_negative_scale_step_entry)) + ((dst_negative_scale_step_entry) + (dst_negative_scale_step_entry)))))) /\ (((((exists ff_h_pvs_step_entrypositive. ff_h_pvs_step_entrypositive + S (dst_positive_step_entry) = S ((S (l)) * dst_positive_scale_step_entry)) /\ exists ff_q_pvs_step_entrypositive. dst_positive_code_step_entry = ff_q_pvs_step_entrypositive * S ((S (l)) * dst_positive_scale_step_entry) + (dst_positive_step_entry))) /\ (((((exists ff_h_pvs_step_entrynegative. ff_h_pvs_step_entrynegative + S (dst_negative_step_entry) = S ((S (l)) * dst_negative_scale_step_entry)) /\ exists ff_q_pvs_step_entrynegative. dst_negative_code_step_entry = ff_q_pvs_step_entrynegative * S ((S (l)) * dst_negative_scale_step_entry) + (dst_negative_step_entry))) /\ (exists ge_balance_positive_step_entryvalue ge_balance_negative_step_entryvalue. (((((b) = 2 * (ge_balance_positive_step_entryvalue) /\ (ge_balance_negative_step_entryvalue) = 0) \/ exists ge_signed_half_step_entryvaluedecode. (((b) = 2 * ge_signed_half_step_entryvaluedecode + 1 /\ (ge_balance_positive_step_entryvalue) = 0) /\ (ge_balance_negative_step_entryvalue) = S ge_signed_half_step_entryvaluedecode))) /\ ((dst_positive_step_entry) + ge_balance_negative_step_entryvalue = (dst_negative_step_entry) + ge_balance_positive_step_entryvalue))))))))) -> (exists dsa_ap_step_add dsa_an_step_add dsa_bp_step_add dsa_bn_step_add dsa_cp_step_add dsa_cn_step_add. (((((a) = 2 * (dsa_ap_step_add) /\ (dsa_an_step_add) = 0) \/ exists ge_signed_half_step_addleft. (((a) = 2 * ge_signed_half_step_addleft + 1 /\ (dsa_ap_step_add) = 0) /\ (dsa_an_step_add) = S ge_signed_half_step_addleft))) /\ ((((((b) = 2 * (dsa_bp_step_add) /\ (dsa_bn_step_add) = 0) \/ exists ge_signed_half_step_addright. (((b) = 2 * ge_signed_half_step_addright + 1 /\ (dsa_bp_step_add) = 0) /\ (dsa_bn_step_add) = S ge_signed_half_step_addright))) /\ ((((((c) = 2 * (dsa_cp_step_add) /\ (dsa_cn_step_add) = 0) \/ exists ge_signed_half_step_addoutput. (((c) = 2 * ge_signed_half_step_addoutput + 1 /\ (dsa_cp_step_add) = 0) /\ (dsa_cn_step_add) = S ge_signed_half_step_addoutput))) /\ ((dsa_ap_step_add + dsa_bp_step_add) + dsa_cn_step_add = (dsa_an_step_add + dsa_bn_step_add) + dsa_cp_step_add))))))) -> (exists dst_positive_code_step_result dst_positive_scale_step_result dst_negative_code_step_result dst_negative_scale_step_result dst_positive_sum_step_result dst_negative_sum_step_result. (((F) = (((((dst_positive_code_step_result) + (dst_positive_scale_step_result)) * S ((dst_positive_code_step_result) + (dst_positive_scale_step_result)) + ((dst_positive_scale_step_result) + (dst_positive_scale_step_result))) + (((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result)))) * S ((((dst_positive_code_step_result) + (dst_positive_scale_step_result)) * S ((dst_positive_code_step_result) + (dst_positive_scale_step_result)) + ((dst_positive_scale_step_result) + (dst_positive_scale_step_result))) + (((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result)))) + ((((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result))) + (((dst_negative_code_step_result) + (dst_negative_scale_step_result)) * S ((dst_negative_code_step_result) + (dst_negative_scale_step_result)) + ((dst_negative_scale_step_result) + (dst_negative_scale_step_result)))))) /\ (((exists fs_u_dst_step_resultpositive fs_v_dst_step_resultpositive. ((((exists fs_h_dst_step_resultpositive_body_start. fs_h_dst_step_resultpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_start. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_start * S ((S (0)) * fs_v_dst_step_resultpositive) + (0))) /\ ((((exists fs_h_dst_step_resultpositive_body_terminal. fs_h_dst_step_resultpositive_body_terminal + S (dst_positive_sum_step_result) = S ((S (S l)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_terminal. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_terminal * S ((S (S l)) * fs_v_dst_step_resultpositive) + (dst_positive_sum_step_result))) /\ forall fs_i_dst_step_resultpositive_body_steps. (exists fs_lt_dst_step_resultpositive_body_steps_bound. fs_lt_dst_step_resultpositive_body_steps_bound + S fs_i_dst_step_resultpositive_body_steps = S l) -> exists fs_a_dst_step_resultpositive_body_steps fs_r_dst_step_resultpositive_body_steps fs_s_dst_step_resultpositive_body_steps. ((((exists fs_h_dst_step_resultpositive_body_steps_summand. fs_h_dst_step_resultpositive_body_steps_summand + S (fs_a_dst_step_resultpositive_body_steps) = S ((S (fs_i_dst_step_resultpositive_body_steps)) * dst_positive_scale_step_result)) /\ exists fs_q_dst_step_resultpositive_body_steps_summand. dst_positive_code_step_result = fs_q_dst_step_resultpositive_body_steps_summand * S ((S (fs_i_dst_step_resultpositive_body_steps)) * dst_positive_scale_step_result) + (fs_a_dst_step_resultpositive_body_steps))) /\ ((((exists fs_h_dst_step_resultpositive_body_steps_partial. fs_h_dst_step_resultpositive_body_steps_partial + S (fs_r_dst_step_resultpositive_body_steps) = S ((S (fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_steps_partial. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_steps_partial * S ((S (fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive) + (fs_r_dst_step_resultpositive_body_steps))) /\ ((((exists fs_h_dst_step_resultpositive_body_steps_successor. fs_h_dst_step_resultpositive_body_steps_successor + S (fs_s_dst_step_resultpositive_body_steps) = S ((S (S fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive)) /\ exists fs_q_dst_step_resultpositive_body_steps_successor. fs_u_dst_step_resultpositive = fs_q_dst_step_resultpositive_body_steps_successor * S ((S (S fs_i_dst_step_resultpositive_body_steps)) * fs_v_dst_step_resultpositive) + (fs_s_dst_step_resultpositive_body_steps))) /\ fs_s_dst_step_resultpositive_body_steps = fs_r_dst_step_resultpositive_body_steps + fs_a_dst_step_resultpositive_body_steps)))))) /\ (((exists fs_u_dst_step_resultnegative fs_v_dst_step_resultnegative. ((((exists fs_h_dst_step_resultnegative_body_start. fs_h_dst_step_resultnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_start. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_start * S ((S (0)) * fs_v_dst_step_resultnegative) + (0))) /\ ((((exists fs_h_dst_step_resultnegative_body_terminal. fs_h_dst_step_resultnegative_body_terminal + S (dst_negative_sum_step_result) = S ((S (S l)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_terminal. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_terminal * S ((S (S l)) * fs_v_dst_step_resultnegative) + (dst_negative_sum_step_result))) /\ forall fs_i_dst_step_resultnegative_body_steps. (exists fs_lt_dst_step_resultnegative_body_steps_bound. fs_lt_dst_step_resultnegative_body_steps_bound + S fs_i_dst_step_resultnegative_body_steps = S l) -> exists fs_a_dst_step_resultnegative_body_steps fs_r_dst_step_resultnegative_body_steps fs_s_dst_step_resultnegative_body_steps. ((((exists fs_h_dst_step_resultnegative_body_steps_summand. fs_h_dst_step_resultnegative_body_steps_summand + S (fs_a_dst_step_resultnegative_body_steps) = S ((S (fs_i_dst_step_resultnegative_body_steps)) * dst_negative_scale_step_result)) /\ exists fs_q_dst_step_resultnegative_body_steps_summand. dst_negative_code_step_result = fs_q_dst_step_resultnegative_body_steps_summand * S ((S (fs_i_dst_step_resultnegative_body_steps)) * dst_negative_scale_step_result) + (fs_a_dst_step_resultnegative_body_steps))) /\ ((((exists fs_h_dst_step_resultnegative_body_steps_partial. fs_h_dst_step_resultnegative_body_steps_partial + S (fs_r_dst_step_resultnegative_body_steps) = S ((S (fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_steps_partial. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_steps_partial * S ((S (fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative) + (fs_r_dst_step_resultnegative_body_steps))) /\ ((((exists fs_h_dst_step_resultnegative_body_steps_successor. fs_h_dst_step_resultnegative_body_steps_successor + S (fs_s_dst_step_resultnegative_body_steps) = S ((S (S fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative)) /\ exists fs_q_dst_step_resultnegative_body_steps_successor. fs_u_dst_step_resultnegative = fs_q_dst_step_resultnegative_body_steps_successor * S ((S (S fs_i_dst_step_resultnegative_body_steps)) * fs_v_dst_step_resultnegative) + (fs_s_dst_step_resultnegative_body_steps))) /\ fs_s_dst_step_resultnegative_body_steps = fs_r_dst_step_resultnegative_body_steps + fs_a_dst_step_resultnegative_body_steps)))))) /\ (exists ge_balance_positive_step_resultresult ge_balance_negative_step_resultresult. (((((c) = 2 * (ge_balance_positive_step_resultresult) /\ (ge_balance_negative_step_resultresult) = 0) \/ exists ge_signed_half_step_resultresultdecode. (((c) = 2 * ge_signed_half_step_resultresultdecode + 1 /\ (ge_balance_positive_step_resultresult) = 0) /\ (ge_balance_negative_step_resultresult) = S ge_signed_half_step_resultresultdecode))) /\ ((dst_positive_sum_step_result) + ge_balance_negative_step_resultresult = (dst_negative_sum_step_result) + ge_balance_positive_step_resultresult)))))))))

Complete tactic proof in conservative notation

All 70 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.

Read the argument

Proof checkpoints

70 script commands · 9 reading checkpoints · 1 local claims

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.

Named ingredients (3)

divisor signed table at to components divisor signed sum from components divisor natural sum successor intro

01Fix variables and assumptionsL1–8

Work with arbitrary variables or the premises of the current implication.

L1
intro F
L2
intro l
L3
intro a
L4
intro b
L5
intro c
L6
intro hs
L7
intro he
L8
intro hadd

02Separate the logical casesL9–17

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

L9
cases hs
L10
cases hs_witness
L11
cases hs_witness_witness
L12
cases hs_witness_witness_witness
L13
cases hs_witness_witness_witness_witness
L14
cases hs_witness_witness_witness_witness_witness
L15
cases hs_witness_witness_witness_witness_witness_witness
L16
cases hs_witness_witness_witness_witness_witness_witness_right
L17
cases hs_witness_witness_witness_witness_witness_witness_right_right

03Establish hentryL18–27

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor signed table at to components.

L18
have hentry : ∃ p. ∃ n. BetaAt(x,x1,l,p) ∧ (BetaAt(x2,x3,l,n) ∧ SignedBalance(b,p,n))Definitions: BetaAt(x,x1,l,p)BetaAt(x2,x3,l,n)SignedBalance(b,p,n)Original native command in the exact edition
L19
specialize divisor_signed_table_at_to_components (F)
L20
specialize divisor_signed_table_at_to_components (x)
L21
specialize divisor_signed_table_at_to_components (x1)
L22
specialize divisor_signed_table_at_to_components (x2)
L23
specialize divisor_signed_table_at_to_components (x3)
L24
specialize divisor_signed_table_at_to_components (l)
L25
specialize divisor_signed_table_at_to_components (b)
L26
apply divisor_signed_table_at_to_components
L27
exact hs_witness_witness_witness_witness_witness_witness_left

04Use earlier factsL28–28