PA00F4

eisenstein_transposed_column_decoded_choice

Alpha v34 checked-use theorem · independently closed; not Stable

A decoded entry of any genuine transposed column has the exact Eisenstein cell orientation.

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

Exact expanded PA statement

forall p q h bb bc i z e k j a. (forall etc_row_index_fubini_total_decoded_choice_column. (exists edt_lt_gap_fubini_total_decoded_choice_column_bound. edt_lt_gap_fubini_total_decoded_choice_column_bound + S (etc_row_index_fubini_total_decoded_choice_column) = k) -> exists etc_bit_fubini_total_decoded_choice_column. ((((exists ff_h_etc_fubini_total_decoded_choice_column_decoded. ff_h_etc_fubini_total_decoded_choice_column_decoded + S (etc_bit_fubini_total_decoded_choice_column) = S ((S (etc_row_index_fubini_total_decoded_choice_column)) * e)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_decoded. z = ff_q_etc_fubini_total_decoded_choice_column_decoded * S ((S (etc_row_index_fubini_total_decoded_choice_column)) * e) + (etc_bit_fubini_total_decoded_choice_column))) /\ (exists etc_count_fubini_total_decoded_choice_column_witness etc_row_code_fubini_total_decoded_choice_column_witness etc_row_scale_fubini_total_decoded_choice_column_witness. ((((((exists ff_h_etc_fubini_total_decoded_choice_column_witness_outer_entry. ff_h_etc_fubini_total_decoded_choice_column_witness_outer_entry + S (etc_count_fubini_total_decoded_choice_column_witness) = S ((S (etc_row_index_fubini_total_decoded_choice_column)) * bc)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_outer_entry. bb = ff_q_etc_fubini_total_decoded_choice_column_witness_outer_entry * S ((S (etc_row_index_fubini_total_decoded_choice_column)) * bc) + (etc_count_fubini_total_decoded_choice_column_witness))) /\ (forall eri_column_etc_fubini_total_decoded_choice_column_witness_row. (exists eri_gap_etc_fubini_total_decoded_choice_column_witness_row_bound. eri_gap_etc_fubini_total_decoded_choice_column_witness_row_bound + S (eri_column_etc_fubini_total_decoded_choice_column_witness_row) = h) -> exists eri_bit_etc_fubini_total_decoded_choice_column_witness_row. ((((exists ff_h_eri_etc_fubini_total_decoded_choice_column_witness_row_decoded. ff_h_eri_etc_fubini_total_decoded_choice_column_witness_row_decoded + S (eri_bit_etc_fubini_total_decoded_choice_column_witness_row) = S ((S (eri_column_etc_fubini_total_decoded_choice_column_witness_row)) * etc_row_scale_fubini_total_decoded_choice_column_witness)) /\ exists ff_q_eri_etc_fubini_total_decoded_choice_column_witness_row_decoded. etc_row_code_fubini_total_decoded_choice_column_witness = ff_q_eri_etc_fubini_total_decoded_choice_column_witness_row_decoded * S ((S (eri_column_etc_fubini_total_decoded_choice_column_witness_row)) * etc_row_scale_fubini_total_decoded_choice_column_witness) + (eri_bit_etc_fubini_total_decoded_choice_column_witness_row))) /\ (((eri_bit_etc_fubini_total_decoded_choice_column_witness_row = 0 /\ ((exists eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_left. eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_left + S (p * S etc_row_index_fubini_total_decoded_choice_column) = q * S eri_column_etc_fubini_total_decoded_choice_column_witness_row) /\ ~(exists eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_right. eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_right + S (q * S eri_column_etc_fubini_total_decoded_choice_column_witness_row) = p * S etc_row_index_fubini_total_decoded_choice_column))) \/ (eri_bit_etc_fubini_total_decoded_choice_column_witness_row = 1 /\ ((exists eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_right. eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_right + S (q * S eri_column_etc_fubini_total_decoded_choice_column_witness_row) = p * S etc_row_index_fubini_total_decoded_choice_column) /\ ~(exists eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_left. eri_gap_etc_fubini_total_decoded_choice_column_witness_row_choice_left + S (p * S etc_row_index_fubini_total_decoded_choice_column) = q * S eri_column_etc_fubini_total_decoded_choice_column_witness_row)))))))) /\ (((exists ff_u_etc_fubini_total_decoded_choice_column_witness_count_relation_sum ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum. ((((exists ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_start. ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_start + S (0) = S ((S (0)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_start. ff_u_etc_fubini_total_decoded_choice_column_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_start * S ((S (0)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum) + (0))) /\ ((((exists ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_terminal. ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_terminal + S (etc_count_fubini_total_decoded_choice_column_witness) = S ((S (h)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_terminal. ff_u_etc_fubini_total_decoded_choice_column_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_terminal * S ((S (h)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum) + (etc_count_fubini_total_decoded_choice_column_witness))) /\ forall ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum. (exists ff_lt_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_bound. ff_lt_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_bound + S ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum = h) -> exists ff_a_etc_fubini_total_decoded_choice_column_witness_count_relation_sum ff_r_etc_fubini_total_decoded_choice_column_witness_count_relation_sum ff_s_etc_fubini_total_decoded_choice_column_witness_count_relation_sum. ((((exists ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_summand. ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_summand + S (ff_a_etc_fubini_total_decoded_choice_column_witness_count_relation_sum) = S ((S (ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) * etc_row_scale_fubini_total_decoded_choice_column_witness)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_summand. etc_row_code_fubini_total_decoded_choice_column_witness = ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_summand * S ((S (ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) * etc_row_scale_fubini_total_decoded_choice_column_witness) + (ff_a_etc_fubini_total_decoded_choice_column_witness_count_relation_sum))) /\ ((((exists ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_partial. ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_partial + S (ff_r_etc_fubini_total_decoded_choice_column_witness_count_relation_sum) = S ((S (ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_partial. ff_u_etc_fubini_total_decoded_choice_column_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_partial * S ((S (ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum) + (ff_r_etc_fubini_total_decoded_choice_column_witness_count_relation_sum))) /\ ((((exists ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_successor. ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_successor + S (ff_s_etc_fubini_total_decoded_choice_column_witness_count_relation_sum) = S ((S (S ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_successor. ff_u_etc_fubini_total_decoded_choice_column_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_sum_successor * S ((S (S ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_column_witness_count_relation_sum) + (ff_s_etc_fubini_total_decoded_choice_column_witness_count_relation_sum))) /\ ff_s_etc_fubini_total_decoded_choice_column_witness_count_relation_sum = ff_r_etc_fubini_total_decoded_choice_column_witness_count_relation_sum + ff_a_etc_fubini_total_decoded_choice_column_witness_count_relation_sum)))))) /\ (forall ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_bits. (exists ff_lt_etc_fubini_total_decoded_choice_column_witness_count_relation_bits_bound. ff_lt_etc_fubini_total_decoded_choice_column_witness_count_relation_bits_bound + S ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_bits = h) -> exists ff_bit_etc_fubini_total_decoded_choice_column_witness_count_relation_bits. ((((exists ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_bits_decoded. ff_h_etc_fubini_total_decoded_choice_column_witness_count_relation_bits_decoded + S (ff_bit_etc_fubini_total_decoded_choice_column_witness_count_relation_bits) = S ((S (ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_bits)) * etc_row_scale_fubini_total_decoded_choice_column_witness)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_bits_decoded. etc_row_code_fubini_total_decoded_choice_column_witness = ff_q_etc_fubini_total_decoded_choice_column_witness_count_relation_bits_decoded * S ((S (ff_i_etc_fubini_total_decoded_choice_column_witness_count_relation_bits)) * etc_row_scale_fubini_total_decoded_choice_column_witness) + (ff_bit_etc_fubini_total_decoded_choice_column_witness_count_relation_bits))) /\ (ff_bit_etc_fubini_total_decoded_choice_column_witness_count_relation_bits = 0 \/ ff_bit_etc_fubini_total_decoded_choice_column_witness_count_relation_bits = 1)))))) /\ (((exists ff_h_etc_fubini_total_decoded_choice_column_witness_inner_entry. ff_h_etc_fubini_total_decoded_choice_column_witness_inner_entry + S (etc_bit_fubini_total_decoded_choice_column) = S ((S (i)) * etc_row_scale_fubini_total_decoded_choice_column_witness)) /\ exists ff_q_etc_fubini_total_decoded_choice_column_witness_inner_entry. etc_row_code_fubini_total_decoded_choice_column_witness = ff_q_etc_fubini_total_decoded_choice_column_witness_inner_entry * S ((S (i)) * etc_row_scale_fubini_total_decoded_choice_column_witness) + (etc_bit_fubini_total_decoded_choice_column))))))) -> (exists edt_lt_gap_fubini_total_decoded_choice_fixed_bound. edt_lt_gap_fubini_total_decoded_choice_fixed_bound + S (i) = h) -> (exists edt_lt_gap_fubini_total_row_bound. edt_lt_gap_fubini_total_row_bound + S (j) = k) -> (((exists ff_h_fubini_total_decoded_choice_entry. ff_h_fubini_total_decoded_choice_entry + S (a) = S ((S (j)) * e)) /\ exists ff_q_fubini_total_decoded_choice_entry. z = ff_q_fubini_total_decoded_choice_entry * S ((S (j)) * e) + (a))) -> (((a = 0 /\ ((exists eri_gap_fubini_total_decoded_choice_result_left. eri_gap_fubini_total_decoded_choice_result_left + S (p * S j) = q * S i) /\ ~(exists eri_gap_fubini_total_decoded_choice_result_right. eri_gap_fubini_total_decoded_choice_result_right + S (q * S i) = p * S j))) \/ (a = 1 /\ ((exists eri_gap_fubini_total_decoded_choice_result_right. eri_gap_fubini_total_decoded_choice_result_right + S (q * S i) = p * S j) /\ ~(exists eri_gap_fubini_total_decoded_choice_result_left. eri_gap_fubini_total_decoded_choice_result_left + S (p * S j) = q * S i)))))

Structural proof guide

Generated structural guide

A decoded entry of any genuine transposed column has the exact Eisenstein cell orientation.

Use the direct prerequisites beta_at_unique, eisenstein_row_indicator_decoded_choice as previously established PA formulas.

The proof proceeds by case analysis (8), intermediate claims (3), equality transport (2).

Referenced ingredients

PA002F beta_at_unique PA00EA eisenstein_row_indicator_decoded_choice

Proof neighborhood

Direct dependencies

PA002F beta_at_unique PA00EA eisenstein_row_indicator_decoded_choice

Direct dependents

PA00F6 eisenstein_transposed_column_counts_extensional

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Alpha-v25 checked-use theorem is independently kernel-checked when replayed; it is not a Stable theorem.

Read the argument

Proof checkpoints

52 script commands · 10 reading checkpoints · 3 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.

Named ingredients (2)
beta at uniqueeisenstein row indicator decoded choice

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro q
  3. L3
    intro h
  4. L4
    intro bb
  5. L5
    intro bc
  6. L6
    intro i
  7. L7
    intro z
  8. L8
    intro e
  9. L9
    intro k
  10. L10
    intro j
02Fix variables and assumptionsL11–15

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

  1. L11
    intro a
  2. L12
    intro hcolumn
  3. L13
    intro hi
  4. L14
    intro hj
  5. L15
    intro ha
03Establish hstoredL16–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hcolumn.

  1. L16
    have hstored : ∃ d. BetaAt(z,e,j,d) ∧ (∃ x. ∃ y. ∃ n. BetaAt(bb,bc,j,x) ∧ (∀ m. Lt(m,h) → ∃ k. BetaAt(y,n,m,k) ∧ (k = 0 ∧ (Lt(p · S j,q · S m) ∧ ¬Lt(q · S m,p · S j)) ∨ k = 1 ∧ (Lt(q · S m,p · S j) ∧ ¬Lt(p · S j,q · S m)))) ∧ BitCount(y,n,h,x) ∧ BetaAt(y,n,i,d))Definitions: LtBetaAtBitCount
  2. L17
    specialize hcolumn j
  3. L18
    apply hcolumn
  4. L19
    exact hj
04Separate the logical casesL20–21

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

  1. L20
    cases hstored
  2. L21
    cases hstored_witness
05Establish hdaL22–30

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L22
    have hda : x = a
  2. L23
    specialize beta_at_unique z
  3. L24
    specialize beta_at_unique e
  4. L25
    specialize beta_at_unique j
  5. L26
    specialize beta_at_unique x
  6. L27
    specialize beta_at_unique a
  7. L28
    apply beta_at_unique
  8. L29
    exact hstored_witness_left
  9. L30
    exact ha
06Separate the logical casesL31–36

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

  1. L31
    cases hstored_witness_right
  2. L32
    cases hstored_witness_right_witness
  3. L33
    cases hstored_witness_right_witness_witness
  4. L34
    cases hstored_witness_right_witness_witness_witness
  5. L35
    cases hstored_witness_right_witness_witness_witness_left
  6. L36
    cases hstored_witness_right_witness_witness_witness_left_left
07Establish hchoiceL37–46

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply eisenstein row indicator decoded choice.

  1. L37
    have hchoice : x = 0 ∧ (Lt(p · S j,q · S i) ∧ ¬Lt(q · S i,p · S j)) ∨ x = 1 ∧ (Lt(q · S i,p · S j) ∧ ¬Lt(p · S j,q · S i))Definitions: Lt
  2. L38
    specialize eisenstein_row_indicator_decoded_choice q
  3. L39
    specialize eisenstein_row_indicator_decoded_choice p
  4. L40
    specialize eisenstein_row_indicator_decoded_choice j
  5. L41
    specialize eisenstein_row_indicator_decoded_choice x2
  6. L42
    specialize eisenstein_row_indicator_decoded_choice x3
  7. L43
    specialize eisenstein_row_indicator_decoded_choice h
  8. L44
    specialize eisenstein_row_indicator_decoded_choice i
  9. L45
    specialize eisenstein_row_indicator_decoded_choice x
  10. L46
    apply eisenstein_row_indicator_decoded_choice
08Use earlier factsL47–49

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L47
    exact hstored_witness_right_witness_witness_witness_left_left_right
  2. L48
    exact hi
  3. L49
    exact hstored_witness_right_witness_witness_witness_right
09Calculate and transport equalitiesL50–51

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L50
    rewrite hda at hchoice
  2. L51
    rewrite hda at hchoice
10Use earlier factsL52–52

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L52
    exact hchoice

Library-wide reading audit

Original exact command ledger · 52 lines
  1. 0001intro p
  2. 0002intro q
  3. 0003intro h
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro i
  7. 0007intro z
  8. 0008intro e
  9. 0009intro k
  10. 0010intro j
  11. 0011intro a
  12. 0012intro hcolumn
  13. 0013intro hi
  14. 0014intro hj
  15. 0015intro ha
  16. 0016have hstored : exists d. ((((exists ff_h_fubini_total_decoded_choice_stored_entry. ff_h_fubini_total_decoded_choice_stored_entry + S (d) = S ((S (j)) * e)) /\ exists ff_q_fubini_total_decoded_choice_stored_entry. z = ff_q_fubini_total_decoded_choice_stored_entry * S ((S (j)) * e) + (d))) /\ (exists etc_count_fubini_total_decoded_choice_stored_witness etc_row_code_fubini_total_decoded_choice_stored_witness etc_row_scale_fubini_total_decoded_choice_stored_witness. ((((((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_outer_entry. ff_h_etc_fubini_total_decoded_choice_stored_witness_outer_entry + S (etc_count_fubini_total_decoded_choice_stored_witness) = S ((S (j)) * bc)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_outer_entry. bb = ff_q_etc_fubini_total_decoded_choice_stored_witness_outer_entry * S ((S (j)) * bc) + (etc_count_fubini_total_decoded_choice_stored_witness))) /\ (forall eri_column_etc_fubini_total_decoded_choice_stored_witness_row. (exists eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_bound. eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_bound + S (eri_column_etc_fubini_total_decoded_choice_stored_witness_row) = h) -> exists eri_bit_etc_fubini_total_decoded_choice_stored_witness_row. ((((exists ff_h_eri_etc_fubini_total_decoded_choice_stored_witness_row_decoded. ff_h_eri_etc_fubini_total_decoded_choice_stored_witness_row_decoded + S (eri_bit_etc_fubini_total_decoded_choice_stored_witness_row) = S ((S (eri_column_etc_fubini_total_decoded_choice_stored_witness_row)) * etc_row_scale_fubini_total_decoded_choice_stored_witness)) /\ exists ff_q_eri_etc_fubini_total_decoded_choice_stored_witness_row_decoded. etc_row_code_fubini_total_decoded_choice_stored_witness = ff_q_eri_etc_fubini_total_decoded_choice_stored_witness_row_decoded * S ((S (eri_column_etc_fubini_total_decoded_choice_stored_witness_row)) * etc_row_scale_fubini_total_decoded_choice_stored_witness) + (eri_bit_etc_fubini_total_decoded_choice_stored_witness_row))) /\ (((eri_bit_etc_fubini_total_decoded_choice_stored_witness_row = 0 /\ ((exists eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_left. eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_left + S (p * S j) = q * S eri_column_etc_fubini_total_decoded_choice_stored_witness_row) /\ ~(exists eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_right. eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_right + S (q * S eri_column_etc_fubini_total_decoded_choice_stored_witness_row) = p * S j))) \/ (eri_bit_etc_fubini_total_decoded_choice_stored_witness_row = 1 /\ ((exists eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_right. eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_right + S (q * S eri_column_etc_fubini_total_decoded_choice_stored_witness_row) = p * S j) /\ ~(exists eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_left. eri_gap_etc_fubini_total_decoded_choice_stored_witness_row_choice_left + S (p * S j) = q * S eri_column_etc_fubini_total_decoded_choice_stored_witness_row)))))))) /\ (((exists ff_u_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum. ((((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_start. ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_start + S (0) = S ((S (0)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_start. ff_u_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_start * S ((S (0)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum) + (0))) /\ ((((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_terminal. ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_terminal + S (etc_count_fubini_total_decoded_choice_stored_witness) = S ((S (h)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_terminal. ff_u_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_terminal * S ((S (h)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum) + (etc_count_fubini_total_decoded_choice_stored_witness))) /\ forall ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum. (exists ff_lt_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_bound. ff_lt_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_bound + S ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum = h) -> exists ff_a_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum ff_r_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum ff_s_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum. ((((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_summand. ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_summand + S (ff_a_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum) = S ((S (ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) * etc_row_scale_fubini_total_decoded_choice_stored_witness)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_summand. etc_row_code_fubini_total_decoded_choice_stored_witness = ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_summand * S ((S (ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) * etc_row_scale_fubini_total_decoded_choice_stored_witness) + (ff_a_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum))) /\ ((((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_partial. ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_partial + S (ff_r_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum) = S ((S (ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_partial. ff_u_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_partial * S ((S (ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum) + (ff_r_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum))) /\ ((((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_successor. ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_successor + S (ff_s_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum) = S ((S (S ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_successor. ff_u_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum = ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum_successor * S ((S (S ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)) * ff_v_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum) + (ff_s_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum))) /\ ff_s_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum = ff_r_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum + ff_a_etc_fubini_total_decoded_choice_stored_witness_count_relation_sum)))))) /\ (forall ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits. (exists ff_lt_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits_bound. ff_lt_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits_bound + S ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits = h) -> exists ff_bit_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits. ((((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits_decoded. ff_h_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits_decoded + S (ff_bit_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits) = S ((S (ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits)) * etc_row_scale_fubini_total_decoded_choice_stored_witness)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits_decoded. etc_row_code_fubini_total_decoded_choice_stored_witness = ff_q_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits_decoded * S ((S (ff_i_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits)) * etc_row_scale_fubini_total_decoded_choice_stored_witness) + (ff_bit_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits))) /\ (ff_bit_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits = 0 \/ ff_bit_etc_fubini_total_decoded_choice_stored_witness_count_relation_bits = 1)))))) /\ (((exists ff_h_etc_fubini_total_decoded_choice_stored_witness_inner_entry. ff_h_etc_fubini_total_decoded_choice_stored_witness_inner_entry + S (d) = S ((S (i)) * etc_row_scale_fubini_total_decoded_choice_stored_witness)) /\ exists ff_q_etc_fubini_total_decoded_choice_stored_witness_inner_entry. etc_row_code_fubini_total_decoded_choice_stored_witness = ff_q_etc_fubini_total_decoded_choice_stored_witness_inner_entry * S ((S (i)) * etc_row_scale_fubini_total_decoded_choice_stored_witness) + (d))))))
  17. 0017specialize hcolumn j
  18. 0018apply hcolumn
  19. 0019exact hj
  20. 0020cases hstored
  21. 0021cases hstored_witness
  22. 0022have hda : x = a
  23. 0023specialize beta_at_unique z
  24. 0024specialize beta_at_unique e
  25. 0025specialize beta_at_unique j
  26. 0026specialize beta_at_unique x
  27. 0027specialize beta_at_unique a
  28. 0028apply beta_at_unique
  29. 0029exact hstored_witness_left
  30. 0030exact ha
  31. 0031cases hstored_witness_right
  32. 0032cases hstored_witness_right_witness
  33. 0033cases hstored_witness_right_witness_witness
  34. 0034cases hstored_witness_right_witness_witness_witness
  35. 0035cases hstored_witness_right_witness_witness_witness_left
  36. 0036cases hstored_witness_right_witness_witness_witness_left_left
  37. 0037have hchoice : ((x = 0 /\ ((exists eri_gap_fubini_total_decoded_choice_stored_choice_left. eri_gap_fubini_total_decoded_choice_stored_choice_left + S (p * S j) = q * S i) /\ ~(exists eri_gap_fubini_total_decoded_choice_stored_choice_right. eri_gap_fubini_total_decoded_choice_stored_choice_right + S (q * S i) = p * S j))) \/ (x = 1 /\ ((exists eri_gap_fubini_total_decoded_choice_stored_choice_right. eri_gap_fubini_total_decoded_choice_stored_choice_right + S (q * S i) = p * S j) /\ ~(exists eri_gap_fubini_total_decoded_choice_stored_choice_left. eri_gap_fubini_total_decoded_choice_stored_choice_left + S (p * S j) = q * S i))))
  38. 0038specialize eisenstein_row_indicator_decoded_choice q
  39. 0039specialize eisenstein_row_indicator_decoded_choice p
  40. 0040specialize eisenstein_row_indicator_decoded_choice j
  41. 0041specialize eisenstein_row_indicator_decoded_choice x2
  42. 0042specialize eisenstein_row_indicator_decoded_choice x3
  43. 0043specialize eisenstein_row_indicator_decoded_choice h
  44. 0044specialize eisenstein_row_indicator_decoded_choice i
  45. 0045specialize eisenstein_row_indicator_decoded_choice x
  46. 0046apply eisenstein_row_indicator_decoded_choice
  47. 0047exact hstored_witness_right_witness_witness_witness_left_left_right
  48. 0048exact hi
  49. 0049exact hstored_witness_right_witness_witness_witness_right
  50. 0050rewrite hda at hchoice
  51. 0051rewrite hda at hchoice
  52. 0052exact hchoice