BT00T7 · Bertrand theorem

beta_pascal_table_prefix_exists

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Every finite width and height has a nested beta Pascal table.

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.

Statement with defined notation

∀ w. ∀ r. ∃ bb. ∃ bc. ∃ sb. ∃ sc. ∀ x. Lt(x,r) → ∃ y. ∃ z. BetaAt(bb,bc,x,y) ∧ (BetaAt(sb,sc,x,z) ∧ (x = 0 ∧ (∀ n. Lt(n,w) → ∃ m. BetaAt(y,z,n,m) ∧ (n = 0 ∧ m = 1 ∨ (∃ k. n = S k ∧ m = 0))) ∨ (∃ n. ∃ m. ∃ k. x = S n ∧ (BetaAt(bb,bc,n,m) ∧ (BetaAt(sb,sc,n,k) ∧ (∀ i. Lt(i,w) → ∃ j. BetaAt(y,z,i,j) ∧ (i = 0 ∧ j = 1 ∨ (∃ u. ∃ v. ∃ x0. i = S u ∧ (BetaAt(m,k,u,v) ∧ (BetaAt(m,k,S u,x0) ∧ j = v + x0))))))))))

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0002 Lt PD0013 BetaAt

11 occurrences

In local proof propositions

PD0002 Lt PD0013 BetaAt

22 occurrences

Exact expanded native-PA statement
forall w r. exists bb bc sb sc. (forall bcf_row_index_bptpx_result. (exists bcf_lt_gap_bptpx_result_row_bound. bcf_lt_gap_bptpx_result_row_bound + S (bcf_row_index_bptpx_result) = r) -> exists bcf_row_code_bptpx_result bcf_row_scale_bptpx_result. ((((exists bcf_height_bptpx_result_decoded_row_code. bcf_height_bptpx_result_decoded_row_code + S (bcf_row_code_bptpx_result) = S ((S (bcf_row_index_bptpx_result)) * bc)) /\ exists bcf_quotient_bptpx_result_decoded_row_code. bb = bcf_quotient_bptpx_result_decoded_row_code * S ((S (bcf_row_index_bptpx_result)) * bc) + (bcf_row_code_bptpx_result))) /\ ((((exists bcf_height_bptpx_result_decoded_row_scale. bcf_height_bptpx_result_decoded_row_scale + S (bcf_row_scale_bptpx_result) = S ((S (bcf_row_index_bptpx_result)) * sc)) /\ exists bcf_quotient_bptpx_result_decoded_row_scale. sb = bcf_quotient_bptpx_result_decoded_row_scale * S ((S (bcf_row_index_bptpx_result)) * sc) + (bcf_row_scale_bptpx_result))) /\ ((bcf_row_index_bptpx_result = 0 /\ (forall bcf_index_bptpx_result_zero_row. (exists bcf_lt_gap_bptpx_result_zero_row_bound. bcf_lt_gap_bptpx_result_zero_row_bound + S (bcf_index_bptpx_result_zero_row) = w) -> exists bcf_value_bptpx_result_zero_row. ((((exists bcf_height_bptpx_result_zero_row_entry. bcf_height_bptpx_result_zero_row_entry + S (bcf_value_bptpx_result_zero_row) = S ((S (bcf_index_bptpx_result_zero_row)) * bcf_row_scale_bptpx_result)) /\ exists bcf_quotient_bptpx_result_zero_row_entry. bcf_row_code_bptpx_result = bcf_quotient_bptpx_result_zero_row_entry * S ((S (bcf_index_bptpx_result_zero_row)) * bcf_row_scale_bptpx_result) + (bcf_value_bptpx_result_zero_row))) /\ ((bcf_index_bptpx_result_zero_row = 0 /\ bcf_value_bptpx_result_zero_row = 1) \/ exists bcf_predecessor_bptpx_result_zero_row. bcf_index_bptpx_result_zero_row = S bcf_predecessor_bptpx_result_zero_row /\ bcf_value_bptpx_result_zero_row = 0)))) \/ exists bcf_predecessor_bptpx_result bcf_previous_code_bptpx_result bcf_previous_scale_bptpx_result. bcf_row_index_bptpx_result = S bcf_predecessor_bptpx_result /\ ((((exists bcf_height_bptpx_result_decoded_previous_code. bcf_height_bptpx_result_decoded_previous_code + S (bcf_previous_code_bptpx_result) = S ((S (bcf_predecessor_bptpx_result)) * bc)) /\ exists bcf_quotient_bptpx_result_decoded_previous_code. bb = bcf_quotient_bptpx_result_decoded_previous_code * S ((S (bcf_predecessor_bptpx_result)) * bc) + (bcf_previous_code_bptpx_result))) /\ ((((exists bcf_height_bptpx_result_decoded_previous_scale. bcf_height_bptpx_result_decoded_previous_scale + S (bcf_previous_scale_bptpx_result) = S ((S (bcf_predecessor_bptpx_result)) * sc)) /\ exists bcf_quotient_bptpx_result_decoded_previous_scale. sb = bcf_quotient_bptpx_result_decoded_previous_scale * S ((S (bcf_predecessor_bptpx_result)) * sc) + (bcf_previous_scale_bptpx_result))) /\ (forall bcf_index_bptpx_result_row_step. (exists bcf_lt_gap_bptpx_result_row_step_bound. bcf_lt_gap_bptpx_result_row_step_bound + S (bcf_index_bptpx_result_row_step) = w) -> exists bcf_value_bptpx_result_row_step. ((((exists bcf_height_bptpx_result_row_step_entry. bcf_height_bptpx_result_row_step_entry + S (bcf_value_bptpx_result_row_step) = S ((S (bcf_index_bptpx_result_row_step)) * bcf_row_scale_bptpx_result)) /\ exists bcf_quotient_bptpx_result_row_step_entry. bcf_row_code_bptpx_result = bcf_quotient_bptpx_result_row_step_entry * S ((S (bcf_index_bptpx_result_row_step)) * bcf_row_scale_bptpx_result) + (bcf_value_bptpx_result_row_step))) /\ ((bcf_index_bptpx_result_row_step = 0 /\ bcf_value_bptpx_result_row_step = 1) \/ exists bcf_predecessor_bptpx_result_row_step bcf_left_bptpx_result_row_step bcf_right_bptpx_result_row_step. bcf_index_bptpx_result_row_step = S bcf_predecessor_bptpx_result_row_step /\ ((((exists bcf_height_bptpx_result_row_step_previous_left. bcf_height_bptpx_result_row_step_previous_left + S (bcf_left_bptpx_result_row_step) = S ((S (bcf_predecessor_bptpx_result_row_step)) * bcf_previous_scale_bptpx_result)) /\ exists bcf_quotient_bptpx_result_row_step_previous_left. bcf_previous_code_bptpx_result = bcf_quotient_bptpx_result_row_step_previous_left * S ((S (bcf_predecessor_bptpx_result_row_step)) * bcf_previous_scale_bptpx_result) + (bcf_left_bptpx_result_row_step))) /\ ((((exists bcf_height_bptpx_result_row_step_previous_right. bcf_height_bptpx_result_row_step_previous_right + S (bcf_right_bptpx_result_row_step) = S ((S (S (bcf_predecessor_bptpx_result_row_step))) * bcf_previous_scale_bptpx_result)) /\ exists bcf_quotient_bptpx_result_row_step_previous_right. bcf_previous_code_bptpx_result = bcf_quotient_bptpx_result_row_step_previous_right * S ((S (S (bcf_predecessor_bptpx_result_row_step))) * bcf_previous_scale_bptpx_result) + (bcf_right_bptpx_result_row_step))) /\ bcf_value_bptpx_result_row_step = bcf_left_bptpx_result_row_step + bcf_right_bptpx_result_row_step)))))))))))

Proof neighborhood

Direct theorem prerequisites

BT000L add_eq_zero_right BT000C succ_ne_zero BT00T6 beta_pascal_table_prefix_extend

Direct theorem dependents

BT00T8 choose_exists

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

34 script commands · 9 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.

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)
add eq zero rightsucc ne zerobeta pascal table prefix extend
01Fix variables and assumptionsL1–1

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

  1. L1
    intro w
02Induction on rL2–2

Split the argument into the base and successor obligations. The induction hypothesis is available only in the successor branch.

  1. L2
    induction r
03Construct an explicit witnessL3–6

Supply the displayed value, then prove that it has the required property.

  1. L3
    exists 0
  2. L4
    exists 0
  3. L5
    exists 0
  4. L6
    exists 0
04Fix variables and assumptionsL7–8

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

  1. L7
    intro i
  2. L8
    intro hi
05Separate the logical casesL9–10

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

  1. L9
    exfalso
  2. L10
    cases hi
06Establish hsiL11–18

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add eq zero right.

  1. L11
    have hsi : S i = 0
  2. L12
    specialize add_eq_zero_right x
  3. L13
    specialize add_eq_zero_right (S i)
  4. L14
    apply add_eq_zero_right
  5. L15
    exact hi_witness
  6. L16
    specialize succ_ne_zero i
  7. L17
    apply succ_ne_zero
  8. L18
    exact hsi
07Establish hpreviousL19–20

Establish this local claim before using it. It is not an additional assumption.

  1. L19
    have hprevious : ∃ bb. ∃ bc. ∃ sb. ∃ sc. ∀ x. Lt(x,r) → ∃ y. ∃ z. BetaAt(bb,bc,x,y) ∧ (BetaAt(sb,sc,x,z) ∧ (x = 0 ∧ (∀ n. Lt(n,w) → ∃ m. BetaAt(y,z,n,m) ∧ (n = 0 ∧ m = 1 ∨ (∃ k. n = S k ∧ m = 0))) ∨ (∃ n. ∃ m. ∃ k. x = S n ∧ (BetaAt(bb,bc,n,m) ∧ (BetaAt(sb,sc,n,k) ∧ (∀ i. Lt(i,w) → ∃ j. BetaAt(y,z,i,j) ∧ (i = 0 ∧ j = 1 ∨ (∃ u. ∃ v. ∃ x0. i = S u ∧ (BetaAt(m,k,u,v) ∧ (BetaAt(m,k,S u,x0) ∧ j = v + x0))))))))))Definitions: Lt(x,r)BetaAt(bb,bc,x,y)BetaAt(sb,sc,x,z)Lt(n,w)BetaAt(y,z,n,m)BetaAt(bb,bc,n,m)BetaAt(sb,sc,n,k)Lt(i,w)BetaAt(y,z,i,j)BetaAt(m,k,u,v)BetaAt(m,k,S u,x0)Original native command in the exact edition
  2. L20
    exact IH
08Separate the logical casesL21–24

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

  1. L21
    cases hprevious
  2. L22
    cases hprevious_witness
  3. L23
    cases hprevious_witness_witness
  4. L24
    cases hprevious_witness_witness_witness
09Establish hnextL25–34

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta pascal table prefix extend.

  1. L25
    have hnext : ∃ bb. ∃ bc. ∃ sb. ∃ sc. ∀ x. Lt(x,S r) → ∃ y. ∃ z. BetaAt(bb,bc,x,y) ∧ (BetaAt(sb,sc,x,z) ∧ (x = 0 ∧ (∀ n. Lt(n,w) → ∃ m. BetaAt(y,z,n,m) ∧ (n = 0 ∧ m = 1 ∨ (∃ k. n = S k ∧ m = 0))) ∨ (∃ n. ∃ m. ∃ k. x = S n ∧ (BetaAt(bb,bc,n,m) ∧ (BetaAt(sb,sc,n,k) ∧ (∀ i. Lt(i,w) → ∃ j. BetaAt(y,z,i,j) ∧ (i = 0 ∧ j = 1 ∨ (∃ u. ∃ v. ∃ x0. i = S u ∧ (BetaAt(m,k,u,v) ∧ (BetaAt(m,k,S u,x0) ∧ j = v + x0))))))))))Definitions: Lt(x,S r)BetaAt(bb,bc,x,y)BetaAt(sb,sc,x,z)Lt(n,w)BetaAt(y,z,n,m)BetaAt(bb,bc,n,m)BetaAt(sb,sc,n,k)Lt(i,w)BetaAt(y,z,i,j)BetaAt(m,k,u,v)BetaAt(m,k,S u,x0)Original native command in the exact edition
  2. L26
    specialize beta_pascal_table_prefix_extend x
  3. L27
    specialize beta_pascal_table_prefix_extend x1
  4. L28
    specialize beta_pascal_table_prefix_extend x2
  5. L29
    specialize beta_pascal_table_prefix_extend x3
  6. L30
    specialize beta_pascal_table_prefix_extend w
  7. L31
    specialize beta_pascal_table_prefix_extend r
  8. L32
    apply beta_pascal_table_prefix_extend
  9. L33
    exact hprevious_witness_witness_witness_witness
  10. L34
    exact hnext

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro w
  2. 0002induction r
  3. 0003exists 0
  4. 0004exists 0
  5. 0005exists 0
  6. 0006exists 0
  7. 0007intro i
  8. 0008intro hi
  9. 0009exfalso
  10. 0010cases hi
  11. 0011have hsi : S i = 0
  12. 0012specialize add_eq_zero_right x
  13. 0013specialize add_eq_zero_right (S i)
  14. 0014apply add_eq_zero_right
  15. 0015exact hi_witness
  16. 0016specialize succ_ne_zero i
  17. 0017apply succ_ne_zero
  18. 0018exact hsi
  19. 0019have hprevious : ∃ bb. ∃ bc. ∃ sb. ∃ sc. ∀ x. Lt(x,r) → ∃ y. ∃ z. BetaAt(bb,bc,x,y) ∧ (BetaAt(sb,sc,x,z) ∧ (x = 0 ∧ (∀ n. Lt(n,w) → ∃ m. BetaAt(y,z,n,m) ∧ (n = 0 ∧ m = 1 ∨ (∃ k. n = S k ∧ m = 0))) ∨ (∃ n. ∃ m. ∃ k. x = S n ∧ (BetaAt(bb,bc,n,m) ∧ (BetaAt(sb,sc,n,k) ∧ (∀ i. Lt(i,w) → ∃ j. BetaAt(y,z,i,j) ∧ (i = 0 ∧ j = 1 ∨ (∃ u. ∃ v. ∃ x0. i = S u ∧ (BetaAt(m,k,u,v) ∧ (BetaAt(m,k,S u,x0) ∧ j = v + x0))))))))))
    Exact native replay linehave hprevious : exists bb bc sb sc. (forall bcf_row_index_bptpx_previous. (exists bcf_lt_gap_bptpx_previous_row_bound. bcf_lt_gap_bptpx_previous_row_bound + S (bcf_row_index_bptpx_previous) = r) -> exists bcf_row_code_bptpx_previous bcf_row_scale_bptpx_previous. ((((exists bcf_height_bptpx_previous_decoded_row_code. bcf_height_bptpx_previous_decoded_row_code + S (bcf_row_code_bptpx_previous) = S ((S (bcf_row_index_bptpx_previous)) * bc)) /\ exists bcf_quotient_bptpx_previous_decoded_row_code. bb = bcf_quotient_bptpx_previous_decoded_row_code * S ((S (bcf_row_index_bptpx_previous)) * bc) + (bcf_row_code_bptpx_previous))) /\ ((((exists bcf_height_bptpx_previous_decoded_row_scale. bcf_height_bptpx_previous_decoded_row_scale + S (bcf_row_scale_bptpx_previous) = S ((S (bcf_row_index_bptpx_previous)) * sc)) /\ exists bcf_quotient_bptpx_previous_decoded_row_scale. sb = bcf_quotient_bptpx_previous_decoded_row_scale * S ((S (bcf_row_index_bptpx_previous)) * sc) + (bcf_row_scale_bptpx_previous))) /\ ((bcf_row_index_bptpx_previous = 0 /\ (forall bcf_index_bptpx_previous_zero_row. (exists bcf_lt_gap_bptpx_previous_zero_row_bound. bcf_lt_gap_bptpx_previous_zero_row_bound + S (bcf_index_bptpx_previous_zero_row) = w) -> exists bcf_value_bptpx_previous_zero_row. ((((exists bcf_height_bptpx_previous_zero_row_entry. bcf_height_bptpx_previous_zero_row_entry + S (bcf_value_bptpx_previous_zero_row) = S ((S (bcf_index_bptpx_previous_zero_row)) * bcf_row_scale_bptpx_previous)) /\ exists bcf_quotient_bptpx_previous_zero_row_entry. bcf_row_code_bptpx_previous = bcf_quotient_bptpx_previous_zero_row_entry * S ((S (bcf_index_bptpx_previous_zero_row)) * bcf_row_scale_bptpx_previous) + (bcf_value_bptpx_previous_zero_row))) /\ ((bcf_index_bptpx_previous_zero_row = 0 /\ bcf_value_bptpx_previous_zero_row = 1) \/ exists bcf_predecessor_bptpx_previous_zero_row. bcf_index_bptpx_previous_zero_row = S bcf_predecessor_bptpx_previous_zero_row /\ bcf_value_bptpx_previous_zero_row = 0)))) \/ exists bcf_predecessor_bptpx_previous bcf_previous_code_bptpx_previous bcf_previous_scale_bptpx_previous. bcf_row_index_bptpx_previous = S bcf_predecessor_bptpx_previous /\ ((((exists bcf_height_bptpx_previous_decoded_previous_code. bcf_height_bptpx_previous_decoded_previous_code + S (bcf_previous_code_bptpx_previous) = S ((S (bcf_predecessor_bptpx_previous)) * bc)) /\ exists bcf_quotient_bptpx_previous_decoded_previous_code. bb = bcf_quotient_bptpx_previous_decoded_previous_code * S ((S (bcf_predecessor_bptpx_previous)) * bc) + (bcf_previous_code_bptpx_previous))) /\ ((((exists bcf_height_bptpx_previous_decoded_previous_scale. bcf_height_bptpx_previous_decoded_previous_scale + S (bcf_previous_scale_bptpx_previous) = S ((S (bcf_predecessor_bptpx_previous)) * sc)) /\ exists bcf_quotient_bptpx_previous_decoded_previous_scale. sb = bcf_quotient_bptpx_previous_decoded_previous_scale * S ((S (bcf_predecessor_bptpx_previous)) * sc) + (bcf_previous_scale_bptpx_previous))) /\ (forall bcf_index_bptpx_previous_row_step. (exists bcf_lt_gap_bptpx_previous_row_step_bound. bcf_lt_gap_bptpx_previous_row_step_bound + S (bcf_index_bptpx_previous_row_step) = w) -> exists bcf_value_bptpx_previous_row_step. ((((exists bcf_height_bptpx_previous_row_step_entry. bcf_height_bptpx_previous_row_step_entry + S (bcf_value_bptpx_previous_row_step) = S ((S (bcf_index_bptpx_previous_row_step)) * bcf_row_scale_bptpx_previous)) /\ exists bcf_quotient_bptpx_previous_row_step_entry. bcf_row_code_bptpx_previous = bcf_quotient_bptpx_previous_row_step_entry * S ((S (bcf_index_bptpx_previous_row_step)) * bcf_row_scale_bptpx_previous) + (bcf_value_bptpx_previous_row_step))) /\ ((bcf_index_bptpx_previous_row_step = 0 /\ bcf_value_bptpx_previous_row_step = 1) \/ exists bcf_predecessor_bptpx_previous_row_step bcf_left_bptpx_previous_row_step bcf_right_bptpx_previous_row_step. bcf_index_bptpx_previous_row_step = S bcf_predecessor_bptpx_previous_row_step /\ ((((exists bcf_height_bptpx_previous_row_step_previous_left. bcf_height_bptpx_previous_row_step_previous_left + S (bcf_left_bptpx_previous_row_step) = S ((S (bcf_predecessor_bptpx_previous_row_step)) * bcf_previous_scale_bptpx_previous)) /\ exists bcf_quotient_bptpx_previous_row_step_previous_left. bcf_previous_code_bptpx_previous = bcf_quotient_bptpx_previous_row_step_previous_left * S ((S (bcf_predecessor_bptpx_previous_row_step)) * bcf_previous_scale_bptpx_previous) + (bcf_left_bptpx_previous_row_step))) /\ ((((exists bcf_height_bptpx_previous_row_step_previous_right. bcf_height_bptpx_previous_row_step_previous_right + S (bcf_right_bptpx_previous_row_step) = S ((S (S (bcf_predecessor_bptpx_previous_row_step))) * bcf_previous_scale_bptpx_previous)) /\ exists bcf_quotient_bptpx_previous_row_step_previous_right. bcf_previous_code_bptpx_previous = bcf_quotient_bptpx_previous_row_step_previous_right * S ((S (S (bcf_predecessor_bptpx_previous_row_step))) * bcf_previous_scale_bptpx_previous) + (bcf_right_bptpx_previous_row_step))) /\ bcf_value_bptpx_previous_row_step = bcf_left_bptpx_previous_row_step + bcf_right_bptpx_previous_row_step)))))))))))
  20. 0020exact IH
  21. 0021cases hprevious
  22. 0022cases hprevious_witness
  23. 0023cases hprevious_witness_witness
  24. 0024cases hprevious_witness_witness_witness
  25. 0025have hnext : ∃ bb. ∃ bc. ∃ sb. ∃ sc. ∀ x. Lt(x,S r) → ∃ y. ∃ z. BetaAt(bb,bc,x,y) ∧ (BetaAt(sb,sc,x,z) ∧ (x = 0 ∧ (∀ n. Lt(n,w) → ∃ m. BetaAt(y,z,n,m) ∧ (n = 0 ∧ m = 1 ∨ (∃ k. n = S k ∧ m = 0))) ∨ (∃ n. ∃ m. ∃ k. x = S n ∧ (BetaAt(bb,bc,n,m) ∧ (BetaAt(sb,sc,n,k) ∧ (∀ i. Lt(i,w) → ∃ j. BetaAt(y,z,i,j) ∧ (i = 0 ∧ j = 1 ∨ (∃ u. ∃ v. ∃ x0. i = S u ∧ (BetaAt(m,k,u,v) ∧ (BetaAt(m,k,S u,x0) ∧ j = v + x0))))))))))
    Exact native replay linehave hnext : exists bb bc sb sc. (forall bcf_row_index_bptpx_successor. (exists bcf_lt_gap_bptpx_successor_row_bound. bcf_lt_gap_bptpx_successor_row_bound + S (bcf_row_index_bptpx_successor) = S (r)) -> exists bcf_row_code_bptpx_successor bcf_row_scale_bptpx_successor. ((((exists bcf_height_bptpx_successor_decoded_row_code. bcf_height_bptpx_successor_decoded_row_code + S (bcf_row_code_bptpx_successor) = S ((S (bcf_row_index_bptpx_successor)) * bc)) /\ exists bcf_quotient_bptpx_successor_decoded_row_code. bb = bcf_quotient_bptpx_successor_decoded_row_code * S ((S (bcf_row_index_bptpx_successor)) * bc) + (bcf_row_code_bptpx_successor))) /\ ((((exists bcf_height_bptpx_successor_decoded_row_scale. bcf_height_bptpx_successor_decoded_row_scale + S (bcf_row_scale_bptpx_successor) = S ((S (bcf_row_index_bptpx_successor)) * sc)) /\ exists bcf_quotient_bptpx_successor_decoded_row_scale. sb = bcf_quotient_bptpx_successor_decoded_row_scale * S ((S (bcf_row_index_bptpx_successor)) * sc) + (bcf_row_scale_bptpx_successor))) /\ ((bcf_row_index_bptpx_successor = 0 /\ (forall bcf_index_bptpx_successor_zero_row. (exists bcf_lt_gap_bptpx_successor_zero_row_bound. bcf_lt_gap_bptpx_successor_zero_row_bound + S (bcf_index_bptpx_successor_zero_row) = w) -> exists bcf_value_bptpx_successor_zero_row. ((((exists bcf_height_bptpx_successor_zero_row_entry. bcf_height_bptpx_successor_zero_row_entry + S (bcf_value_bptpx_successor_zero_row) = S ((S (bcf_index_bptpx_successor_zero_row)) * bcf_row_scale_bptpx_successor)) /\ exists bcf_quotient_bptpx_successor_zero_row_entry. bcf_row_code_bptpx_successor = bcf_quotient_bptpx_successor_zero_row_entry * S ((S (bcf_index_bptpx_successor_zero_row)) * bcf_row_scale_bptpx_successor) + (bcf_value_bptpx_successor_zero_row))) /\ ((bcf_index_bptpx_successor_zero_row = 0 /\ bcf_value_bptpx_successor_zero_row = 1) \/ exists bcf_predecessor_bptpx_successor_zero_row. bcf_index_bptpx_successor_zero_row = S bcf_predecessor_bptpx_successor_zero_row /\ bcf_value_bptpx_successor_zero_row = 0)))) \/ exists bcf_predecessor_bptpx_successor bcf_previous_code_bptpx_successor bcf_previous_scale_bptpx_successor. bcf_row_index_bptpx_successor = S bcf_predecessor_bptpx_successor /\ ((((exists bcf_height_bptpx_successor_decoded_previous_code. bcf_height_bptpx_successor_decoded_previous_code + S (bcf_previous_code_bptpx_successor) = S ((S (bcf_predecessor_bptpx_successor)) * bc)) /\ exists bcf_quotient_bptpx_successor_decoded_previous_code. bb = bcf_quotient_bptpx_successor_decoded_previous_code * S ((S (bcf_predecessor_bptpx_successor)) * bc) + (bcf_previous_code_bptpx_successor))) /\ ((((exists bcf_height_bptpx_successor_decoded_previous_scale. bcf_height_bptpx_successor_decoded_previous_scale + S (bcf_previous_scale_bptpx_successor) = S ((S (bcf_predecessor_bptpx_successor)) * sc)) /\ exists bcf_quotient_bptpx_successor_decoded_previous_scale. sb = bcf_quotient_bptpx_successor_decoded_previous_scale * S ((S (bcf_predecessor_bptpx_successor)) * sc) + (bcf_previous_scale_bptpx_successor))) /\ (forall bcf_index_bptpx_successor_row_step. (exists bcf_lt_gap_bptpx_successor_row_step_bound. bcf_lt_gap_bptpx_successor_row_step_bound + S (bcf_index_bptpx_successor_row_step) = w) -> exists bcf_value_bptpx_successor_row_step. ((((exists bcf_height_bptpx_successor_row_step_entry. bcf_height_bptpx_successor_row_step_entry + S (bcf_value_bptpx_successor_row_step) = S ((S (bcf_index_bptpx_successor_row_step)) * bcf_row_scale_bptpx_successor)) /\ exists bcf_quotient_bptpx_successor_row_step_entry. bcf_row_code_bptpx_successor = bcf_quotient_bptpx_successor_row_step_entry * S ((S (bcf_index_bptpx_successor_row_step)) * bcf_row_scale_bptpx_successor) + (bcf_value_bptpx_successor_row_step))) /\ ((bcf_index_bptpx_successor_row_step = 0 /\ bcf_value_bptpx_successor_row_step = 1) \/ exists bcf_predecessor_bptpx_successor_row_step bcf_left_bptpx_successor_row_step bcf_right_bptpx_successor_row_step. bcf_index_bptpx_successor_row_step = S bcf_predecessor_bptpx_successor_row_step /\ ((((exists bcf_height_bptpx_successor_row_step_previous_left. bcf_height_bptpx_successor_row_step_previous_left + S (bcf_left_bptpx_successor_row_step) = S ((S (bcf_predecessor_bptpx_successor_row_step)) * bcf_previous_scale_bptpx_successor)) /\ exists bcf_quotient_bptpx_successor_row_step_previous_left. bcf_previous_code_bptpx_successor = bcf_quotient_bptpx_successor_row_step_previous_left * S ((S (bcf_predecessor_bptpx_successor_row_step)) * bcf_previous_scale_bptpx_successor) + (bcf_left_bptpx_successor_row_step))) /\ ((((exists bcf_height_bptpx_successor_row_step_previous_right. bcf_height_bptpx_successor_row_step_previous_right + S (bcf_right_bptpx_successor_row_step) = S ((S (S (bcf_predecessor_bptpx_successor_row_step))) * bcf_previous_scale_bptpx_successor)) /\ exists bcf_quotient_bptpx_successor_row_step_previous_right. bcf_previous_code_bptpx_successor = bcf_quotient_bptpx_successor_row_step_previous_right * S ((S (S (bcf_predecessor_bptpx_successor_row_step))) * bcf_previous_scale_bptpx_successor) + (bcf_right_bptpx_successor_row_step))) /\ bcf_value_bptpx_successor_row_step = bcf_left_bptpx_successor_row_step + bcf_right_bptpx_successor_row_step)))))))))))
  26. 0026specialize beta_pascal_table_prefix_extend x
  27. 0027specialize beta_pascal_table_prefix_extend x1
  28. 0028specialize beta_pascal_table_prefix_extend x2
  29. 0029specialize beta_pascal_table_prefix_extend x3
  30. 0030specialize beta_pascal_table_prefix_extend w
  31. 0031specialize beta_pascal_table_prefix_extend r
  32. 0032apply beta_pascal_table_prefix_extend
  33. 0033exact hprevious_witness_witness_witness_witness
  34. 0034exact hnext