BT00T5 · Bertrand theorem

beta_pascal_row_step_exists

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

Every previous beta row has a finite Pascal successor row.

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

∀ pb. ∀ pc. ∀ w. ∃ b. ∃ c. ∀ x. Lt(x,w) → ∃ y. BetaAt(b,c,x,y) ∧ (x = 0 ∧ y = 1 ∨ (∃ z. ∃ n. ∃ m. x = S z ∧ (BetaAt(pb,pc,z,n) ∧ (BetaAt(pb,pc,S z,m) ∧ y = n + m))))

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

4 occurrences

In local proof propositions

PD0002 Lt PD0013 BetaAt

8 occurrences

Exact expanded native-PA statement
forall pb pc w. exists b c. (forall bcf_index_bpsrx_result. (exists bcf_lt_gap_bpsrx_result_bound. bcf_lt_gap_bpsrx_result_bound + S (bcf_index_bpsrx_result) = w) -> exists bcf_value_bpsrx_result. ((((exists bcf_height_bpsrx_result_entry. bcf_height_bpsrx_result_entry + S (bcf_value_bpsrx_result) = S ((S (bcf_index_bpsrx_result)) * c)) /\ exists bcf_quotient_bpsrx_result_entry. b = bcf_quotient_bpsrx_result_entry * S ((S (bcf_index_bpsrx_result)) * c) + (bcf_value_bpsrx_result))) /\ ((bcf_index_bpsrx_result = 0 /\ bcf_value_bpsrx_result = 1) \/ exists bcf_predecessor_bpsrx_result bcf_left_bpsrx_result bcf_right_bpsrx_result. bcf_index_bpsrx_result = S bcf_predecessor_bpsrx_result /\ ((((exists bcf_height_bpsrx_result_previous_left. bcf_height_bpsrx_result_previous_left + S (bcf_left_bpsrx_result) = S ((S (bcf_predecessor_bpsrx_result)) * pc)) /\ exists bcf_quotient_bpsrx_result_previous_left. pb = bcf_quotient_bpsrx_result_previous_left * S ((S (bcf_predecessor_bpsrx_result)) * pc) + (bcf_left_bpsrx_result))) /\ ((((exists bcf_height_bpsrx_result_previous_right. bcf_height_bpsrx_result_previous_right + S (bcf_right_bpsrx_result) = S ((S (S (bcf_predecessor_bpsrx_result))) * pc)) /\ exists bcf_quotient_bpsrx_result_previous_right. pb = bcf_quotient_bpsrx_result_previous_right * S ((S (S (bcf_predecessor_bpsrx_result))) * pc) + (bcf_right_bpsrx_result))) /\ bcf_value_bpsrx_result = bcf_left_bpsrx_result + bcf_right_bpsrx_result)))))

Proof neighborhood

Direct theorem prerequisites

BT000L add_eq_zero_right BT000C succ_ne_zero BT00T4 beta_pascal_row_step_extend

Direct theorem dependents

BT00T6 beta_pascal_table_prefix_extend

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

30 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 row step extend
01Fix variables and assumptionsL1–2

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

  1. L1
    intro pb
  2. L2
    intro pc
02Induction on wL3–3

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

  1. L3
    induction w
03Construct an explicit witnessL4–5

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

  1. L4
    exists 0
  2. L5
    exists 0
04Fix variables and assumptionsL6–7

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

  1. L6
    intro i
  2. L7
    intro hi
05Separate the logical casesL8–9

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

  1. L8
    exfalso
  2. L9
    cases hi
06Establish hsiL10–17

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

  1. L10
    have hsi : S i = 0
  2. L11
    specialize add_eq_zero_right x
  3. L12
    specialize add_eq_zero_right (S i)
  4. L13
    apply add_eq_zero_right
  5. L14
    exact hi_witness
  6. L15
    specialize succ_ne_zero i
  7. L16
    apply succ_ne_zero
  8. L17
    exact hsi
07Establish hpreviousL18–19

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

  1. L18
    have hprevious : ∃ b. ∃ c. ∀ x. Lt(x,w) → ∃ y. BetaAt(b,c,x,y) ∧ (x = 0 ∧ y = 1 ∨ (∃ z. ∃ n. ∃ m. x = S z ∧ (BetaAt(pb,pc,z,n) ∧ (BetaAt(pb,pc,S z,m) ∧ y = n + m))))Definitions: Lt(x,w)BetaAt(b,c,x,y)BetaAt(pb,pc,z,n)BetaAt(pb,pc,S z,m)Original native command in the exact edition
  2. L19
    exact IH
08Separate the logical casesL20–21

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

  1. L20
    cases hprevious
  2. L21
    cases hprevious_witness
09Establish hnextL22–30

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

  1. L22
    have hnext : ∃ b. ∃ c. ∀ x. Lt(x,S w) → ∃ y. BetaAt(b,c,x,y) ∧ (x = 0 ∧ y = 1 ∨ (∃ z. ∃ n. ∃ m. x = S z ∧ (BetaAt(pb,pc,z,n) ∧ (BetaAt(pb,pc,S z,m) ∧ y = n + m))))Definitions: Lt(x,S w)BetaAt(b,c,x,y)BetaAt(pb,pc,z,n)BetaAt(pb,pc,S z,m)Original native command in the exact edition
  2. L23
    specialize beta_pascal_row_step_extend pb
  3. L24
    specialize beta_pascal_row_step_extend pc
  4. L25
    specialize beta_pascal_row_step_extend x
  5. L26
    specialize beta_pascal_row_step_extend x1
  6. L27
    specialize beta_pascal_row_step_extend w
  7. L28
    apply beta_pascal_row_step_extend
  8. L29
    exact hprevious_witness_witness
  9. L30
    exact hnext

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro pb
  2. 0002intro pc
  3. 0003induction w
  4. 0004exists 0
  5. 0005exists 0
  6. 0006intro i
  7. 0007intro hi
  8. 0008exfalso
  9. 0009cases hi
  10. 0010have hsi : S i = 0
  11. 0011specialize add_eq_zero_right x
  12. 0012specialize add_eq_zero_right (S i)
  13. 0013apply add_eq_zero_right
  14. 0014exact hi_witness
  15. 0015specialize succ_ne_zero i
  16. 0016apply succ_ne_zero
  17. 0017exact hsi
  18. 0018have hprevious : ∃ b. ∃ c. ∀ x. Lt(x,w) → ∃ y. BetaAt(b,c,x,y) ∧ (x = 0 ∧ y = 1 ∨ (∃ z. ∃ n. ∃ m. x = S z ∧ (BetaAt(pb,pc,z,n) ∧ (BetaAt(pb,pc,S z,m) ∧ y = n + m))))
    Exact native replay linehave hprevious : exists b c. (forall bcf_index_bpsrx_previous. (exists bcf_lt_gap_bpsrx_previous_bound. bcf_lt_gap_bpsrx_previous_bound + S (bcf_index_bpsrx_previous) = w) -> exists bcf_value_bpsrx_previous. ((((exists bcf_height_bpsrx_previous_entry. bcf_height_bpsrx_previous_entry + S (bcf_value_bpsrx_previous) = S ((S (bcf_index_bpsrx_previous)) * c)) /\ exists bcf_quotient_bpsrx_previous_entry. b = bcf_quotient_bpsrx_previous_entry * S ((S (bcf_index_bpsrx_previous)) * c) + (bcf_value_bpsrx_previous))) /\ ((bcf_index_bpsrx_previous = 0 /\ bcf_value_bpsrx_previous = 1) \/ exists bcf_predecessor_bpsrx_previous bcf_left_bpsrx_previous bcf_right_bpsrx_previous. bcf_index_bpsrx_previous = S bcf_predecessor_bpsrx_previous /\ ((((exists bcf_height_bpsrx_previous_previous_left. bcf_height_bpsrx_previous_previous_left + S (bcf_left_bpsrx_previous) = S ((S (bcf_predecessor_bpsrx_previous)) * pc)) /\ exists bcf_quotient_bpsrx_previous_previous_left. pb = bcf_quotient_bpsrx_previous_previous_left * S ((S (bcf_predecessor_bpsrx_previous)) * pc) + (bcf_left_bpsrx_previous))) /\ ((((exists bcf_height_bpsrx_previous_previous_right. bcf_height_bpsrx_previous_previous_right + S (bcf_right_bpsrx_previous) = S ((S (S (bcf_predecessor_bpsrx_previous))) * pc)) /\ exists bcf_quotient_bpsrx_previous_previous_right. pb = bcf_quotient_bpsrx_previous_previous_right * S ((S (S (bcf_predecessor_bpsrx_previous))) * pc) + (bcf_right_bpsrx_previous))) /\ bcf_value_bpsrx_previous = bcf_left_bpsrx_previous + bcf_right_bpsrx_previous)))))
  19. 0019exact IH
  20. 0020cases hprevious
  21. 0021cases hprevious_witness
  22. 0022have hnext : ∃ b. ∃ c. ∀ x. Lt(x,S w) → ∃ y. BetaAt(b,c,x,y) ∧ (x = 0 ∧ y = 1 ∨ (∃ z. ∃ n. ∃ m. x = S z ∧ (BetaAt(pb,pc,z,n) ∧ (BetaAt(pb,pc,S z,m) ∧ y = n + m))))
    Exact native replay linehave hnext : exists b c. (forall bcf_index_bpsrx_successor. (exists bcf_lt_gap_bpsrx_successor_bound. bcf_lt_gap_bpsrx_successor_bound + S (bcf_index_bpsrx_successor) = S (w)) -> exists bcf_value_bpsrx_successor. ((((exists bcf_height_bpsrx_successor_entry. bcf_height_bpsrx_successor_entry + S (bcf_value_bpsrx_successor) = S ((S (bcf_index_bpsrx_successor)) * c)) /\ exists bcf_quotient_bpsrx_successor_entry. b = bcf_quotient_bpsrx_successor_entry * S ((S (bcf_index_bpsrx_successor)) * c) + (bcf_value_bpsrx_successor))) /\ ((bcf_index_bpsrx_successor = 0 /\ bcf_value_bpsrx_successor = 1) \/ exists bcf_predecessor_bpsrx_successor bcf_left_bpsrx_successor bcf_right_bpsrx_successor. bcf_index_bpsrx_successor = S bcf_predecessor_bpsrx_successor /\ ((((exists bcf_height_bpsrx_successor_previous_left. bcf_height_bpsrx_successor_previous_left + S (bcf_left_bpsrx_successor) = S ((S (bcf_predecessor_bpsrx_successor)) * pc)) /\ exists bcf_quotient_bpsrx_successor_previous_left. pb = bcf_quotient_bpsrx_successor_previous_left * S ((S (bcf_predecessor_bpsrx_successor)) * pc) + (bcf_left_bpsrx_successor))) /\ ((((exists bcf_height_bpsrx_successor_previous_right. bcf_height_bpsrx_successor_previous_right + S (bcf_right_bpsrx_successor) = S ((S (S (bcf_predecessor_bpsrx_successor))) * pc)) /\ exists bcf_quotient_bpsrx_successor_previous_right. pb = bcf_quotient_bpsrx_successor_previous_right * S ((S (S (bcf_predecessor_bpsrx_successor))) * pc) + (bcf_right_bpsrx_successor))) /\ bcf_value_bpsrx_successor = bcf_left_bpsrx_successor + bcf_right_bpsrx_successor)))))
  23. 0023specialize beta_pascal_row_step_extend pb
  24. 0024specialize beta_pascal_row_step_extend pc
  25. 0025specialize beta_pascal_row_step_extend x
  26. 0026specialize beta_pascal_row_step_extend x1
  27. 0027specialize beta_pascal_row_step_extend w
  28. 0028apply beta_pascal_row_step_extend
  29. 0029exact hprevious_witness_witness
  30. 0030exact hnext