PA009Y · theorem

beta_product_double_succ_decompose

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

A product of length S(S k) decomposes into its k-prefix and its final two factors.

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

∀ b. ∀ c. ∀ k. ∀ l. ∀ Q. l = S S k → Product(b,c,l,Q) → ∃ x. ∃ y. ∃ z. BetaAt(b,c,k,x) ∧ (BetaAt(b,c,S k,y) ∧ (Product(b,c,k,z) ∧ Q = z · x · y))

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0013 BetaAt PD0014 Product

4 occurrences

In local proof propositions

PD0013 BetaAt PD0014 Product

4 occurrences

Exact expanded native-PA statement
forall b c k l Q. l = S (S k) -> (exists wpp_trace_code_two_product wpp_trace_scale_two_product. ((((exists wpp_beta_height_two_product_start. wpp_beta_height_two_product_start + S (1) = S ((S (0)) * wpp_trace_scale_two_product)) /\ exists wpp_beta_quotient_two_product_start. wpp_trace_code_two_product = wpp_beta_quotient_two_product_start * S ((S (0)) * wpp_trace_scale_two_product) + (1))) /\ ((((exists wpp_beta_height_two_product_terminal. wpp_beta_height_two_product_terminal + S (Q) = S ((S (l)) * wpp_trace_scale_two_product)) /\ exists wpp_beta_quotient_two_product_terminal. wpp_trace_code_two_product = wpp_beta_quotient_two_product_terminal * S ((S (l)) * wpp_trace_scale_two_product) + (Q))) /\ forall wpp_index_two_product. (exists wpp_gap_two_product_bound. wpp_gap_two_product_bound + S (wpp_index_two_product) = l) -> exists wpp_factor_two_product wpp_prefix_two_product wpp_successor_two_product. ((((exists wpp_beta_height_two_product_factor. wpp_beta_height_two_product_factor + S (wpp_factor_two_product) = S ((S (wpp_index_two_product)) * c)) /\ exists wpp_beta_quotient_two_product_factor. b = wpp_beta_quotient_two_product_factor * S ((S (wpp_index_two_product)) * c) + (wpp_factor_two_product))) /\ ((((exists wpp_beta_height_two_product_prefix. wpp_beta_height_two_product_prefix + S (wpp_prefix_two_product) = S ((S (wpp_index_two_product)) * wpp_trace_scale_two_product)) /\ exists wpp_beta_quotient_two_product_prefix. wpp_trace_code_two_product = wpp_beta_quotient_two_product_prefix * S ((S (wpp_index_two_product)) * wpp_trace_scale_two_product) + (wpp_prefix_two_product))) /\ ((((exists wpp_beta_height_two_product_successor. wpp_beta_height_two_product_successor + S (wpp_successor_two_product) = S ((S (S (wpp_index_two_product))) * wpp_trace_scale_two_product)) /\ exists wpp_beta_quotient_two_product_successor. wpp_trace_code_two_product = wpp_beta_quotient_two_product_successor * S ((S (S (wpp_index_two_product))) * wpp_trace_scale_two_product) + (wpp_successor_two_product))) /\ wpp_successor_two_product = wpp_prefix_two_product * wpp_factor_two_product)))))) -> (exists wpp_left_factor_two_result wpp_right_factor_two_result wpp_prefix_product_two_result. (((exists wpp_beta_height_two_result_left_entry. wpp_beta_height_two_result_left_entry + S (wpp_left_factor_two_result) = S ((S (k)) * c)) /\ exists wpp_beta_quotient_two_result_left_entry. b = wpp_beta_quotient_two_result_left_entry * S ((S (k)) * c) + (wpp_left_factor_two_result))) /\ ((((exists wpp_beta_height_two_result_right_entry. wpp_beta_height_two_result_right_entry + S (wpp_right_factor_two_result) = S ((S (S (k))) * c)) /\ exists wpp_beta_quotient_two_result_right_entry. b = wpp_beta_quotient_two_result_right_entry * S ((S (S (k))) * c) + (wpp_right_factor_two_result))) /\ ((exists wpp_trace_code_two_result_prefix wpp_trace_scale_two_result_prefix. ((((exists wpp_beta_height_two_result_prefix_start. wpp_beta_height_two_result_prefix_start + S (1) = S ((S (0)) * wpp_trace_scale_two_result_prefix)) /\ exists wpp_beta_quotient_two_result_prefix_start. wpp_trace_code_two_result_prefix = wpp_beta_quotient_two_result_prefix_start * S ((S (0)) * wpp_trace_scale_two_result_prefix) + (1))) /\ ((((exists wpp_beta_height_two_result_prefix_terminal. wpp_beta_height_two_result_prefix_terminal + S (wpp_prefix_product_two_result) = S ((S (k)) * wpp_trace_scale_two_result_prefix)) /\ exists wpp_beta_quotient_two_result_prefix_terminal. wpp_trace_code_two_result_prefix = wpp_beta_quotient_two_result_prefix_terminal * S ((S (k)) * wpp_trace_scale_two_result_prefix) + (wpp_prefix_product_two_result))) /\ forall wpp_index_two_result_prefix. (exists wpp_gap_two_result_prefix_bound. wpp_gap_two_result_prefix_bound + S (wpp_index_two_result_prefix) = k) -> exists wpp_factor_two_result_prefix wpp_prefix_two_result_prefix wpp_successor_two_result_prefix. ((((exists wpp_beta_height_two_result_prefix_factor. wpp_beta_height_two_result_prefix_factor + S (wpp_factor_two_result_prefix) = S ((S (wpp_index_two_result_prefix)) * c)) /\ exists wpp_beta_quotient_two_result_prefix_factor. b = wpp_beta_quotient_two_result_prefix_factor * S ((S (wpp_index_two_result_prefix)) * c) + (wpp_factor_two_result_prefix))) /\ ((((exists wpp_beta_height_two_result_prefix_prefix. wpp_beta_height_two_result_prefix_prefix + S (wpp_prefix_two_result_prefix) = S ((S (wpp_index_two_result_prefix)) * wpp_trace_scale_two_result_prefix)) /\ exists wpp_beta_quotient_two_result_prefix_prefix. wpp_trace_code_two_result_prefix = wpp_beta_quotient_two_result_prefix_prefix * S ((S (wpp_index_two_result_prefix)) * wpp_trace_scale_two_result_prefix) + (wpp_prefix_two_result_prefix))) /\ ((((exists wpp_beta_height_two_result_prefix_successor. wpp_beta_height_two_result_prefix_successor + S (wpp_successor_two_result_prefix) = S ((S (S (wpp_index_two_result_prefix))) * wpp_trace_scale_two_result_prefix)) /\ exists wpp_beta_quotient_two_result_prefix_successor. wpp_trace_code_two_result_prefix = wpp_beta_quotient_two_result_prefix_successor * S ((S (S (wpp_index_two_result_prefix))) * wpp_trace_scale_two_result_prefix) + (wpp_successor_two_result_prefix))) /\ wpp_successor_two_result_prefix = wpp_prefix_two_result_prefix * wpp_factor_two_result_prefix)))))) /\ Q = (wpp_prefix_product_two_result * wpp_left_factor_two_result) * wpp_right_factor_two_result)))

Proof neighborhood

Direct theorem prerequisites

PA004A beta_product_succ_decompose

Direct theorem dependents

PA00A0 beta_adjacent_target_pairs_product_power PA00B9 beta_adjacent_unit_pairs_product_one

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

44 script commands · 14 reading checkpoints · 2 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 (1)
beta product succ decompose
01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro k
  4. L4
    intro l
  5. L5
    intro Q
  6. L6
    intro hlength
  7. L7
    intro hproduct
02Calculate and transport equalitiesL8–10

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

  1. L8
    rewrite hlength at hproduct
  2. L9
    rewrite hlength at hproduct
  3. L10
    rewrite hlength at hproduct
03Establish houterL11–17

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

  1. L11
    have houter : ∃ wpp_factor_outer_decomposition. ∃ wpp_prefix_product_outer_decomposition. BetaAt(b,c,S k,wpp_factor_outer_decomposition) ∧ (Product(b,c,S k,wpp_prefix_product_outer_decomposition) ∧ Q = wpp_prefix_product_outer_decomposition · wpp_factor_outer_decomposition)Definitions: BetaAt(b,c,S k,wpp_factor_outer_decomposition)Product(b,c,S k,wpp_prefix_product_outer_decomposition)Original native command in the exact edition
  2. L12
    specialize beta_product_succ_decompose b
  3. L13
    specialize beta_product_succ_decompose c
  4. L14
    specialize beta_product_succ_decompose (S k)
  5. L15
    specialize beta_product_succ_decompose Q
  6. L16
    apply beta_product_succ_decompose
  7. L17
    exact hproduct
04Separate the logical casesL18–21

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

  1. L18
    cases houter
  2. L19
    cases houter_witness
  3. L20
    cases houter_witness_witness
  4. L21
    cases houter_witness_witness_right
05Establish hinnerL22–28

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

  1. L22
    have hinner : ∃ wpp_factor_inner_decomposition. ∃ wpp_prefix_product_inner_decomposition. BetaAt(b,c,k,wpp_factor_inner_decomposition) ∧ (Product(b,c,k,wpp_prefix_product_inner_decomposition) ∧ x1 = wpp_prefix_product_inner_decomposition · wpp_factor_inner_decomposition)Definitions: BetaAt(b,c,k,wpp_factor_inner_decomposition)Product(b,c,k,wpp_prefix_product_inner_decomposition)Original native command in the exact edition
  2. L23
    specialize beta_product_succ_decompose b
  3. L24
    specialize beta_product_succ_decompose c
  4. L25
    specialize beta_product_succ_decompose k
  5. L26
    specialize beta_product_succ_decompose x1
  6. L27
    apply beta_product_succ_decompose
  7. L28
    exact houter_witness_witness_right_left
06Separate the logical casesL29–32

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

  1. L29
    cases hinner
  2. L30
    cases hinner_witness
  3. L31
    cases hinner_witness_witness
  4. L32
    cases hinner_witness_witness_right
07Construct an explicit witnessL33–35

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

  1. L33
    exists x2
  2. L34
    exists x
  3. L35
    exists x3
08Separate the logical casesL36–36

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

  1. L36
    split
09Use earlier factsL37–37

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

  1. L37
    exact hinner_witness_witness_left
10Separate the logical casesL38–38

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

  1. L38
    split
11Use earlier factsL39–39

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

  1. L39
    exact houter_witness_witness_left
12Separate the logical casesL40–40

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

  1. L40
    split
13Use earlier factsL41–41

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

  1. L41
    exact hinner_witness_witness_right_left
14Calculate and transport equalitiesL42–44

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

  1. L42
    rewrite houter_witness_witness_right_right
  2. L43
    rewrite hinner_witness_witness_right_right
  3. L44
    refl

Library-wide reading audit

Original defined command ledger · 44 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro k
  4. 0004intro l
  5. 0005intro Q
  6. 0006intro hlength
  7. 0007intro hproduct
  8. 0008rewrite hlength at hproduct
  9. 0009rewrite hlength at hproduct
  10. 0010rewrite hlength at hproduct
  11. 0011have houter : ∃ wpp_factor_outer_decomposition. ∃ wpp_prefix_product_outer_decomposition. BetaAt(b,c,S k,wpp_factor_outer_decomposition) ∧ (Product(b,c,S k,wpp_prefix_product_outer_decomposition) ∧ Q = wpp_prefix_product_outer_decomposition · wpp_factor_outer_decomposition)
    Exact native replay linehave houter : exists wpp_factor_outer_decomposition wpp_prefix_product_outer_decomposition. (((exists wpp_beta_height_outer_decomposition_entry. wpp_beta_height_outer_decomposition_entry + S (wpp_factor_outer_decomposition) = S ((S (S k)) * c)) /\ exists wpp_beta_quotient_outer_decomposition_entry. b = wpp_beta_quotient_outer_decomposition_entry * S ((S (S k)) * c) + (wpp_factor_outer_decomposition))) /\ ((exists wpp_trace_code_outer_decomposition_prefix wpp_trace_scale_outer_decomposition_prefix. ((((exists wpp_beta_height_outer_decomposition_prefix_start. wpp_beta_height_outer_decomposition_prefix_start + S (1) = S ((S (0)) * wpp_trace_scale_outer_decomposition_prefix)) /\ exists wpp_beta_quotient_outer_decomposition_prefix_start. wpp_trace_code_outer_decomposition_prefix = wpp_beta_quotient_outer_decomposition_prefix_start * S ((S (0)) * wpp_trace_scale_outer_decomposition_prefix) + (1))) /\ ((((exists wpp_beta_height_outer_decomposition_prefix_terminal. wpp_beta_height_outer_decomposition_prefix_terminal + S (wpp_prefix_product_outer_decomposition) = S ((S (S k)) * wpp_trace_scale_outer_decomposition_prefix)) /\ exists wpp_beta_quotient_outer_decomposition_prefix_terminal. wpp_trace_code_outer_decomposition_prefix = wpp_beta_quotient_outer_decomposition_prefix_terminal * S ((S (S k)) * wpp_trace_scale_outer_decomposition_prefix) + (wpp_prefix_product_outer_decomposition))) /\ forall wpp_index_outer_decomposition_prefix. (exists wpp_gap_outer_decomposition_prefix_bound. wpp_gap_outer_decomposition_prefix_bound + S (wpp_index_outer_decomposition_prefix) = S k) -> exists wpp_factor_outer_decomposition_prefix wpp_prefix_outer_decomposition_prefix wpp_successor_outer_decomposition_prefix. ((((exists wpp_beta_height_outer_decomposition_prefix_factor. wpp_beta_height_outer_decomposition_prefix_factor + S (wpp_factor_outer_decomposition_prefix) = S ((S (wpp_index_outer_decomposition_prefix)) * c)) /\ exists wpp_beta_quotient_outer_decomposition_prefix_factor. b = wpp_beta_quotient_outer_decomposition_prefix_factor * S ((S (wpp_index_outer_decomposition_prefix)) * c) + (wpp_factor_outer_decomposition_prefix))) /\ ((((exists wpp_beta_height_outer_decomposition_prefix_prefix. wpp_beta_height_outer_decomposition_prefix_prefix + S (wpp_prefix_outer_decomposition_prefix) = S ((S (wpp_index_outer_decomposition_prefix)) * wpp_trace_scale_outer_decomposition_prefix)) /\ exists wpp_beta_quotient_outer_decomposition_prefix_prefix. wpp_trace_code_outer_decomposition_prefix = wpp_beta_quotient_outer_decomposition_prefix_prefix * S ((S (wpp_index_outer_decomposition_prefix)) * wpp_trace_scale_outer_decomposition_prefix) + (wpp_prefix_outer_decomposition_prefix))) /\ ((((exists wpp_beta_height_outer_decomposition_prefix_successor. wpp_beta_height_outer_decomposition_prefix_successor + S (wpp_successor_outer_decomposition_prefix) = S ((S (S (wpp_index_outer_decomposition_prefix))) * wpp_trace_scale_outer_decomposition_prefix)) /\ exists wpp_beta_quotient_outer_decomposition_prefix_successor. wpp_trace_code_outer_decomposition_prefix = wpp_beta_quotient_outer_decomposition_prefix_successor * S ((S (S (wpp_index_outer_decomposition_prefix))) * wpp_trace_scale_outer_decomposition_prefix) + (wpp_successor_outer_decomposition_prefix))) /\ wpp_successor_outer_decomposition_prefix = wpp_prefix_outer_decomposition_prefix * wpp_factor_outer_decomposition_prefix)))))) /\ Q = wpp_prefix_product_outer_decomposition * wpp_factor_outer_decomposition)
  12. 0012specialize beta_product_succ_decompose b
  13. 0013specialize beta_product_succ_decompose c
  14. 0014specialize beta_product_succ_decompose (S k)
  15. 0015specialize beta_product_succ_decompose Q
  16. 0016apply beta_product_succ_decompose
  17. 0017exact hproduct
  18. 0018cases houter
  19. 0019cases houter_witness
  20. 0020cases houter_witness_witness
  21. 0021cases houter_witness_witness_right
  22. 0022have hinner : ∃ wpp_factor_inner_decomposition. ∃ wpp_prefix_product_inner_decomposition. BetaAt(b,c,k,wpp_factor_inner_decomposition) ∧ (Product(b,c,k,wpp_prefix_product_inner_decomposition) ∧ x1 = wpp_prefix_product_inner_decomposition · wpp_factor_inner_decomposition)
    Exact native replay linehave hinner : exists wpp_factor_inner_decomposition wpp_prefix_product_inner_decomposition. (((exists wpp_beta_height_inner_decomposition_entry. wpp_beta_height_inner_decomposition_entry + S (wpp_factor_inner_decomposition) = S ((S (k)) * c)) /\ exists wpp_beta_quotient_inner_decomposition_entry. b = wpp_beta_quotient_inner_decomposition_entry * S ((S (k)) * c) + (wpp_factor_inner_decomposition))) /\ ((exists wpp_trace_code_inner_decomposition_prefix wpp_trace_scale_inner_decomposition_prefix. ((((exists wpp_beta_height_inner_decomposition_prefix_start. wpp_beta_height_inner_decomposition_prefix_start + S (1) = S ((S (0)) * wpp_trace_scale_inner_decomposition_prefix)) /\ exists wpp_beta_quotient_inner_decomposition_prefix_start. wpp_trace_code_inner_decomposition_prefix = wpp_beta_quotient_inner_decomposition_prefix_start * S ((S (0)) * wpp_trace_scale_inner_decomposition_prefix) + (1))) /\ ((((exists wpp_beta_height_inner_decomposition_prefix_terminal. wpp_beta_height_inner_decomposition_prefix_terminal + S (wpp_prefix_product_inner_decomposition) = S ((S (k)) * wpp_trace_scale_inner_decomposition_prefix)) /\ exists wpp_beta_quotient_inner_decomposition_prefix_terminal. wpp_trace_code_inner_decomposition_prefix = wpp_beta_quotient_inner_decomposition_prefix_terminal * S ((S (k)) * wpp_trace_scale_inner_decomposition_prefix) + (wpp_prefix_product_inner_decomposition))) /\ forall wpp_index_inner_decomposition_prefix. (exists wpp_gap_inner_decomposition_prefix_bound. wpp_gap_inner_decomposition_prefix_bound + S (wpp_index_inner_decomposition_prefix) = k) -> exists wpp_factor_inner_decomposition_prefix wpp_prefix_inner_decomposition_prefix wpp_successor_inner_decomposition_prefix. ((((exists wpp_beta_height_inner_decomposition_prefix_factor. wpp_beta_height_inner_decomposition_prefix_factor + S (wpp_factor_inner_decomposition_prefix) = S ((S (wpp_index_inner_decomposition_prefix)) * c)) /\ exists wpp_beta_quotient_inner_decomposition_prefix_factor. b = wpp_beta_quotient_inner_decomposition_prefix_factor * S ((S (wpp_index_inner_decomposition_prefix)) * c) + (wpp_factor_inner_decomposition_prefix))) /\ ((((exists wpp_beta_height_inner_decomposition_prefix_prefix. wpp_beta_height_inner_decomposition_prefix_prefix + S (wpp_prefix_inner_decomposition_prefix) = S ((S (wpp_index_inner_decomposition_prefix)) * wpp_trace_scale_inner_decomposition_prefix)) /\ exists wpp_beta_quotient_inner_decomposition_prefix_prefix. wpp_trace_code_inner_decomposition_prefix = wpp_beta_quotient_inner_decomposition_prefix_prefix * S ((S (wpp_index_inner_decomposition_prefix)) * wpp_trace_scale_inner_decomposition_prefix) + (wpp_prefix_inner_decomposition_prefix))) /\ ((((exists wpp_beta_height_inner_decomposition_prefix_successor. wpp_beta_height_inner_decomposition_prefix_successor + S (wpp_successor_inner_decomposition_prefix) = S ((S (S (wpp_index_inner_decomposition_prefix))) * wpp_trace_scale_inner_decomposition_prefix)) /\ exists wpp_beta_quotient_inner_decomposition_prefix_successor. wpp_trace_code_inner_decomposition_prefix = wpp_beta_quotient_inner_decomposition_prefix_successor * S ((S (S (wpp_index_inner_decomposition_prefix))) * wpp_trace_scale_inner_decomposition_prefix) + (wpp_successor_inner_decomposition_prefix))) /\ wpp_successor_inner_decomposition_prefix = wpp_prefix_inner_decomposition_prefix * wpp_factor_inner_decomposition_prefix)))))) /\ x1 = wpp_prefix_product_inner_decomposition * wpp_factor_inner_decomposition)
  23. 0023specialize beta_product_succ_decompose b
  24. 0024specialize beta_product_succ_decompose c
  25. 0025specialize beta_product_succ_decompose k
  26. 0026specialize beta_product_succ_decompose x1
  27. 0027apply beta_product_succ_decompose
  28. 0028exact houter_witness_witness_right_left
  29. 0029cases hinner
  30. 0030cases hinner_witness
  31. 0031cases hinner_witness_witness
  32. 0032cases hinner_witness_witness_right
  33. 0033exists x2
  34. 0034exists x
  35. 0035exists x3
  36. 0036split
  37. 0037exact hinner_witness_witness_left
  38. 0038split
  39. 0039exact houter_witness_witness_left
  40. 0040split
  41. 0041exact hinner_witness_witness_right_left
  42. 0042rewrite houter_witness_witness_right_right
  43. 0043rewrite hinner_witness_witness_right_right
  44. 0044refl