PA008W

injective_prefix_zero

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

Decoded-prefix injectivity is vacuous at length zero.

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 b c. (forall wpo_injective_left_wpoi_zero_injective wpo_injective_right_wpoi_zero_injective wpo_injective_value_wpoi_zero_injective. (exists wpo_gap_wpoi_zero_injective_left_bound. wpo_gap_wpoi_zero_injective_left_bound + S (wpo_injective_left_wpoi_zero_injective) = 0) -> (exists wpo_gap_wpoi_zero_injective_right_bound. wpo_gap_wpoi_zero_injective_right_bound + S (wpo_injective_right_wpoi_zero_injective) = 0) -> (((exists wpo_beta_height_wpoi_zero_injective_left_entry. wpo_beta_height_wpoi_zero_injective_left_entry + S (wpo_injective_value_wpoi_zero_injective) = S ((S (wpo_injective_left_wpoi_zero_injective)) * c)) /\ exists wpo_beta_quotient_wpoi_zero_injective_left_entry. b = wpo_beta_quotient_wpoi_zero_injective_left_entry * S ((S (wpo_injective_left_wpoi_zero_injective)) * c) + (wpo_injective_value_wpoi_zero_injective))) -> (((exists wpo_beta_height_wpoi_zero_injective_right_entry. wpo_beta_height_wpoi_zero_injective_right_entry + S (wpo_injective_value_wpoi_zero_injective) = S ((S (wpo_injective_right_wpoi_zero_injective)) * c)) /\ exists wpo_beta_quotient_wpoi_zero_injective_right_entry. b = wpo_beta_quotient_wpoi_zero_injective_right_entry * S ((S (wpo_injective_right_wpoi_zero_injective)) * c) + (wpo_injective_value_wpoi_zero_injective))) -> wpo_injective_left_wpoi_zero_injective = wpo_injective_right_wpoi_zero_injective)

Structural proof guide

Generated structural guide

Decoded-prefix injectivity is vacuous at length zero.

Use the direct prerequisites add_eq_zero_right, succ_ne_zero as previously established PA formulas.

The proof proceeds by case analysis (1), intermediate claims (1).

Referenced ingredients

PA0004 add_eq_zero_right PA0005 succ_ne_zero

Proof neighborhood

Direct dependencies

PA0004 add_eq_zero_right PA0005 succ_ne_zero

Direct dependents

PA008X scaled_pair_order_state_zero PA00AA pair_order_state_zero

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

19 script commands · 3 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.

Named ingredients (2)
add eq zero rightsucc ne zero
01Fix variables and assumptionsL1–9

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro q
  4. L4
    intro r
  5. L5
    intro w
  6. L6
    intro hq
  7. L7
    intro hr
  8. L8
    intro hleft
  9. L9
    intro hright
02Separate the logical casesL10–11

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

  1. L10
    exfalso
  2. L11
    cases hq
03Establish hsqL12–19

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

  1. L12
    have hsq : S q = 0
  2. L13
    specialize add_eq_zero_right x
  3. L14
    specialize add_eq_zero_right (S q)
  4. L15
    apply add_eq_zero_right
  5. L16
    exact hq_witness
  6. L17
    specialize succ_ne_zero q
  7. L18
    apply succ_ne_zero
  8. L19
    exact hsq

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro q
  4. 0004intro r
  5. 0005intro w
  6. 0006intro hq
  7. 0007intro hr
  8. 0008intro hleft
  9. 0009intro hright
  10. 0010exfalso
  11. 0011cases hq
  12. 0012have hsq : S q = 0
  13. 0013specialize add_eq_zero_right x
  14. 0014specialize add_eq_zero_right (S q)
  15. 0015apply add_eq_zero_right
  16. 0016exact hq_witness
  17. 0017specialize succ_ne_zero q
  18. 0018apply succ_ne_zero
  19. 0019exact hsq