PA000W

le_eq_or_lt

Stable checked-use theorem · independently closed

A witnessed inequality is either equality or a witnessed strict inequality.

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 a b. (exists k. k + a = b) -> a = b \/ exists k. k + S a = b

Structural proof guide

Generated structural guide

A witnessed inequality is either equality or a witnessed strict inequality.

Use the direct prerequisites zero_or_succ, zero_add, add_succ_left as previously established PA formulas.

The proof proceeds by case analysis (3), equality transport (3).

Referenced ingredients

PA0008 zero_or_succ PA0001 zero_add PA000E add_succ_left

Proof neighborhood

Direct dependencies

PA0008 zero_or_succ PA0001 zero_add PA000E add_succ_left

Direct dependents

PA001K gcd_balanced_bezout_exists_up_to PA001U beta_exclusive_accumulated_product_step PA002G beta_exclusive_recode_congruence_step PA002Y beta_range_succ_extend PA0035 gcd_exists_up_to PA003D finite_lt_succ_eq_or_lt PA003G beta_prefix_sum_trace_exists PA003W beta_prefix_product_trace_exists PA0044 beta_repeat_succ_extend PA005L quadratic_residue_search_up_to PA00B4 finite_bounded_nonendpoint_injective_coverage PA00BB pair_order_terminal_state_magnitude_range PA00DX eisenstein_initial_segment_bit_count_functional

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

Read the argument

Proof checkpoints

21 script commands · 16 reading checkpoints · 0 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 (3)
zero or succzero addadd succ left
01Fix variables and assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
02Separate the logical casesL4–4

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

  1. L4
    cases h
03Use earlier factsL5–5

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

  1. L5
    specialize zero_or_succ x
04Separate the logical casesL6–7

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

  1. L6
    cases zero_or_succ
  2. L7
    left
05Calculate and transport equalitiesL8–8

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

  1. L8
    rewrite zero_or_succ_left at h_witness
06Use earlier factsL9–9

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

  1. L9
    specialize zero_add a
07Calculate and transport equalitiesL10–10

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

  1. L10
    rewrite zero_add at h_witness
08Use earlier factsL11–11

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

  1. L11
    exact h_witness
09Separate the logical casesL12–13

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

  1. L12
    cases zero_or_succ_right
  2. L13
    right
10Construct an explicit witnessL14–14

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

  1. L14
    exists x1
11Calculate and transport equalitiesL15–16

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

  1. L15
    trans S x1 + a
  2. L16
    trans S (x1 + a)
12Use earlier factsL17–17

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

  1. L17
    apply PA4
13Calculate and transport equalitiesL18–18

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

  1. L18
    symm
14Use earlier factsL19–19

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

  1. L19
    apply add_succ_left
15Calculate and transport equalitiesL20–20

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

  1. L20
    rewrite <- zero_or_succ_right_witness
16Use earlier factsL21–21

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

  1. L21
    exact h_witness

Library-wide reading audit

Original exact command ledger · 21 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004cases h
  5. 0005specialize zero_or_succ x
  6. 0006cases zero_or_succ
  7. 0007left
  8. 0008rewrite zero_or_succ_left at h_witness
  9. 0009specialize zero_add a
  10. 0010rewrite zero_add at h_witness
  11. 0011exact h_witness
  12. 0012cases zero_or_succ_right
  13. 0013right
  14. 0014exists x1
  15. 0015trans S x1 + a
  16. 0016trans S (x1 + a)
  17. 0017apply PA4
  18. 0018symm
  19. 0019apply add_succ_left
  20. 0020rewrite <- zero_or_succ_right_witness
  21. 0021exact h_witness