PA003A

lt_not_le

Stable checked-use theorem · independently closed

A strict inequality excludes the reverse weak 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 + S a = b) -> ~ (exists k. k + b = a)

Structural proof guide

Generated structural guide

A strict inequality excludes the reverse weak inequality.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by structural induction (1), case analysis (2), intermediate claims (1), equality transport (5).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

PA0063 prime_bounded_nonzero_mod_inverse PA0072 gauss_half_range_signed_choices PA0082 prime_positive_bounded_product_coprime PA0089 bounded_nonzero_not_divides PA008B prime_mul_index_map_exists_up_to PA008K prime_range_product_coprime PA009C prime_scaled_inverse_unique PA00AG prime_bounded_square_one_cases PA00D8 distinct_odd_prime_half_products_ne PA00D9 distinct_odd_prime_half_cell_oriented PA00DN prime_nondivisor_bounded_scaled_remainder_nonzero PA00DR odd_half_division_quotient_bounded PA00DS nonzero_remainder_division_positive_multiple_threshold 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

34 script commands · 11 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.

01Establish hbadL1–2

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

  1. L1
    have hbad : forall z u v. ~(z = u + (v + S z))
  2. L2
    intro z
02Induction on zL3–12

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

  1. L3
    induction z
  2. L4
    intro u
  3. L5
    intro v
  4. L6
    intro hz
  5. L7
    apply PA1
  6. L8
    symm
  7. L9
    rewrite PA4 at hz
  8. L10
    rewrite PA4 at hz
  9. L11
    exact hz
  10. L12
    intro u
03Fix variables and assumptionsL13–14

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

  1. L13
    intro v
  2. L14
    intro hz
04Use earlier factsL15–18

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

  1. L15
    specialize IH u
  2. L16
    specialize IH v
  3. L17
    apply IH
  4. L18
    apply PA2
05Calculate and transport equalitiesL19–20

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

  1. L19
    rewrite PA4 at hz
  2. L20
    rewrite PA4 at hz
06Use earlier factsL21–21

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

  1. L21
    exact hz
07Fix variables and assumptionsL22–25

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

  1. L22
    intro a
  2. L23
    intro b
  3. L24
    intro hab
  4. L25
    intro hba
08Separate the logical casesL26–27

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

  1. L26
    cases hab
  2. L27
    cases hba
09Use earlier factsL28–31

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

  1. L28
    specialize hbad a
  2. L29
    specialize hbad x1
  3. L30
    specialize hbad x
  4. L31
    apply hbad
10Calculate and transport equalitiesL32–33

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

  1. L32
    symm
  2. L33
    rewrite <- hab_witness at hba_witness
11Use earlier factsL34–34

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

  1. L34
    exact hba_witness

Library-wide reading audit

Original exact command ledger · 34 lines
  1. 0001have hbad : forall z u v. ~(z = u + (v + S z))
  2. 0002intro z
  3. 0003induction z
  4. 0004intro u
  5. 0005intro v
  6. 0006intro hz
  7. 0007apply PA1
  8. 0008symm
  9. 0009rewrite PA4 at hz
  10. 0010rewrite PA4 at hz
  11. 0011exact hz
  12. 0012intro u
  13. 0013intro v
  14. 0014intro hz
  15. 0015specialize IH u
  16. 0016specialize IH v
  17. 0017apply IH
  18. 0018apply PA2
  19. 0019rewrite PA4 at hz
  20. 0020rewrite PA4 at hz
  21. 0021exact hz
  22. 0022intro a
  23. 0023intro b
  24. 0024intro hab
  25. 0025intro hba
  26. 0026cases hab
  27. 0027cases hba
  28. 0028specialize hbad a
  29. 0029specialize hbad x1
  30. 0030specialize hbad x
  31. 0031apply hbad
  32. 0032symm
  33. 0033rewrite <- hab_witness at hba_witness
  34. 0034exact hba_witness