PA0033 · theorem

lt_of_le_of_lt

Stable checked-use theorem · independently closed

Weak order followed by strict order remains strict.

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

∀ a. ∀ b. ∀ c. Le(a,b) → Lt(b,c) → Lt(a,c)

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

PD0001 Le PD0002 Lt

3 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall a b c. (exists k. k + a = b) -> (exists k. k + S b = c) -> exists k. k + S a = c

Proof neighborhood

Direct theorem prerequisites

PA0009 add_assoc

Direct theorem dependents

PA0034 beta_half_range_entry_bounds PA007C gauss_signed_half_magnitude_injective PA00AG prime_bounded_square_one_cases PA00C6 odd_signed_division_branch_exact PA00D8 distinct_odd_prime_half_products_ne PA00DP distinct_primes_own_odd_half_scaled_remainder_nonzero PA00DS nonzero_remainder_division_positive_multiple_threshold

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

13 script commands · 7 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.

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)
add assoc
01Fix variables and assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro hab
  5. L5
    intro hbc
02Separate the logical casesL6–7

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

  1. L6
    cases hab
  2. L7
    cases hbc
03Construct an explicit witnessL8–8

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

  1. L8
    exists x1 + x
04Calculate and transport equalitiesL9–9

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

  1. L9
    trans x1 + (x + S a)
05Use earlier factsL10–10

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

  1. L10
    apply add_assoc
06Calculate and transport equalitiesL11–12

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

  1. L11
    rewrite PA4
  2. L12
    rewrite hab_witness
07Use earlier factsL13–13

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

  1. L13
    exact hbc_witness

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro hab
  5. 0005intro hbc
  6. 0006cases hab
  7. 0007cases hbc
  8. 0008exists x1 + x
  9. 0009trans x1 + (x + S a)
  10. 0010apply add_assoc
  11. 0011rewrite PA4
  12. 0012rewrite hab_witness
  13. 0013exact hbc_witness