PA000X · theorem

lt_to_le

Stable checked-use theorem · independently closed

A witnessed strict inequality entails the corresponding 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.

Statement with defined notation

∀ a. ∀ b. Lt(a,b) → Le(a,b)

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

2 occurrences

In local proof propositions

none

0 occurrences

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

Proof neighborhood

Direct theorem prerequisites

PA000E add_succ_left

Direct theorem dependents

PA001U beta_exclusive_accumulated_product_step PA002G beta_exclusive_recode_congruence_step PA008B prime_mul_index_map_exists_up_to PA008S prime_scaled_inverse_prefix_exists_bounded PA009Q pair_index_left_below_double PA00A6 prime_inverse_prefix_exists_bounded PA00D9 distinct_odd_prime_half_cell_oriented

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

11 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 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
03Construct an explicit witnessL5–5

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

  1. L5
    exists S x
04Calculate and transport equalitiesL6–7

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

  1. L6
    trans x + S a
  2. L7
    trans S (x + a)
05Use earlier factsL8–8

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

  1. L8
    apply add_succ_left
06Calculate and transport equalitiesL9–9

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

  1. L9
    symm
07Use earlier factsL10–11

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

  1. L10
    apply PA4
  2. L11
    exact h_witness

Library-wide reading audit

Original defined command ledger · 11 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004cases h
  5. 0005exists S x
  6. 0006trans x + S a
  7. 0007trans S (x + a)
  8. 0008apply add_succ_left
  9. 0009symm
  10. 0010apply PA4
  11. 0011exact h_witness