PA006O

lt_trans

Stable checked-use theorem · independently closed

Strict order is transitive.

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

Structural proof guide

Generated structural guide

Strict order is transitive.

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

The proof proceeds by case analysis (2).

Referenced ingredients

PA0009 add_assoc PA000E add_succ_left

Proof neighborhood

Direct dependencies

PA0009 add_assoc PA000E add_succ_left

Direct dependents

PA00DS nonzero_remainder_division_positive_multiple_threshold

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

20 script commands · 9 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 (2)
add assocadd succ left
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 + S 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 + (S 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 equalitiesL11–13

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

  1. L11
    trans x1 + S (x + S a)
  2. L12
    congr
  3. L13
    refl
07Use earlier factsL14–14

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

  1. L14
    apply add_succ_left
08Calculate and transport equalitiesL15–18

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

  1. L15
    trans x1 + S b
  2. L16
    congr
  3. L17
    refl
  4. L18
    congr
09Use earlier factsL19–20

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

  1. L19
    exact hab_witness
  2. L20
    exact hbc_witness

Library-wide reading audit

Original exact command ledger · 20 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 + S x
  9. 0009trans x1 + (S x + S a)
  10. 0010apply add_assoc
  11. 0011trans x1 + S (x + S a)
  12. 0012congr
  13. 0013refl
  14. 0014apply add_succ_left
  15. 0015trans x1 + S b
  16. 0016congr
  17. 0017refl
  18. 0018congr
  19. 0019exact hab_witness
  20. 0020exact hbc_witness