PA003K

add_le_add_left

Stable checked-use theorem · independently closed

Adding the same left summand preserves the witness-defined order.

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 + a = b) -> exists r. r + (c + a) = c + b

Structural proof guide

Generated structural guide

Adding the same left summand preserves the witness-defined order.

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

The proof proceeds by case analysis (1).

Referenced ingredients

PA0009 add_assoc PA000F add_comm

Proof neighborhood

Direct dependencies

PA0009 add_assoc PA000F add_comm

Direct dependents

PA007C gauss_signed_half_magnitude_injective PA009Q pair_index_left_below_double PA009R pair_index_right_below_double

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

18 script commands · 11 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 comm
01Fix variables and assumptionsL1–4

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 h
02Separate the logical casesL5–5

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

  1. L5
    cases h
03Construct an explicit witnessL6–6

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

  1. L6
    exists x
04Calculate and transport equalitiesL7–8

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

  1. L7
    trans (x + c) + a
  2. L8
    symm
05Use earlier factsL9–9

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

  1. L9
    apply add_assoc
06Calculate and transport equalitiesL10–11

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

  1. L10
    trans (c + x) + a
  2. L11
    congr
07Use earlier factsL12–12

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

  1. L12
    apply add_comm
08Calculate and transport equalitiesL13–14

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

  1. L13
    refl
  2. L14
    trans c + (x + a)
09Use earlier factsL15–15

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

  1. L15
    apply add_assoc
10Calculate and transport equalitiesL16–17

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

  1. L16
    congr
  2. L17
    refl
11Use earlier factsL18–18

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

  1. L18
    exact h_witness

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro h
  5. 0005cases h
  6. 0006exists x
  7. 0007trans (x + c) + a
  8. 0008symm
  9. 0009apply add_assoc
  10. 0010trans (c + x) + a
  11. 0011congr
  12. 0012apply add_comm
  13. 0013refl
  14. 0014trans c + (x + a)
  15. 0015apply add_assoc
  16. 0016congr
  17. 0017refl
  18. 0018exact h_witness