DL006C

integer_span_pair_equal_transitive

Equality of signed natural pairs is transitive by genuine cancellative arithmetic, not equality of the chosen positive and negative components.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

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.

This branch proves the finite determinant/rank/span substrate. It does not claim Smith or Hermite normal form, lattice index equals determinant, determinant multiplicativity, lattice reduction, or geometry-of-numbers theorems.

Exact theorem in conservative defined notation

∀ a. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. a + d = c + b → c + f = e + d → a + f = e + b

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

none

Actual proof prerequisites

add_right_cancel · checked external prerequisitefour_square_euler_add_swap_last · checked external prerequisite
Original expanded first-order statement 
forall a b c d e f. a + d = c + b -> c + f = e + d -> a + f = e + b

Complete tactic proof in conservative notation

All 25 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

25 script commands · 12 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.

01Fix variables and assumptionsL1–8

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 d
  5. L5
    intro e
  6. L6
    intro f
  7. L7
    intro hfirst
  8. L8
    intro hsecond
02Use earlier factsL9–12

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

  1. L9
    specialize add_right_cancel (a + f)
  2. L10
    specialize add_right_cancel (e + b)
  3. L11
    specialize add_right_cancel (d)
  4. L12
    apply add_right_cancel
03Calculate and transport equalitiesL13–13

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

  1. L13
    trans (a + d) + f
04Use earlier factsL14–14

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

  1. L14
    apply four_square_euler_add_swap_last
05Calculate and transport equalitiesL15–16

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

  1. L15
    trans (c + b) + f
  2. L16
    congr
06Use earlier factsL17–17

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

  1. L17
    exact hfirst
07Calculate and transport equalitiesL18–19

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

  1. L18
    refl
  2. L19
    trans (c + f) + b
08Use earlier factsL20–20

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

  1. L20
    apply four_square_euler_add_swap_last
09Calculate and transport equalitiesL21–22

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

  1. L21
    trans (e + d) + b
  2. L22
    congr
10Use earlier factsL23–23

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

  1. L23
    exact hsecond
11Calculate and transport equalitiesL24–24

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

  1. L24
    refl
12Use earlier factsL25–25

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

  1. L25
    apply four_square_euler_add_swap_last

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007intro hfirst
  8. 0008intro hsecond
  9. 0009specialize add_right_cancel (a + f)
  10. 0010specialize add_right_cancel (e + b)
  11. 0011specialize add_right_cancel (d)
  12. 0012apply add_right_cancel
  13. 0013trans (a + d) + f
  14. 0014apply four_square_euler_add_swap_last
  15. 0015trans (c + b) + f
  16. 0016congr
  17. 0017exact hfirst
  18. 0018refl
  19. 0019trans (c + f) + b
  20. 0020apply four_square_euler_add_swap_last
  21. 0021trans (e + d) + b
  22. 0022congr
  23. 0023exact hsecond
  24. 0024refl
  25. 0025apply four_square_euler_add_swap_last