FS001V · theorem body

four_square_descent_add_le_add

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

Two explicitly witnessed natural weak inequalities add constructively.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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. ∀ d. Le(a,b) → Le(c,d) → Le(a + c,b + d)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0001 Le

In local proof propositions

none
Exact expanded first-order statement
forall a b c d. (exists gap. gap + a = b) -> (exists gap. gap + c = d) -> exists gap. gap + (a + c) = b + d

Proof neighborhood

Direct theorem prerequisites

add_shuffle_middle · Alpha closed

Direct theorem dependents

FS001X four_square_descent_odd_half_norm_strict

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

14 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.

01Fix variables and assumptionsL1–6

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 hfirst
  6. L6
    intro hsecond
02Separate the logical casesL7–8

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

  1. L7
    cases hfirst
  2. L8
    cases hsecond
03Construct an explicit witnessL9–9

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

  1. L9
    exists x + x1
04Calculate and transport equalitiesL10–10

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

  1. L10
    trans (x + a) + (x1 + c)
05Use earlier factsL11–11

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

  1. L11
    apply add_shuffle_middle
06Calculate and transport equalitiesL12–12

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

  1. L12
    congr
07Use earlier factsL13–14

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

  1. L13
    exact hfirst_witness
  2. L14
    exact hsecond_witness

Library-wide reading audit

Original defined command ledger · 14 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro hfirst
  6. 0006intro hsecond
  7. 0007cases hfirst
  8. 0008cases hsecond
  9. 0009exists x + x1
  10. 0010trans (x + a) + (x1 + c)
  11. 0011apply add_shuffle_middle
  12. 0012congr
  13. 0013exact hfirst_witness
  14. 0014exact hsecond_witness