BA002E

cf_convergent_state_code_constructor

Conservative existential sharing constructs the exact five-coordinate state without adding a pairing function to HA.

Alpha v34 checked-use · first admitted v29 · 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.

The initial 0/1 convergent is included: u is natural, not necessarily positive. Comparison denominators are strictly smaller and positive. Signed competitors are represented by an arbitrary difference rp−rn. Approximation inequalities are proved from the trace, never stored as assumptions in Convergent.

Exact theorem in conservative defined notation

∀ s. ∀ u. ∀ U. ∀ v. ∀ V. ∀ l. ∀ r. ∀ m. ∀ z. NaturalPair(l,u,U) → NaturalPair(r,v,V) → NaturalPair(m,l,r) → NaturalPair(z,s,m) → ConvergentMatrixCode(s,u,U,v,V,z)

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

Definition DAG

NaturalPair(z,a,b) · 4ConvergentMatrixCode(s,u,U,v,V,z) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall s u U v V l r m z. l = ((u) + (U)) * S ((u) + (U)) + ((U) + (U)) -> r = ((v) + (V)) * S ((v) + (V)) + ((V) + (V)) -> m = ((l) + (r)) * S ((l) + (r)) + ((r) + (r)) -> z = ((s) + (m)) * S ((s) + (m)) + ((m) + (m)) -> (exists cfc_left_state_constructor cfc_right_state_constructor cfc_matrix_state_constructor. ((cfc_left_state_constructor = ((u) + (U)) * S ((u) + (U)) + ((U) + (U))) /\ ((cfc_right_state_constructor = ((v) + (V)) * S ((v) + (V)) + ((V) + (V))) /\ ((cfc_matrix_state_constructor = ((cfc_left_state_constructor) + (cfc_right_state_constructor)) * S ((cfc_left_state_constructor) + (cfc_right_state_constructor)) + ((cfc_right_state_constructor) + (cfc_right_state_constructor))) /\ ((z) = ((s) + (cfc_matrix_state_constructor)) * S ((s) + (cfc_matrix_state_constructor)) + ((cfc_matrix_state_constructor) + (cfc_matrix_state_constructor)))))))

Complete tactic proof in conservative notation

All 23 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

23 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.

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 assumptionsL1–10

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro s
  2. L2
    intro u
  3. L3
    intro U
  4. L4
    intro v
  5. L5
    intro V
  6. L6
    intro l
  7. L7
    intro r
  8. L8
    intro m
  9. L9
    intro z
  10. L10
    intro hl
02Fix variables and assumptionsL11–13

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro hr
  2. L12
    intro hm
  3. L13
    intro hz
03Construct an explicit witnessL14–16

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

  1. L14
    exists l
  2. L15
    exists r
  3. L16
    exists m
04Separate the logical casesL17–17

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

  1. L17
    split
05Use earlier factsL18–18

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

  1. L18
    exact hl
06Separate the logical casesL19–19

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

  1. L19
    split
07Use earlier factsL20–20

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

  1. L20
    exact hr
08Separate the logical casesL21–21

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

  1. L21
    split
09Use earlier factsL22–23

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

  1. L22
    exact hm
  2. L23
    exact hz

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro s
  2. 0002intro u
  3. 0003intro U
  4. 0004intro v
  5. 0005intro V
  6. 0006intro l
  7. 0007intro r
  8. 0008intro m
  9. 0009intro z
  10. 0010intro hl
  11. 0011intro hr
  12. 0012intro hm
  13. 0013intro hz
  14. 0014exists l
  15. 0015exists r
  16. 0016exists m
  17. 0017split
  18. 0018exact hl
  19. 0019split
  20. 0020exact hr
  21. 0021split
  22. 0022exact hm
  23. 0023exact hz