BA0037

cf_convergent_matrix_entry_transport

Ordinary simultaneous equality transport preserves the actual four terminal matrix entries of the finite computation.

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. ∀ h. ∀ e. ∀ k. ∀ u. ∀ U. ∀ v. ∀ V. ∀ p. ∀ P. ∀ n. ∀ N. u = p → U = P → v = n → V = N → ConvergentMatrixTrace(s,h,e,k,p,P,n,N) → ConvergentMatrixTrace(s,h,e,k,u,U,v,V)

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

Definition DAG

ConvergentMatrixTrace(s,h,e,k,u,U,v,V) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall s h e k u U v V p P n N. u = p -> U = P -> v = n -> V = N -> (exists cfc_tail_entries_source. ((exists cfc_state_entries_sourceinitial. ((exists cfc_left_entries_sourceinitialcode cfc_right_entries_sourceinitialcode cfc_matrix_entries_sourceinitialcode. ((cfc_left_entries_sourceinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_entries_sourceinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_entries_sourceinitialcode = ((cfc_left_entries_sourceinitialcode) + (cfc_right_entries_sourceinitialcode)) * S ((cfc_left_entries_sourceinitialcode) + (cfc_right_entries_sourceinitialcode)) + ((cfc_right_entries_sourceinitialcode) + (cfc_right_entries_sourceinitialcode))) /\ ((cfc_state_entries_sourceinitial) = ((cfc_tail_entries_source) + (cfc_matrix_entries_sourceinitialcode)) * S ((cfc_tail_entries_source) + (cfc_matrix_entries_sourceinitialcode)) + ((cfc_matrix_entries_sourceinitialcode) + (cfc_matrix_entries_sourceinitialcode))))))) /\ (((exists ff_h_entries_sourceinitialentry. ff_h_entries_sourceinitialentry + S (cfc_state_entries_sourceinitial) = S ((S (0)) * e)) /\ exists ff_q_entries_sourceinitialentry. h = ff_q_entries_sourceinitialentry * S ((S (0)) * e) + (cfc_state_entries_sourceinitial))))) /\ ((exists cfc_state_entries_sourceterminal. ((exists cfc_left_entries_sourceterminalcode cfc_right_entries_sourceterminalcode cfc_matrix_entries_sourceterminalcode. ((cfc_left_entries_sourceterminalcode = ((p) + (P)) * S ((p) + (P)) + ((P) + (P))) /\ ((cfc_right_entries_sourceterminalcode = ((n) + (N)) * S ((n) + (N)) + ((N) + (N))) /\ ((cfc_matrix_entries_sourceterminalcode = ((cfc_left_entries_sourceterminalcode) + (cfc_right_entries_sourceterminalcode)) * S ((cfc_left_entries_sourceterminalcode) + (cfc_right_entries_sourceterminalcode)) + ((cfc_right_entries_sourceterminalcode) + (cfc_right_entries_sourceterminalcode))) /\ ((cfc_state_entries_sourceterminal) = ((s) + (cfc_matrix_entries_sourceterminalcode)) * S ((s) + (cfc_matrix_entries_sourceterminalcode)) + ((cfc_matrix_entries_sourceterminalcode) + (cfc_matrix_entries_sourceterminalcode))))))) /\ (((exists ff_h_entries_sourceterminalentry. ff_h_entries_sourceterminalentry + S (cfc_state_entries_sourceterminal) = S ((S (k)) * e)) /\ exists ff_q_entries_sourceterminalentry. h = ff_q_entries_sourceterminalentry * S ((S (k)) * e) + (cfc_state_entries_sourceterminal))))) /\ (forall cfc_index_entries_source. (exists cfba_gap_entries_sourcebound. cfba_gap_entries_sourcebound + S (cfc_index_entries_source) = (k)) -> exists cfc_old_entries_source cfc_a_entries_source cfc_b_entries_source cfc_c_entries_source cfc_d_entries_source cfc_new_entries_source cfc_quotient_entries_source. ((exists cfc_state_entries_sourceprevious. ((exists cfc_left_entries_sourcepreviouscode cfc_right_entries_sourcepreviouscode cfc_matrix_entries_sourcepreviouscode. ((cfc_left_entries_sourcepreviouscode = ((cfc_a_entries_source) + (cfc_b_entries_source)) * S ((cfc_a_entries_source) + (cfc_b_entries_source)) + ((cfc_b_entries_source) + (cfc_b_entries_source))) /\ ((cfc_right_entries_sourcepreviouscode = ((cfc_c_entries_source) + (cfc_d_entries_source)) * S ((cfc_c_entries_source) + (cfc_d_entries_source)) + ((cfc_d_entries_source) + (cfc_d_entries_source))) /\ ((cfc_matrix_entries_sourcepreviouscode = ((cfc_left_entries_sourcepreviouscode) + (cfc_right_entries_sourcepreviouscode)) * S ((cfc_left_entries_sourcepreviouscode) + (cfc_right_entries_sourcepreviouscode)) + ((cfc_right_entries_sourcepreviouscode) + (cfc_right_entries_sourcepreviouscode))) /\ ((cfc_state_entries_sourceprevious) = ((cfc_old_entries_source) + (cfc_matrix_entries_sourcepreviouscode)) * S ((cfc_old_entries_source) + (cfc_matrix_entries_sourcepreviouscode)) + ((cfc_matrix_entries_sourcepreviouscode) + (cfc_matrix_entries_sourcepreviouscode))))))) /\ (((exists ff_h_entries_sourcepreviousentry. ff_h_entries_sourcepreviousentry + S (cfc_state_entries_sourceprevious) = S ((S (cfc_index_entries_source)) * e)) /\ exists ff_q_entries_sourcepreviousentry. h = ff_q_entries_sourcepreviousentry * S ((S (cfc_index_entries_source)) * e) + (cfc_state_entries_sourceprevious))))) /\ ((exists cfc_state_entries_sourcefollowing. ((exists cfc_left_entries_sourcefollowingcode cfc_right_entries_sourcefollowingcode cfc_matrix_entries_sourcefollowingcode. ((cfc_left_entries_sourcefollowingcode = (((cfc_quotient_entries_source * cfc_a_entries_source + cfc_c_entries_source)) + ((cfc_quotient_entries_source * cfc_b_entries_source + cfc_d_entries_source))) * S (((cfc_quotient_entries_source * cfc_a_entries_source + cfc_c_entries_source)) + ((cfc_quotient_entries_source * cfc_b_entries_source + cfc_d_entries_source))) + (((cfc_quotient_entries_source * cfc_b_entries_source + cfc_d_entries_source)) + ((cfc_quotient_entries_source * cfc_b_entries_source + cfc_d_entries_source)))) /\ ((cfc_right_entries_sourcefollowingcode = ((cfc_a_entries_source) + (cfc_b_entries_source)) * S ((cfc_a_entries_source) + (cfc_b_entries_source)) + ((cfc_b_entries_source) + (cfc_b_entries_source))) /\ ((cfc_matrix_entries_sourcefollowingcode = ((cfc_left_entries_sourcefollowingcode) + (cfc_right_entries_sourcefollowingcode)) * S ((cfc_left_entries_sourcefollowingcode) + (cfc_right_entries_sourcefollowingcode)) + ((cfc_right_entries_sourcefollowingcode) + (cfc_right_entries_sourcefollowingcode))) /\ ((cfc_state_entries_sourcefollowing) = ((cfc_new_entries_source) + (cfc_matrix_entries_sourcefollowingcode)) * S ((cfc_new_entries_source) + (cfc_matrix_entries_sourcefollowingcode)) + ((cfc_matrix_entries_sourcefollowingcode) + (cfc_matrix_entries_sourcefollowingcode))))))) /\ (((exists ff_h_entries_sourcefollowingentry. ff_h_entries_sourcefollowingentry + S (cfc_state_entries_sourcefollowing) = S ((S (S cfc_index_entries_source)) * e)) /\ exists ff_q_entries_sourcefollowingentry. h = ff_q_entries_sourcefollowingentry * S ((S (S cfc_index_entries_source)) * e) + (cfc_state_entries_sourcefollowing))))) /\ (cfc_new_entries_source = S ((cfc_quotient_entries_source + cfc_old_entries_source) * S (cfc_quotient_entries_source + cfc_old_entries_source) + (cfc_old_entries_source + cfc_old_entries_source))))))))) -> (exists cfc_tail_entries_target. ((exists cfc_state_entries_targetinitial. ((exists cfc_left_entries_targetinitialcode cfc_right_entries_targetinitialcode cfc_matrix_entries_targetinitialcode. ((cfc_left_entries_targetinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_entries_targetinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_entries_targetinitialcode = ((cfc_left_entries_targetinitialcode) + (cfc_right_entries_targetinitialcode)) * S ((cfc_left_entries_targetinitialcode) + (cfc_right_entries_targetinitialcode)) + ((cfc_right_entries_targetinitialcode) + (cfc_right_entries_targetinitialcode))) /\ ((cfc_state_entries_targetinitial) = ((cfc_tail_entries_target) + (cfc_matrix_entries_targetinitialcode)) * S ((cfc_tail_entries_target) + (cfc_matrix_entries_targetinitialcode)) + ((cfc_matrix_entries_targetinitialcode) + (cfc_matrix_entries_targetinitialcode))))))) /\ (((exists ff_h_entries_targetinitialentry. ff_h_entries_targetinitialentry + S (cfc_state_entries_targetinitial) = S ((S (0)) * e)) /\ exists ff_q_entries_targetinitialentry. h = ff_q_entries_targetinitialentry * S ((S (0)) * e) + (cfc_state_entries_targetinitial))))) /\ ((exists cfc_state_entries_targetterminal. ((exists cfc_left_entries_targetterminalcode cfc_right_entries_targetterminalcode cfc_matrix_entries_targetterminalcode. ((cfc_left_entries_targetterminalcode = ((u) + (U)) * S ((u) + (U)) + ((U) + (U))) /\ ((cfc_right_entries_targetterminalcode = ((v) + (V)) * S ((v) + (V)) + ((V) + (V))) /\ ((cfc_matrix_entries_targetterminalcode = ((cfc_left_entries_targetterminalcode) + (cfc_right_entries_targetterminalcode)) * S ((cfc_left_entries_targetterminalcode) + (cfc_right_entries_targetterminalcode)) + ((cfc_right_entries_targetterminalcode) + (cfc_right_entries_targetterminalcode))) /\ ((cfc_state_entries_targetterminal) = ((s) + (cfc_matrix_entries_targetterminalcode)) * S ((s) + (cfc_matrix_entries_targetterminalcode)) + ((cfc_matrix_entries_targetterminalcode) + (cfc_matrix_entries_targetterminalcode))))))) /\ (((exists ff_h_entries_targetterminalentry. ff_h_entries_targetterminalentry + S (cfc_state_entries_targetterminal) = S ((S (k)) * e)) /\ exists ff_q_entries_targetterminalentry. h = ff_q_entries_targetterminalentry * S ((S (k)) * e) + (cfc_state_entries_targetterminal))))) /\ (forall cfc_index_entries_target. (exists cfba_gap_entries_targetbound. cfba_gap_entries_targetbound + S (cfc_index_entries_target) = (k)) -> exists cfc_old_entries_target cfc_a_entries_target cfc_b_entries_target cfc_c_entries_target cfc_d_entries_target cfc_new_entries_target cfc_quotient_entries_target. ((exists cfc_state_entries_targetprevious. ((exists cfc_left_entries_targetpreviouscode cfc_right_entries_targetpreviouscode cfc_matrix_entries_targetpreviouscode. ((cfc_left_entries_targetpreviouscode = ((cfc_a_entries_target) + (cfc_b_entries_target)) * S ((cfc_a_entries_target) + (cfc_b_entries_target)) + ((cfc_b_entries_target) + (cfc_b_entries_target))) /\ ((cfc_right_entries_targetpreviouscode = ((cfc_c_entries_target) + (cfc_d_entries_target)) * S ((cfc_c_entries_target) + (cfc_d_entries_target)) + ((cfc_d_entries_target) + (cfc_d_entries_target))) /\ ((cfc_matrix_entries_targetpreviouscode = ((cfc_left_entries_targetpreviouscode) + (cfc_right_entries_targetpreviouscode)) * S ((cfc_left_entries_targetpreviouscode) + (cfc_right_entries_targetpreviouscode)) + ((cfc_right_entries_targetpreviouscode) + (cfc_right_entries_targetpreviouscode))) /\ ((cfc_state_entries_targetprevious) = ((cfc_old_entries_target) + (cfc_matrix_entries_targetpreviouscode)) * S ((cfc_old_entries_target) + (cfc_matrix_entries_targetpreviouscode)) + ((cfc_matrix_entries_targetpreviouscode) + (cfc_matrix_entries_targetpreviouscode))))))) /\ (((exists ff_h_entries_targetpreviousentry. ff_h_entries_targetpreviousentry + S (cfc_state_entries_targetprevious) = S ((S (cfc_index_entries_target)) * e)) /\ exists ff_q_entries_targetpreviousentry. h = ff_q_entries_targetpreviousentry * S ((S (cfc_index_entries_target)) * e) + (cfc_state_entries_targetprevious))))) /\ ((exists cfc_state_entries_targetfollowing. ((exists cfc_left_entries_targetfollowingcode cfc_right_entries_targetfollowingcode cfc_matrix_entries_targetfollowingcode. ((cfc_left_entries_targetfollowingcode = (((cfc_quotient_entries_target * cfc_a_entries_target + cfc_c_entries_target)) + ((cfc_quotient_entries_target * cfc_b_entries_target + cfc_d_entries_target))) * S (((cfc_quotient_entries_target * cfc_a_entries_target + cfc_c_entries_target)) + ((cfc_quotient_entries_target * cfc_b_entries_target + cfc_d_entries_target))) + (((cfc_quotient_entries_target * cfc_b_entries_target + cfc_d_entries_target)) + ((cfc_quotient_entries_target * cfc_b_entries_target + cfc_d_entries_target)))) /\ ((cfc_right_entries_targetfollowingcode = ((cfc_a_entries_target) + (cfc_b_entries_target)) * S ((cfc_a_entries_target) + (cfc_b_entries_target)) + ((cfc_b_entries_target) + (cfc_b_entries_target))) /\ ((cfc_matrix_entries_targetfollowingcode = ((cfc_left_entries_targetfollowingcode) + (cfc_right_entries_targetfollowingcode)) * S ((cfc_left_entries_targetfollowingcode) + (cfc_right_entries_targetfollowingcode)) + ((cfc_right_entries_targetfollowingcode) + (cfc_right_entries_targetfollowingcode))) /\ ((cfc_state_entries_targetfollowing) = ((cfc_new_entries_target) + (cfc_matrix_entries_targetfollowingcode)) * S ((cfc_new_entries_target) + (cfc_matrix_entries_targetfollowingcode)) + ((cfc_matrix_entries_targetfollowingcode) + (cfc_matrix_entries_targetfollowingcode))))))) /\ (((exists ff_h_entries_targetfollowingentry. ff_h_entries_targetfollowingentry + S (cfc_state_entries_targetfollowing) = S ((S (S cfc_index_entries_target)) * e)) /\ exists ff_q_entries_targetfollowingentry. h = ff_q_entries_targetfollowingentry * S ((S (S cfc_index_entries_target)) * e) + (cfc_state_entries_targetfollowing))))) /\ (cfc_new_entries_target = S ((cfc_quotient_entries_target + cfc_old_entries_target) * S (cfc_quotient_entries_target + cfc_old_entries_target) + (cfc_old_entries_target + cfc_old_entries_target)))))))))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 5 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 h
  3. L3
    intro e
  4. L4
    intro k
  5. L5
    intro u
  6. L6
    intro U
  7. L7
    intro v
  8. L8
    intro V
  9. L9
    intro p
  10. L10
    intro P
02Fix variables and assumptionsL11–17

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

  1. L11
    intro n
  2. L12
    intro N
  3. L13
    intro hu
  4. L14
    intro hU
  5. L15
    intro hv
  6. L16
    intro hV
  7. L17
    intro ht
03Calculate and transport equalitiesL18–27

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

  1. L18
    rewrite hu
  2. L19
    rewrite hu
  3. L20
    rewrite hU
  4. L21
    rewrite hU
  5. L22
    rewrite hU
  6. L23
    rewrite hU
  7. L24
    rewrite hv
  8. L25
    rewrite hv
  9. L26
    rewrite hV
  10. L27
    rewrite hV
04Calculate and transport equalitiesL28–29

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

  1. L28
    rewrite hV
  2. L29
    rewrite hV
05Use earlier factsL30–30

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

  1. L30
    exact ht

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro s
  2. 0002intro h
  3. 0003intro e
  4. 0004intro k
  5. 0005intro u
  6. 0006intro U
  7. 0007intro v
  8. 0008intro V
  9. 0009intro p
  10. 0010intro P
  11. 0011intro n
  12. 0012intro N
  13. 0013intro hu
  14. 0014intro hU
  15. 0015intro hv
  16. 0016intro hV
  17. 0017intro ht
  18. 0018rewrite hu
  19. 0019rewrite hu
  20. 0020rewrite hU
  21. 0021rewrite hU
  22. 0022rewrite hU
  23. 0023rewrite hU
  24. 0024rewrite hv
  25. 0025rewrite hv
  26. 0026rewrite hV
  27. 0027rewrite hV
  28. 0028rewrite hV
  29. 0029rewrite hV
  30. 0030exact ht