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. ∀ l. ∀ u. ∀ U. ∀ v. ∀ V. k = l → ConvergentMatrixTrace(s,h,e,k,u,U,v,V) → ConvergentMatrixTrace(s,h,e,l,u,U,v,V)Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
none
Original expanded first-order statement
forall s h e k l u U v V. k = l -> (exists cfc_tail_matrix_length_source. ((exists cfc_state_matrix_length_sourceinitial. ((exists cfc_left_matrix_length_sourceinitialcode cfc_right_matrix_length_sourceinitialcode cfc_matrix_matrix_length_sourceinitialcode. ((cfc_left_matrix_length_sourceinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_matrix_length_sourceinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_matrix_length_sourceinitialcode = ((cfc_left_matrix_length_sourceinitialcode) + (cfc_right_matrix_length_sourceinitialcode)) * S ((cfc_left_matrix_length_sourceinitialcode) + (cfc_right_matrix_length_sourceinitialcode)) + ((cfc_right_matrix_length_sourceinitialcode) + (cfc_right_matrix_length_sourceinitialcode))) /\ ((cfc_state_matrix_length_sourceinitial) = ((cfc_tail_matrix_length_source) + (cfc_matrix_matrix_length_sourceinitialcode)) * S ((cfc_tail_matrix_length_source) + (cfc_matrix_matrix_length_sourceinitialcode)) + ((cfc_matrix_matrix_length_sourceinitialcode) + (cfc_matrix_matrix_length_sourceinitialcode))))))) /\ (((exists ff_h_matrix_length_sourceinitialentry. ff_h_matrix_length_sourceinitialentry + S (cfc_state_matrix_length_sourceinitial) = S ((S (0)) * e)) /\ exists ff_q_matrix_length_sourceinitialentry. h = ff_q_matrix_length_sourceinitialentry * S ((S (0)) * e) + (cfc_state_matrix_length_sourceinitial))))) /\ ((exists cfc_state_matrix_length_sourceterminal. ((exists cfc_left_matrix_length_sourceterminalcode cfc_right_matrix_length_sourceterminalcode cfc_matrix_matrix_length_sourceterminalcode. ((cfc_left_matrix_length_sourceterminalcode = ((u) + (U)) * S ((u) + (U)) + ((U) + (U))) /\ ((cfc_right_matrix_length_sourceterminalcode = ((v) + (V)) * S ((v) + (V)) + ((V) + (V))) /\ ((cfc_matrix_matrix_length_sourceterminalcode = ((cfc_left_matrix_length_sourceterminalcode) + (cfc_right_matrix_length_sourceterminalcode)) * S ((cfc_left_matrix_length_sourceterminalcode) + (cfc_right_matrix_length_sourceterminalcode)) + ((cfc_right_matrix_length_sourceterminalcode) + (cfc_right_matrix_length_sourceterminalcode))) /\ ((cfc_state_matrix_length_sourceterminal) = ((s) + (cfc_matrix_matrix_length_sourceterminalcode)) * S ((s) + (cfc_matrix_matrix_length_sourceterminalcode)) + ((cfc_matrix_matrix_length_sourceterminalcode) + (cfc_matrix_matrix_length_sourceterminalcode))))))) /\ (((exists ff_h_matrix_length_sourceterminalentry. ff_h_matrix_length_sourceterminalentry + S (cfc_state_matrix_length_sourceterminal) = S ((S (k)) * e)) /\ exists ff_q_matrix_length_sourceterminalentry. h = ff_q_matrix_length_sourceterminalentry * S ((S (k)) * e) + (cfc_state_matrix_length_sourceterminal))))) /\ (forall cfc_index_matrix_length_source. (exists cfba_gap_matrix_length_sourcebound. cfba_gap_matrix_length_sourcebound + S (cfc_index_matrix_length_source) = (k)) -> exists cfc_old_matrix_length_source cfc_a_matrix_length_source cfc_b_matrix_length_source cfc_c_matrix_length_source cfc_d_matrix_length_source cfc_new_matrix_length_source cfc_quotient_matrix_length_source. ((exists cfc_state_matrix_length_sourceprevious. ((exists cfc_left_matrix_length_sourcepreviouscode cfc_right_matrix_length_sourcepreviouscode cfc_matrix_matrix_length_sourcepreviouscode. ((cfc_left_matrix_length_sourcepreviouscode = ((cfc_a_matrix_length_source) + (cfc_b_matrix_length_source)) * S ((cfc_a_matrix_length_source) + (cfc_b_matrix_length_source)) + ((cfc_b_matrix_length_source) + (cfc_b_matrix_length_source))) /\ ((cfc_right_matrix_length_sourcepreviouscode = ((cfc_c_matrix_length_source) + (cfc_d_matrix_length_source)) * S ((cfc_c_matrix_length_source) + (cfc_d_matrix_length_source)) + ((cfc_d_matrix_length_source) + (cfc_d_matrix_length_source))) /\ ((cfc_matrix_matrix_length_sourcepreviouscode = ((cfc_left_matrix_length_sourcepreviouscode) + (cfc_right_matrix_length_sourcepreviouscode)) * S ((cfc_left_matrix_length_sourcepreviouscode) + (cfc_right_matrix_length_sourcepreviouscode)) + ((cfc_right_matrix_length_sourcepreviouscode) + (cfc_right_matrix_length_sourcepreviouscode))) /\ ((cfc_state_matrix_length_sourceprevious) = ((cfc_old_matrix_length_source) + (cfc_matrix_matrix_length_sourcepreviouscode)) * S ((cfc_old_matrix_length_source) + (cfc_matrix_matrix_length_sourcepreviouscode)) + ((cfc_matrix_matrix_length_sourcepreviouscode) + (cfc_matrix_matrix_length_sourcepreviouscode))))))) /\ (((exists ff_h_matrix_length_sourcepreviousentry. ff_h_matrix_length_sourcepreviousentry + S (cfc_state_matrix_length_sourceprevious) = S ((S (cfc_index_matrix_length_source)) * e)) /\ exists ff_q_matrix_length_sourcepreviousentry. h = ff_q_matrix_length_sourcepreviousentry * S ((S (cfc_index_matrix_length_source)) * e) + (cfc_state_matrix_length_sourceprevious))))) /\ ((exists cfc_state_matrix_length_sourcefollowing. ((exists cfc_left_matrix_length_sourcefollowingcode cfc_right_matrix_length_sourcefollowingcode cfc_matrix_matrix_length_sourcefollowingcode. ((cfc_left_matrix_length_sourcefollowingcode = (((cfc_quotient_matrix_length_source * cfc_a_matrix_length_source + cfc_c_matrix_length_source)) + ((cfc_quotient_matrix_length_source * cfc_b_matrix_length_source + cfc_d_matrix_length_source))) * S (((cfc_quotient_matrix_length_source * cfc_a_matrix_length_source + cfc_c_matrix_length_source)) + ((cfc_quotient_matrix_length_source * cfc_b_matrix_length_source + cfc_d_matrix_length_source))) + (((cfc_quotient_matrix_length_source * cfc_b_matrix_length_source + cfc_d_matrix_length_source)) + ((cfc_quotient_matrix_length_source * cfc_b_matrix_length_source + cfc_d_matrix_length_source)))) /\ ((cfc_right_matrix_length_sourcefollowingcode = ((cfc_a_matrix_length_source) + (cfc_b_matrix_length_source)) * S ((cfc_a_matrix_length_source) + (cfc_b_matrix_length_source)) + ((cfc_b_matrix_length_source) + (cfc_b_matrix_length_source))) /\ ((cfc_matrix_matrix_length_sourcefollowingcode = ((cfc_left_matrix_length_sourcefollowingcode) + (cfc_right_matrix_length_sourcefollowingcode)) * S ((cfc_left_matrix_length_sourcefollowingcode) + (cfc_right_matrix_length_sourcefollowingcode)) + ((cfc_right_matrix_length_sourcefollowingcode) + (cfc_right_matrix_length_sourcefollowingcode))) /\ ((cfc_state_matrix_length_sourcefollowing) = ((cfc_new_matrix_length_source) + (cfc_matrix_matrix_length_sourcefollowingcode)) * S ((cfc_new_matrix_length_source) + (cfc_matrix_matrix_length_sourcefollowingcode)) + ((cfc_matrix_matrix_length_sourcefollowingcode) + (cfc_matrix_matrix_length_sourcefollowingcode))))))) /\ (((exists ff_h_matrix_length_sourcefollowingentry. ff_h_matrix_length_sourcefollowingentry + S (cfc_state_matrix_length_sourcefollowing) = S ((S (S cfc_index_matrix_length_source)) * e)) /\ exists ff_q_matrix_length_sourcefollowingentry. h = ff_q_matrix_length_sourcefollowingentry * S ((S (S cfc_index_matrix_length_source)) * e) + (cfc_state_matrix_length_sourcefollowing))))) /\ (cfc_new_matrix_length_source = S ((cfc_quotient_matrix_length_source + cfc_old_matrix_length_source) * S (cfc_quotient_matrix_length_source + cfc_old_matrix_length_source) + (cfc_old_matrix_length_source + cfc_old_matrix_length_source))))))))) -> (exists cfc_tail_matrix_length_target. ((exists cfc_state_matrix_length_targetinitial. ((exists cfc_left_matrix_length_targetinitialcode cfc_right_matrix_length_targetinitialcode cfc_matrix_matrix_length_targetinitialcode. ((cfc_left_matrix_length_targetinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_matrix_length_targetinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_matrix_length_targetinitialcode = ((cfc_left_matrix_length_targetinitialcode) + (cfc_right_matrix_length_targetinitialcode)) * S ((cfc_left_matrix_length_targetinitialcode) + (cfc_right_matrix_length_targetinitialcode)) + ((cfc_right_matrix_length_targetinitialcode) + (cfc_right_matrix_length_targetinitialcode))) /\ ((cfc_state_matrix_length_targetinitial) = ((cfc_tail_matrix_length_target) + (cfc_matrix_matrix_length_targetinitialcode)) * S ((cfc_tail_matrix_length_target) + (cfc_matrix_matrix_length_targetinitialcode)) + ((cfc_matrix_matrix_length_targetinitialcode) + (cfc_matrix_matrix_length_targetinitialcode))))))) /\ (((exists ff_h_matrix_length_targetinitialentry. ff_h_matrix_length_targetinitialentry + S (cfc_state_matrix_length_targetinitial) = S ((S (0)) * e)) /\ exists ff_q_matrix_length_targetinitialentry. h = ff_q_matrix_length_targetinitialentry * S ((S (0)) * e) + (cfc_state_matrix_length_targetinitial))))) /\ ((exists cfc_state_matrix_length_targetterminal. ((exists cfc_left_matrix_length_targetterminalcode cfc_right_matrix_length_targetterminalcode cfc_matrix_matrix_length_targetterminalcode. ((cfc_left_matrix_length_targetterminalcode = ((u) + (U)) * S ((u) + (U)) + ((U) + (U))) /\ ((cfc_right_matrix_length_targetterminalcode = ((v) + (V)) * S ((v) + (V)) + ((V) + (V))) /\ ((cfc_matrix_matrix_length_targetterminalcode = ((cfc_left_matrix_length_targetterminalcode) + (cfc_right_matrix_length_targetterminalcode)) * S ((cfc_left_matrix_length_targetterminalcode) + (cfc_right_matrix_length_targetterminalcode)) + ((cfc_right_matrix_length_targetterminalcode) + (cfc_right_matrix_length_targetterminalcode))) /\ ((cfc_state_matrix_length_targetterminal) = ((s) + (cfc_matrix_matrix_length_targetterminalcode)) * S ((s) + (cfc_matrix_matrix_length_targetterminalcode)) + ((cfc_matrix_matrix_length_targetterminalcode) + (cfc_matrix_matrix_length_targetterminalcode))))))) /\ (((exists ff_h_matrix_length_targetterminalentry. ff_h_matrix_length_targetterminalentry + S (cfc_state_matrix_length_targetterminal) = S ((S (l)) * e)) /\ exists ff_q_matrix_length_targetterminalentry. h = ff_q_matrix_length_targetterminalentry * S ((S (l)) * e) + (cfc_state_matrix_length_targetterminal))))) /\ (forall cfc_index_matrix_length_target. (exists cfba_gap_matrix_length_targetbound. cfba_gap_matrix_length_targetbound + S (cfc_index_matrix_length_target) = (l)) -> exists cfc_old_matrix_length_target cfc_a_matrix_length_target cfc_b_matrix_length_target cfc_c_matrix_length_target cfc_d_matrix_length_target cfc_new_matrix_length_target cfc_quotient_matrix_length_target. ((exists cfc_state_matrix_length_targetprevious. ((exists cfc_left_matrix_length_targetpreviouscode cfc_right_matrix_length_targetpreviouscode cfc_matrix_matrix_length_targetpreviouscode. ((cfc_left_matrix_length_targetpreviouscode = ((cfc_a_matrix_length_target) + (cfc_b_matrix_length_target)) * S ((cfc_a_matrix_length_target) + (cfc_b_matrix_length_target)) + ((cfc_b_matrix_length_target) + (cfc_b_matrix_length_target))) /\ ((cfc_right_matrix_length_targetpreviouscode = ((cfc_c_matrix_length_target) + (cfc_d_matrix_length_target)) * S ((cfc_c_matrix_length_target) + (cfc_d_matrix_length_target)) + ((cfc_d_matrix_length_target) + (cfc_d_matrix_length_target))) /\ ((cfc_matrix_matrix_length_targetpreviouscode = ((cfc_left_matrix_length_targetpreviouscode) + (cfc_right_matrix_length_targetpreviouscode)) * S ((cfc_left_matrix_length_targetpreviouscode) + (cfc_right_matrix_length_targetpreviouscode)) + ((cfc_right_matrix_length_targetpreviouscode) + (cfc_right_matrix_length_targetpreviouscode))) /\ ((cfc_state_matrix_length_targetprevious) = ((cfc_old_matrix_length_target) + (cfc_matrix_matrix_length_targetpreviouscode)) * S ((cfc_old_matrix_length_target) + (cfc_matrix_matrix_length_targetpreviouscode)) + ((cfc_matrix_matrix_length_targetpreviouscode) + (cfc_matrix_matrix_length_targetpreviouscode))))))) /\ (((exists ff_h_matrix_length_targetpreviousentry. ff_h_matrix_length_targetpreviousentry + S (cfc_state_matrix_length_targetprevious) = S ((S (cfc_index_matrix_length_target)) * e)) /\ exists ff_q_matrix_length_targetpreviousentry. h = ff_q_matrix_length_targetpreviousentry * S ((S (cfc_index_matrix_length_target)) * e) + (cfc_state_matrix_length_targetprevious))))) /\ ((exists cfc_state_matrix_length_targetfollowing. ((exists cfc_left_matrix_length_targetfollowingcode cfc_right_matrix_length_targetfollowingcode cfc_matrix_matrix_length_targetfollowingcode. ((cfc_left_matrix_length_targetfollowingcode = (((cfc_quotient_matrix_length_target * cfc_a_matrix_length_target + cfc_c_matrix_length_target)) + ((cfc_quotient_matrix_length_target * cfc_b_matrix_length_target + cfc_d_matrix_length_target))) * S (((cfc_quotient_matrix_length_target * cfc_a_matrix_length_target + cfc_c_matrix_length_target)) + ((cfc_quotient_matrix_length_target * cfc_b_matrix_length_target + cfc_d_matrix_length_target))) + (((cfc_quotient_matrix_length_target * cfc_b_matrix_length_target + cfc_d_matrix_length_target)) + ((cfc_quotient_matrix_length_target * cfc_b_matrix_length_target + cfc_d_matrix_length_target)))) /\ ((cfc_right_matrix_length_targetfollowingcode = ((cfc_a_matrix_length_target) + (cfc_b_matrix_length_target)) * S ((cfc_a_matrix_length_target) + (cfc_b_matrix_length_target)) + ((cfc_b_matrix_length_target) + (cfc_b_matrix_length_target))) /\ ((cfc_matrix_matrix_length_targetfollowingcode = ((cfc_left_matrix_length_targetfollowingcode) + (cfc_right_matrix_length_targetfollowingcode)) * S ((cfc_left_matrix_length_targetfollowingcode) + (cfc_right_matrix_length_targetfollowingcode)) + ((cfc_right_matrix_length_targetfollowingcode) + (cfc_right_matrix_length_targetfollowingcode))) /\ ((cfc_state_matrix_length_targetfollowing) = ((cfc_new_matrix_length_target) + (cfc_matrix_matrix_length_targetfollowingcode)) * S ((cfc_new_matrix_length_target) + (cfc_matrix_matrix_length_targetfollowingcode)) + ((cfc_matrix_matrix_length_targetfollowingcode) + (cfc_matrix_matrix_length_targetfollowingcode))))))) /\ (((exists ff_h_matrix_length_targetfollowingentry. ff_h_matrix_length_targetfollowingentry + S (cfc_state_matrix_length_targetfollowing) = S ((S (S cfc_index_matrix_length_target)) * e)) /\ exists ff_q_matrix_length_targetfollowingentry. h = ff_q_matrix_length_targetfollowingentry * S ((S (S cfc_index_matrix_length_target)) * e) + (cfc_state_matrix_length_targetfollowing))))) /\ (cfc_new_matrix_length_target = S ((cfc_quotient_matrix_length_target + cfc_old_matrix_length_target) * S (cfc_quotient_matrix_length_target + cfc_old_matrix_length_target) + (cfc_old_matrix_length_target + cfc_old_matrix_length_target)))))))))Complete tactic proof in conservative notation
All 15 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
15 script commands · 4 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
02Fix variables and assumptionsL11–11
Work with arbitrary variables or the premises of the current implication.
- L11
intro ht
03Calculate and transport equalitiesL12–14
04Use earlier factsL15–15
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L15
exact ht