BA004E

continued_fraction_initial_convergent_is_first_quotient

The uniquely determined zeroth convergent is the actual first quotient divided by one; this includes the zero-numerator boundary.

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. ∀ q. ∀ t. ∀ u. ∀ v. ListCell(s,q,t) → Convergent(s,0,u,v) → u = q ∧ v = 1

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

Definition DAG

ListCell(s,q,t) · 1Convergent(s,i,u,v) · 1

Actual proof prerequisites

continued_fraction_first_cell_is_initial_convergentcontinued_fraction_convergent_functional
Original expanded first-order statement 
forall s q t u v. (s = S ((q + t) * S (q + t) + (t + t))) -> (exists cfc_previous_numerator_initial_unique_given cfc_previous_denominator_initial_unique_given cfc_code_initial_unique_given cfc_scale_initial_unique_given. ((~(v = 0)) /\ (exists cfc_tail_initial_unique_givencomputation. ((exists cfc_state_initial_unique_givencomputationinitial. ((exists cfc_left_initial_unique_givencomputationinitialcode cfc_right_initial_unique_givencomputationinitialcode cfc_matrix_initial_unique_givencomputationinitialcode. ((cfc_left_initial_unique_givencomputationinitialcode = ((1) + (0)) * S ((1) + (0)) + ((0) + (0))) /\ ((cfc_right_initial_unique_givencomputationinitialcode = ((0) + (1)) * S ((0) + (1)) + ((1) + (1))) /\ ((cfc_matrix_initial_unique_givencomputationinitialcode = ((cfc_left_initial_unique_givencomputationinitialcode) + (cfc_right_initial_unique_givencomputationinitialcode)) * S ((cfc_left_initial_unique_givencomputationinitialcode) + (cfc_right_initial_unique_givencomputationinitialcode)) + ((cfc_right_initial_unique_givencomputationinitialcode) + (cfc_right_initial_unique_givencomputationinitialcode))) /\ ((cfc_state_initial_unique_givencomputationinitial) = ((cfc_tail_initial_unique_givencomputation) + (cfc_matrix_initial_unique_givencomputationinitialcode)) * S ((cfc_tail_initial_unique_givencomputation) + (cfc_matrix_initial_unique_givencomputationinitialcode)) + ((cfc_matrix_initial_unique_givencomputationinitialcode) + (cfc_matrix_initial_unique_givencomputationinitialcode))))))) /\ (((exists ff_h_initial_unique_givencomputationinitialentry. ff_h_initial_unique_givencomputationinitialentry + S (cfc_state_initial_unique_givencomputationinitial) = S ((S (0)) * cfc_scale_initial_unique_given)) /\ exists ff_q_initial_unique_givencomputationinitialentry. cfc_code_initial_unique_given = ff_q_initial_unique_givencomputationinitialentry * S ((S (0)) * cfc_scale_initial_unique_given) + (cfc_state_initial_unique_givencomputationinitial))))) /\ ((exists cfc_state_initial_unique_givencomputationterminal. ((exists cfc_left_initial_unique_givencomputationterminalcode cfc_right_initial_unique_givencomputationterminalcode cfc_matrix_initial_unique_givencomputationterminalcode. ((cfc_left_initial_unique_givencomputationterminalcode = ((u) + (cfc_previous_numerator_initial_unique_given)) * S ((u) + (cfc_previous_numerator_initial_unique_given)) + ((cfc_previous_numerator_initial_unique_given) + (cfc_previous_numerator_initial_unique_given))) /\ ((cfc_right_initial_unique_givencomputationterminalcode = ((v) + (cfc_previous_denominator_initial_unique_given)) * S ((v) + (cfc_previous_denominator_initial_unique_given)) + ((cfc_previous_denominator_initial_unique_given) + (cfc_previous_denominator_initial_unique_given))) /\ ((cfc_matrix_initial_unique_givencomputationterminalcode = ((cfc_left_initial_unique_givencomputationterminalcode) + (cfc_right_initial_unique_givencomputationterminalcode)) * S ((cfc_left_initial_unique_givencomputationterminalcode) + (cfc_right_initial_unique_givencomputationterminalcode)) + ((cfc_right_initial_unique_givencomputationterminalcode) + (cfc_right_initial_unique_givencomputationterminalcode))) /\ ((cfc_state_initial_unique_givencomputationterminal) = ((s) + (cfc_matrix_initial_unique_givencomputationterminalcode)) * S ((s) + (cfc_matrix_initial_unique_givencomputationterminalcode)) + ((cfc_matrix_initial_unique_givencomputationterminalcode) + (cfc_matrix_initial_unique_givencomputationterminalcode))))))) /\ (((exists ff_h_initial_unique_givencomputationterminalentry. ff_h_initial_unique_givencomputationterminalentry + S (cfc_state_initial_unique_givencomputationterminal) = S ((S (S (0))) * cfc_scale_initial_unique_given)) /\ exists ff_q_initial_unique_givencomputationterminalentry. cfc_code_initial_unique_given = ff_q_initial_unique_givencomputationterminalentry * S ((S (S (0))) * cfc_scale_initial_unique_given) + (cfc_state_initial_unique_givencomputationterminal))))) /\ (forall cfc_index_initial_unique_givencomputation. (exists cfba_gap_initial_unique_givencomputationbound. cfba_gap_initial_unique_givencomputationbound + S (cfc_index_initial_unique_givencomputation) = (S (0))) -> exists cfc_old_initial_unique_givencomputation cfc_a_initial_unique_givencomputation cfc_b_initial_unique_givencomputation cfc_c_initial_unique_givencomputation cfc_d_initial_unique_givencomputation cfc_new_initial_unique_givencomputation cfc_quotient_initial_unique_givencomputation. ((exists cfc_state_initial_unique_givencomputationprevious. ((exists cfc_left_initial_unique_givencomputationpreviouscode cfc_right_initial_unique_givencomputationpreviouscode cfc_matrix_initial_unique_givencomputationpreviouscode. ((cfc_left_initial_unique_givencomputationpreviouscode = ((cfc_a_initial_unique_givencomputation) + (cfc_b_initial_unique_givencomputation)) * S ((cfc_a_initial_unique_givencomputation) + (cfc_b_initial_unique_givencomputation)) + ((cfc_b_initial_unique_givencomputation) + (cfc_b_initial_unique_givencomputation))) /\ ((cfc_right_initial_unique_givencomputationpreviouscode = ((cfc_c_initial_unique_givencomputation) + (cfc_d_initial_unique_givencomputation)) * S ((cfc_c_initial_unique_givencomputation) + (cfc_d_initial_unique_givencomputation)) + ((cfc_d_initial_unique_givencomputation) + (cfc_d_initial_unique_givencomputation))) /\ ((cfc_matrix_initial_unique_givencomputationpreviouscode = ((cfc_left_initial_unique_givencomputationpreviouscode) + (cfc_right_initial_unique_givencomputationpreviouscode)) * S ((cfc_left_initial_unique_givencomputationpreviouscode) + (cfc_right_initial_unique_givencomputationpreviouscode)) + ((cfc_right_initial_unique_givencomputationpreviouscode) + (cfc_right_initial_unique_givencomputationpreviouscode))) /\ ((cfc_state_initial_unique_givencomputationprevious) = ((cfc_old_initial_unique_givencomputation) + (cfc_matrix_initial_unique_givencomputationpreviouscode)) * S ((cfc_old_initial_unique_givencomputation) + (cfc_matrix_initial_unique_givencomputationpreviouscode)) + ((cfc_matrix_initial_unique_givencomputationpreviouscode) + (cfc_matrix_initial_unique_givencomputationpreviouscode))))))) /\ (((exists ff_h_initial_unique_givencomputationpreviousentry. ff_h_initial_unique_givencomputationpreviousentry + S (cfc_state_initial_unique_givencomputationprevious) = S ((S (cfc_index_initial_unique_givencomputation)) * cfc_scale_initial_unique_given)) /\ exists ff_q_initial_unique_givencomputationpreviousentry. cfc_code_initial_unique_given = ff_q_initial_unique_givencomputationpreviousentry * S ((S (cfc_index_initial_unique_givencomputation)) * cfc_scale_initial_unique_given) + (cfc_state_initial_unique_givencomputationprevious))))) /\ ((exists cfc_state_initial_unique_givencomputationfollowing. ((exists cfc_left_initial_unique_givencomputationfollowingcode cfc_right_initial_unique_givencomputationfollowingcode cfc_matrix_initial_unique_givencomputationfollowingcode. ((cfc_left_initial_unique_givencomputationfollowingcode = (((cfc_quotient_initial_unique_givencomputation * cfc_a_initial_unique_givencomputation + cfc_c_initial_unique_givencomputation)) + ((cfc_quotient_initial_unique_givencomputation * cfc_b_initial_unique_givencomputation + cfc_d_initial_unique_givencomputation))) * S (((cfc_quotient_initial_unique_givencomputation * cfc_a_initial_unique_givencomputation + cfc_c_initial_unique_givencomputation)) + ((cfc_quotient_initial_unique_givencomputation * cfc_b_initial_unique_givencomputation + cfc_d_initial_unique_givencomputation))) + (((cfc_quotient_initial_unique_givencomputation * cfc_b_initial_unique_givencomputation + cfc_d_initial_unique_givencomputation)) + ((cfc_quotient_initial_unique_givencomputation * cfc_b_initial_unique_givencomputation + cfc_d_initial_unique_givencomputation)))) /\ ((cfc_right_initial_unique_givencomputationfollowingcode = ((cfc_a_initial_unique_givencomputation) + (cfc_b_initial_unique_givencomputation)) * S ((cfc_a_initial_unique_givencomputation) + (cfc_b_initial_unique_givencomputation)) + ((cfc_b_initial_unique_givencomputation) + (cfc_b_initial_unique_givencomputation))) /\ ((cfc_matrix_initial_unique_givencomputationfollowingcode = ((cfc_left_initial_unique_givencomputationfollowingcode) + (cfc_right_initial_unique_givencomputationfollowingcode)) * S ((cfc_left_initial_unique_givencomputationfollowingcode) + (cfc_right_initial_unique_givencomputationfollowingcode)) + ((cfc_right_initial_unique_givencomputationfollowingcode) + (cfc_right_initial_unique_givencomputationfollowingcode))) /\ ((cfc_state_initial_unique_givencomputationfollowing) = ((cfc_new_initial_unique_givencomputation) + (cfc_matrix_initial_unique_givencomputationfollowingcode)) * S ((cfc_new_initial_unique_givencomputation) + (cfc_matrix_initial_unique_givencomputationfollowingcode)) + ((cfc_matrix_initial_unique_givencomputationfollowingcode) + (cfc_matrix_initial_unique_givencomputationfollowingcode))))))) /\ (((exists ff_h_initial_unique_givencomputationfollowingentry. ff_h_initial_unique_givencomputationfollowingentry + S (cfc_state_initial_unique_givencomputationfollowing) = S ((S (S cfc_index_initial_unique_givencomputation)) * cfc_scale_initial_unique_given)) /\ exists ff_q_initial_unique_givencomputationfollowingentry. cfc_code_initial_unique_given = ff_q_initial_unique_givencomputationfollowingentry * S ((S (S cfc_index_initial_unique_givencomputation)) * cfc_scale_initial_unique_given) + (cfc_state_initial_unique_givencomputationfollowing))))) /\ (cfc_new_initial_unique_givencomputation = S ((cfc_quotient_initial_unique_givencomputation + cfc_old_initial_unique_givencomputation) * S (cfc_quotient_initial_unique_givencomputation + cfc_old_initial_unique_givencomputation) + (cfc_old_initial_unique_givencomputation + cfc_old_initial_unique_givencomputation))))))))))) -> u = q /\ v = 1

Complete tactic proof in conservative notation

All 20 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

20 script commands · 3 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.

Named ingredients (2)
continued fraction convergent functionalcontinued fraction first cell is initial convergent
01Fix variables and assumptionsL1–7

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

  1. L1
    intro s
  2. L2
    intro q
  3. L3
    intro t
  4. L4
    intro u
  5. L5
    intro v
  6. L6
    intro hcell
  7. L7
    intro hc
02Use earlier factsL8–17

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

  1. L8
    specialize continued_fraction_convergent_functional (s)
  2. L9
    specialize continued_fraction_convergent_functional (0)
  3. L10
    specialize continued_fraction_convergent_functional (u)
  4. L11
    specialize continued_fraction_convergent_functional (v)
  5. L12
    specialize continued_fraction_convergent_functional (q)
  6. L13
    specialize continued_fraction_convergent_functional (1)
  7. L14
    apply continued_fraction_convergent_functional
  8. L15
    exact hc
  9. L16
    specialize continued_fraction_first_cell_is_initial_convergent (s)
  10. L17
    specialize continued_fraction_first_cell_is_initial_convergent (q)
03Use earlier factsL18–20

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

  1. L18
    specialize continued_fraction_first_cell_is_initial_convergent (t)
  2. L19
    apply continued_fraction_first_cell_is_initial_convergent
  3. L20
    exact hcell

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro s
  2. 0002intro q
  3. 0003intro t
  4. 0004intro u
  5. 0005intro v
  6. 0006intro hcell
  7. 0007intro hc
  8. 0008specialize continued_fraction_convergent_functional (s)
  9. 0009specialize continued_fraction_convergent_functional (0)
  10. 0010specialize continued_fraction_convergent_functional (u)
  11. 0011specialize continued_fraction_convergent_functional (v)
  12. 0012specialize continued_fraction_convergent_functional (q)
  13. 0013specialize continued_fraction_convergent_functional (1)
  14. 0014apply continued_fraction_convergent_functional
  15. 0015exact hc
  16. 0016specialize continued_fraction_first_cell_is_initial_convergent (s)
  17. 0017specialize continued_fraction_first_cell_is_initial_convergent (q)
  18. 0018specialize continued_fraction_first_cell_is_initial_convergent (t)
  19. 0019apply continued_fraction_first_cell_is_initial_convergent
  20. 0020exact hcell