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
∀ p. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ l. ∀ i. PowerQuotPrefix(p,n,b,c,l) → PowerQuotPrefix(p,n + n,d,e,l) → Lt(i,l) → ∃ x. ∃ y. ∃ z. BetaAt(b,c,i,x) ∧ (BetaAt(d,e,i,y) ∧ (z = 0 ∧ y = x + x ∨ z = 1 ∧ y = S (x + x)))Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.
Definitions used by this theorem
In the theorem statement
5 occurrences
In local proof propositions
6 occurrences
Exact expanded native-PA statement
forall p n b c d e l i. (forall bls_index_b5ccqc_left. (exists bls_gap_b5ccqc_left_bound. bls_gap_b5ccqc_left_bound + S (bls_index_b5ccqc_left) = (l)) -> exists bls_power_b5ccqc_left bls_quotient_b5ccqc_left bls_remainder_b5ccqc_left. ((exists bpvi_b_bls_b5ccqc_left_power bpvi_c_bls_b5ccqc_left_power. ((forall bpvi_i_bls_b5ccqc_left_power. (exists bpvi_repeat_gap_bls_b5ccqc_left_power. bpvi_repeat_gap_bls_b5ccqc_left_power + S bpvi_i_bls_b5ccqc_left_power = S bls_index_b5ccqc_left) -> (((exists bpvi_h_bls_b5ccqc_left_power_repeat. bpvi_h_bls_b5ccqc_left_power_repeat + S (p) = S ((S (bpvi_i_bls_b5ccqc_left_power)) * bpvi_c_bls_b5ccqc_left_power)) /\ exists bpvi_q_bls_b5ccqc_left_power_repeat. bpvi_b_bls_b5ccqc_left_power = bpvi_q_bls_b5ccqc_left_power_repeat * S ((S (bpvi_i_bls_b5ccqc_left_power)) * bpvi_c_bls_b5ccqc_left_power) + (p)))) /\ (exists bpvi_u_bls_b5ccqc_left_power bpvi_v_bls_b5ccqc_left_power. ((((exists bpvi_h_bls_b5ccqc_left_power_start. bpvi_h_bls_b5ccqc_left_power_start + S (1) = S ((S (0)) * bpvi_v_bls_b5ccqc_left_power)) /\ exists bpvi_q_bls_b5ccqc_left_power_start. bpvi_u_bls_b5ccqc_left_power = bpvi_q_bls_b5ccqc_left_power_start * S ((S (0)) * bpvi_v_bls_b5ccqc_left_power) + (1))) /\ ((((exists bpvi_h_bls_b5ccqc_left_power_terminal. bpvi_h_bls_b5ccqc_left_power_terminal + S (bls_power_b5ccqc_left) = S ((S (S bls_index_b5ccqc_left)) * bpvi_v_bls_b5ccqc_left_power)) /\ exists bpvi_q_bls_b5ccqc_left_power_terminal. bpvi_u_bls_b5ccqc_left_power = bpvi_q_bls_b5ccqc_left_power_terminal * S ((S (S bls_index_b5ccqc_left)) * bpvi_v_bls_b5ccqc_left_power) + (bls_power_b5ccqc_left))) /\ forall bpvi_j_bls_b5ccqc_left_power. (exists bpvi_product_gap_bls_b5ccqc_left_power. bpvi_product_gap_bls_b5ccqc_left_power + S bpvi_j_bls_b5ccqc_left_power = S bls_index_b5ccqc_left) -> exists bpvi_factor_bls_b5ccqc_left_power bpvi_partial_bls_b5ccqc_left_power bpvi_successor_bls_b5ccqc_left_power. ((((exists bpvi_h_bls_b5ccqc_left_power_factor. bpvi_h_bls_b5ccqc_left_power_factor + S (bpvi_factor_bls_b5ccqc_left_power) = S ((S (bpvi_j_bls_b5ccqc_left_power)) * bpvi_c_bls_b5ccqc_left_power)) /\ exists bpvi_q_bls_b5ccqc_left_power_factor. bpvi_b_bls_b5ccqc_left_power = bpvi_q_bls_b5ccqc_left_power_factor * S ((S (bpvi_j_bls_b5ccqc_left_power)) * bpvi_c_bls_b5ccqc_left_power) + (bpvi_factor_bls_b5ccqc_left_power))) /\ ((((exists bpvi_h_bls_b5ccqc_left_power_partial. bpvi_h_bls_b5ccqc_left_power_partial + S (bpvi_partial_bls_b5ccqc_left_power) = S ((S (bpvi_j_bls_b5ccqc_left_power)) * bpvi_v_bls_b5ccqc_left_power)) /\ exists bpvi_q_bls_b5ccqc_left_power_partial. bpvi_u_bls_b5ccqc_left_power = bpvi_q_bls_b5ccqc_left_power_partial * S ((S (bpvi_j_bls_b5ccqc_left_power)) * bpvi_v_bls_b5ccqc_left_power) + (bpvi_partial_bls_b5ccqc_left_power))) /\ ((((exists bpvi_h_bls_b5ccqc_left_power_successor. bpvi_h_bls_b5ccqc_left_power_successor + S (bpvi_successor_bls_b5ccqc_left_power) = S ((S (S bpvi_j_bls_b5ccqc_left_power)) * bpvi_v_bls_b5ccqc_left_power)) /\ exists bpvi_q_bls_b5ccqc_left_power_successor. bpvi_u_bls_b5ccqc_left_power = bpvi_q_bls_b5ccqc_left_power_successor * S ((S (S bpvi_j_bls_b5ccqc_left_power)) * bpvi_v_bls_b5ccqc_left_power) + (bpvi_successor_bls_b5ccqc_left_power))) /\ bpvi_successor_bls_b5ccqc_left_power = bpvi_partial_bls_b5ccqc_left_power * bpvi_factor_bls_b5ccqc_left_power)))))))) /\ ((((exists ff_h_bls_b5ccqc_left_quotient_entry. ff_h_bls_b5ccqc_left_quotient_entry + S (bls_quotient_b5ccqc_left) = S ((S (bls_index_b5ccqc_left)) * c)) /\ exists ff_q_bls_b5ccqc_left_quotient_entry. b = ff_q_bls_b5ccqc_left_quotient_entry * S ((S (bls_index_b5ccqc_left)) * c) + (bls_quotient_b5ccqc_left))) /\ ((n = bls_power_b5ccqc_left * bls_quotient_b5ccqc_left + bls_remainder_b5ccqc_left /\ exists bls_remainder_gap_b5ccqc_left_division. bls_remainder_gap_b5ccqc_left_division + S (bls_remainder_b5ccqc_left) = bls_power_b5ccqc_left))))) -> (forall bls_index_b5ccqc_right. (exists bls_gap_b5ccqc_right_bound. bls_gap_b5ccqc_right_bound + S (bls_index_b5ccqc_right) = (l)) -> exists bls_power_b5ccqc_right bls_quotient_b5ccqc_right bls_remainder_b5ccqc_right. ((exists bpvi_b_bls_b5ccqc_right_power bpvi_c_bls_b5ccqc_right_power. ((forall bpvi_i_bls_b5ccqc_right_power. (exists bpvi_repeat_gap_bls_b5ccqc_right_power. bpvi_repeat_gap_bls_b5ccqc_right_power + S bpvi_i_bls_b5ccqc_right_power = S bls_index_b5ccqc_right) -> (((exists bpvi_h_bls_b5ccqc_right_power_repeat. bpvi_h_bls_b5ccqc_right_power_repeat + S (p) = S ((S (bpvi_i_bls_b5ccqc_right_power)) * bpvi_c_bls_b5ccqc_right_power)) /\ exists bpvi_q_bls_b5ccqc_right_power_repeat. bpvi_b_bls_b5ccqc_right_power = bpvi_q_bls_b5ccqc_right_power_repeat * S ((S (bpvi_i_bls_b5ccqc_right_power)) * bpvi_c_bls_b5ccqc_right_power) + (p)))) /\ (exists bpvi_u_bls_b5ccqc_right_power bpvi_v_bls_b5ccqc_right_power. ((((exists bpvi_h_bls_b5ccqc_right_power_start. bpvi_h_bls_b5ccqc_right_power_start + S (1) = S ((S (0)) * bpvi_v_bls_b5ccqc_right_power)) /\ exists bpvi_q_bls_b5ccqc_right_power_start. bpvi_u_bls_b5ccqc_right_power = bpvi_q_bls_b5ccqc_right_power_start * S ((S (0)) * bpvi_v_bls_b5ccqc_right_power) + (1))) /\ ((((exists bpvi_h_bls_b5ccqc_right_power_terminal. bpvi_h_bls_b5ccqc_right_power_terminal + S (bls_power_b5ccqc_right) = S ((S (S bls_index_b5ccqc_right)) * bpvi_v_bls_b5ccqc_right_power)) /\ exists bpvi_q_bls_b5ccqc_right_power_terminal. bpvi_u_bls_b5ccqc_right_power = bpvi_q_bls_b5ccqc_right_power_terminal * S ((S (S bls_index_b5ccqc_right)) * bpvi_v_bls_b5ccqc_right_power) + (bls_power_b5ccqc_right))) /\ forall bpvi_j_bls_b5ccqc_right_power. (exists bpvi_product_gap_bls_b5ccqc_right_power. bpvi_product_gap_bls_b5ccqc_right_power + S bpvi_j_bls_b5ccqc_right_power = S bls_index_b5ccqc_right) -> exists bpvi_factor_bls_b5ccqc_right_power bpvi_partial_bls_b5ccqc_right_power bpvi_successor_bls_b5ccqc_right_power. ((((exists bpvi_h_bls_b5ccqc_right_power_factor. bpvi_h_bls_b5ccqc_right_power_factor + S (bpvi_factor_bls_b5ccqc_right_power) = S ((S (bpvi_j_bls_b5ccqc_right_power)) * bpvi_c_bls_b5ccqc_right_power)) /\ exists bpvi_q_bls_b5ccqc_right_power_factor. bpvi_b_bls_b5ccqc_right_power = bpvi_q_bls_b5ccqc_right_power_factor * S ((S (bpvi_j_bls_b5ccqc_right_power)) * bpvi_c_bls_b5ccqc_right_power) + (bpvi_factor_bls_b5ccqc_right_power))) /\ ((((exists bpvi_h_bls_b5ccqc_right_power_partial. bpvi_h_bls_b5ccqc_right_power_partial + S (bpvi_partial_bls_b5ccqc_right_power) = S ((S (bpvi_j_bls_b5ccqc_right_power)) * bpvi_v_bls_b5ccqc_right_power)) /\ exists bpvi_q_bls_b5ccqc_right_power_partial. bpvi_u_bls_b5ccqc_right_power = bpvi_q_bls_b5ccqc_right_power_partial * S ((S (bpvi_j_bls_b5ccqc_right_power)) * bpvi_v_bls_b5ccqc_right_power) + (bpvi_partial_bls_b5ccqc_right_power))) /\ ((((exists bpvi_h_bls_b5ccqc_right_power_successor. bpvi_h_bls_b5ccqc_right_power_successor + S (bpvi_successor_bls_b5ccqc_right_power) = S ((S (S bpvi_j_bls_b5ccqc_right_power)) * bpvi_v_bls_b5ccqc_right_power)) /\ exists bpvi_q_bls_b5ccqc_right_power_successor. bpvi_u_bls_b5ccqc_right_power = bpvi_q_bls_b5ccqc_right_power_successor * S ((S (S bpvi_j_bls_b5ccqc_right_power)) * bpvi_v_bls_b5ccqc_right_power) + (bpvi_successor_bls_b5ccqc_right_power))) /\ bpvi_successor_bls_b5ccqc_right_power = bpvi_partial_bls_b5ccqc_right_power * bpvi_factor_bls_b5ccqc_right_power)))))))) /\ ((((exists ff_h_bls_b5ccqc_right_quotient_entry. ff_h_bls_b5ccqc_right_quotient_entry + S (bls_quotient_b5ccqc_right) = S ((S (bls_index_b5ccqc_right)) * e)) /\ exists ff_q_bls_b5ccqc_right_quotient_entry. d = ff_q_bls_b5ccqc_right_quotient_entry * S ((S (bls_index_b5ccqc_right)) * e) + (bls_quotient_b5ccqc_right))) /\ ((n + n = bls_power_b5ccqc_right * bls_quotient_b5ccqc_right + bls_remainder_b5ccqc_right /\ exists bls_remainder_gap_b5ccqc_right_division. bls_remainder_gap_b5ccqc_right_division + S (bls_remainder_b5ccqc_right) = bls_power_b5ccqc_right))))) -> (exists bcf_lt_gap_b5ccqc_bound. bcf_lt_gap_b5ccqc_bound + S (i) = l) -> (exists q Q bit. (((exists fs_h_b5ccqc_result_left. fs_h_b5ccqc_result_left + S (q) = S ((S (i)) * c)) /\ exists fs_q_b5ccqc_result_left. b = fs_q_b5ccqc_result_left * S ((S (i)) * c) + (q))) /\ ((((exists fs_h_b5ccqc_result_right. fs_h_b5ccqc_result_right + S (Q) = S ((S (i)) * e)) /\ exists fs_q_b5ccqc_result_right. d = fs_q_b5ccqc_result_right * S ((S (i)) * e) + (Q))) /\ (((bit = 0 /\ Q = q + q) \/ (bit = 1 /\ Q = S (q + q))))))Proof neighborhood
Direct theorem prerequisites
Direct theorem dependents
Definition-aware tactic body
Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.
Read the argument
Proof checkpoints
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.
Named ingredients (2)
01Fix variables and assumptionsL1–10
02Fix variables and assumptionsL11–11
Work with arbitrary variables or the premises of the current implication.
- L11
intro hi
03Establish hleft_dataL12–15
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hleft.
- L12
have hleft_data : ∃ D. ∃ q. ∃ r. Pow(p,S i,D) ∧ (BetaAt(b,c,i,q) ∧ DivRem(n,D,q,r))Definitions: Pow(p,S i,D)BetaAt(b,c,i,q)DivRem(n,D,q,r)Original native command in the exact edition - L13
specialize hleft i - L14
apply hleft - L15
exact hi
04Separate the logical casesL16–20
05Establish hright_dataL21–24
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hright.
- L21
have hright_data : ∃ D. ∃ q. ∃ r. Pow(p,S i,D) ∧ (BetaAt(d,e,i,q) ∧ DivRem(n + n,D,q,r))Definitions: Pow(p,S i,D)BetaAt(d,e,i,q)DivRem(n + n,D,q,r)Original native command in the exact edition - L22
specialize hright i - L23
apply hright - L24
exact hi
06Separate the logical casesL25–29
07Establish hpower_eqL30–39
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow functional.
- L30
have hpower_eq : x = x3 - L31
specialize pow_functional p - L32
specialize pow_functional (S i) - L33
specialize pow_functional x - L34
specialize pow_functional x3 - L35
apply pow_functional - L36
exact hleft_data_witness_witness_witness_left - L37
exact hright_data_witness_witness_witness_left - L38
rewrite <- hpower_eq at hright_data_witness_witness_witness_right_right - L39
rewrite <- hpower_eq at hright_data_witness_witness_witness_right_right
08Establish hcarryL40–49
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division double quotient bit.
- L40
have hcarry : x4 = x1 + x1 \/ x4 = S (x1 + x1) - L41
specialize division_double_quotient_bit x - L42
specialize division_double_quotient_bit n - L43
specialize division_double_quotient_bit x1 - L44
specialize division_double_quotient_bit x2 - L45
specialize division_double_quotient_bit x4 - L46
specialize division_double_quotient_bit x5 - L47
apply division_double_quotient_bit - L48
exact hleft_data_witness_witness_witness_right_right - L49
exact hright_data_witness_witness_witness_right_right
09Separate the logical casesL50–50
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L50
cases hcarry
10Construct an explicit witnessL51–53
11Separate the logical casesL54–54
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L54
split
12Use earlier factsL55–55
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L55
exact hleft_data_witness_witness_witness_right_left
13Separate the logical casesL56–56
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L56
split
14Use earlier factsL57–57
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L57
exact hright_data_witness_witness_witness_right_left
15Separate the logical casesL58–59
16Calculate and transport equalitiesL60–60
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L60
refl
17Use earlier factsL61–61
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L61
exact hcarry_left
18Construct an explicit witnessL62–64
19Separate the logical casesL65–65
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L65
split
20Use earlier factsL66–66
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L66
exact hleft_data_witness_witness_witness_right_left
21Separate the logical casesL67–67
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L67
split
22Use earlier factsL68–68
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L68
exact hright_data_witness_witness_witness_right_left
23Separate the logical casesL69–70
24Calculate and transport equalitiesL71–71
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L71
refl
25Use earlier factsL72–72
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L72
exact hcarry_right
Original defined command ledger · 72 lines
- 0001
intro p - 0002
intro n - 0003
intro b - 0004
intro c - 0005
intro d - 0006
intro e - 0007
intro l - 0008
intro i - 0009
intro hleft - 0010
intro hright - 0011
intro hi - 0012
have hleft_data : ∃ D. ∃ q. ∃ r. Pow(p,S i,D) ∧ (BetaAt(b,c,i,q) ∧ DivRem(n,D,q,r))Exact native replay line
have hleft_data : exists D q r. (exists bpvi_b_b5ccqc_left_power bpvi_c_b5ccqc_left_power. ((forall bpvi_i_b5ccqc_left_power. (exists bpvi_repeat_gap_b5ccqc_left_power. bpvi_repeat_gap_b5ccqc_left_power + S bpvi_i_b5ccqc_left_power = S i) -> (((exists bpvi_h_b5ccqc_left_power_repeat. bpvi_h_b5ccqc_left_power_repeat + S (p) = S ((S (bpvi_i_b5ccqc_left_power)) * bpvi_c_b5ccqc_left_power)) /\ exists bpvi_q_b5ccqc_left_power_repeat. bpvi_b_b5ccqc_left_power = bpvi_q_b5ccqc_left_power_repeat * S ((S (bpvi_i_b5ccqc_left_power)) * bpvi_c_b5ccqc_left_power) + (p)))) /\ (exists bpvi_u_b5ccqc_left_power bpvi_v_b5ccqc_left_power. ((((exists bpvi_h_b5ccqc_left_power_start. bpvi_h_b5ccqc_left_power_start + S (1) = S ((S (0)) * bpvi_v_b5ccqc_left_power)) /\ exists bpvi_q_b5ccqc_left_power_start. bpvi_u_b5ccqc_left_power = bpvi_q_b5ccqc_left_power_start * S ((S (0)) * bpvi_v_b5ccqc_left_power) + (1))) /\ ((((exists bpvi_h_b5ccqc_left_power_terminal. bpvi_h_b5ccqc_left_power_terminal + S (D) = S ((S (S i)) * bpvi_v_b5ccqc_left_power)) /\ exists bpvi_q_b5ccqc_left_power_terminal. bpvi_u_b5ccqc_left_power = bpvi_q_b5ccqc_left_power_terminal * S ((S (S i)) * bpvi_v_b5ccqc_left_power) + (D))) /\ forall bpvi_j_b5ccqc_left_power. (exists bpvi_product_gap_b5ccqc_left_power. bpvi_product_gap_b5ccqc_left_power + S bpvi_j_b5ccqc_left_power = S i) -> exists bpvi_factor_b5ccqc_left_power bpvi_partial_b5ccqc_left_power bpvi_successor_b5ccqc_left_power. ((((exists bpvi_h_b5ccqc_left_power_factor. bpvi_h_b5ccqc_left_power_factor + S (bpvi_factor_b5ccqc_left_power) = S ((S (bpvi_j_b5ccqc_left_power)) * bpvi_c_b5ccqc_left_power)) /\ exists bpvi_q_b5ccqc_left_power_factor. bpvi_b_b5ccqc_left_power = bpvi_q_b5ccqc_left_power_factor * S ((S (bpvi_j_b5ccqc_left_power)) * bpvi_c_b5ccqc_left_power) + (bpvi_factor_b5ccqc_left_power))) /\ ((((exists bpvi_h_b5ccqc_left_power_partial. bpvi_h_b5ccqc_left_power_partial + S (bpvi_partial_b5ccqc_left_power) = S ((S (bpvi_j_b5ccqc_left_power)) * bpvi_v_b5ccqc_left_power)) /\ exists bpvi_q_b5ccqc_left_power_partial. bpvi_u_b5ccqc_left_power = bpvi_q_b5ccqc_left_power_partial * S ((S (bpvi_j_b5ccqc_left_power)) * bpvi_v_b5ccqc_left_power) + (bpvi_partial_b5ccqc_left_power))) /\ ((((exists bpvi_h_b5ccqc_left_power_successor. bpvi_h_b5ccqc_left_power_successor + S (bpvi_successor_b5ccqc_left_power) = S ((S (S bpvi_j_b5ccqc_left_power)) * bpvi_v_b5ccqc_left_power)) /\ exists bpvi_q_b5ccqc_left_power_successor. bpvi_u_b5ccqc_left_power = bpvi_q_b5ccqc_left_power_successor * S ((S (S bpvi_j_b5ccqc_left_power)) * bpvi_v_b5ccqc_left_power) + (bpvi_successor_b5ccqc_left_power))) /\ bpvi_successor_b5ccqc_left_power = bpvi_partial_b5ccqc_left_power * bpvi_factor_b5ccqc_left_power)))))))) /\ ((((exists fs_h_b5ccqc_left_entry. fs_h_b5ccqc_left_entry + S (q) = S ((S (i)) * c)) /\ exists fs_q_b5ccqc_left_entry. b = fs_q_b5ccqc_left_entry * S ((S (i)) * c) + (q))) /\ (((n) = (D) * (q) + (r) /\ (exists bcf_lt_gap_b5ccqc_left_division_bound. bcf_lt_gap_b5ccqc_left_division_bound + S (r) = D)))) - 0013
specialize hleft i - 0014
apply hleft - 0015
exact hi - 0016
cases hleft_data - 0017
cases hleft_data_witness - 0018
cases hleft_data_witness_witness - 0019
cases hleft_data_witness_witness_witness - 0020
cases hleft_data_witness_witness_witness_right - 0021
have hright_data : ∃ D. ∃ q. ∃ r. Pow(p,S i,D) ∧ (BetaAt(d,e,i,q) ∧ DivRem(n + n,D,q,r))Exact native replay line
have hright_data : exists D q r. (exists bpvi_b_b5ccqc_right_power bpvi_c_b5ccqc_right_power. ((forall bpvi_i_b5ccqc_right_power. (exists bpvi_repeat_gap_b5ccqc_right_power. bpvi_repeat_gap_b5ccqc_right_power + S bpvi_i_b5ccqc_right_power = S i) -> (((exists bpvi_h_b5ccqc_right_power_repeat. bpvi_h_b5ccqc_right_power_repeat + S (p) = S ((S (bpvi_i_b5ccqc_right_power)) * bpvi_c_b5ccqc_right_power)) /\ exists bpvi_q_b5ccqc_right_power_repeat. bpvi_b_b5ccqc_right_power = bpvi_q_b5ccqc_right_power_repeat * S ((S (bpvi_i_b5ccqc_right_power)) * bpvi_c_b5ccqc_right_power) + (p)))) /\ (exists bpvi_u_b5ccqc_right_power bpvi_v_b5ccqc_right_power. ((((exists bpvi_h_b5ccqc_right_power_start. bpvi_h_b5ccqc_right_power_start + S (1) = S ((S (0)) * bpvi_v_b5ccqc_right_power)) /\ exists bpvi_q_b5ccqc_right_power_start. bpvi_u_b5ccqc_right_power = bpvi_q_b5ccqc_right_power_start * S ((S (0)) * bpvi_v_b5ccqc_right_power) + (1))) /\ ((((exists bpvi_h_b5ccqc_right_power_terminal. bpvi_h_b5ccqc_right_power_terminal + S (D) = S ((S (S i)) * bpvi_v_b5ccqc_right_power)) /\ exists bpvi_q_b5ccqc_right_power_terminal. bpvi_u_b5ccqc_right_power = bpvi_q_b5ccqc_right_power_terminal * S ((S (S i)) * bpvi_v_b5ccqc_right_power) + (D))) /\ forall bpvi_j_b5ccqc_right_power. (exists bpvi_product_gap_b5ccqc_right_power. bpvi_product_gap_b5ccqc_right_power + S bpvi_j_b5ccqc_right_power = S i) -> exists bpvi_factor_b5ccqc_right_power bpvi_partial_b5ccqc_right_power bpvi_successor_b5ccqc_right_power. ((((exists bpvi_h_b5ccqc_right_power_factor. bpvi_h_b5ccqc_right_power_factor + S (bpvi_factor_b5ccqc_right_power) = S ((S (bpvi_j_b5ccqc_right_power)) * bpvi_c_b5ccqc_right_power)) /\ exists bpvi_q_b5ccqc_right_power_factor. bpvi_b_b5ccqc_right_power = bpvi_q_b5ccqc_right_power_factor * S ((S (bpvi_j_b5ccqc_right_power)) * bpvi_c_b5ccqc_right_power) + (bpvi_factor_b5ccqc_right_power))) /\ ((((exists bpvi_h_b5ccqc_right_power_partial. bpvi_h_b5ccqc_right_power_partial + S (bpvi_partial_b5ccqc_right_power) = S ((S (bpvi_j_b5ccqc_right_power)) * bpvi_v_b5ccqc_right_power)) /\ exists bpvi_q_b5ccqc_right_power_partial. bpvi_u_b5ccqc_right_power = bpvi_q_b5ccqc_right_power_partial * S ((S (bpvi_j_b5ccqc_right_power)) * bpvi_v_b5ccqc_right_power) + (bpvi_partial_b5ccqc_right_power))) /\ ((((exists bpvi_h_b5ccqc_right_power_successor. bpvi_h_b5ccqc_right_power_successor + S (bpvi_successor_b5ccqc_right_power) = S ((S (S bpvi_j_b5ccqc_right_power)) * bpvi_v_b5ccqc_right_power)) /\ exists bpvi_q_b5ccqc_right_power_successor. bpvi_u_b5ccqc_right_power = bpvi_q_b5ccqc_right_power_successor * S ((S (S bpvi_j_b5ccqc_right_power)) * bpvi_v_b5ccqc_right_power) + (bpvi_successor_b5ccqc_right_power))) /\ bpvi_successor_b5ccqc_right_power = bpvi_partial_b5ccqc_right_power * bpvi_factor_b5ccqc_right_power)))))))) /\ ((((exists fs_h_b5ccqc_right_entry. fs_h_b5ccqc_right_entry + S (q) = S ((S (i)) * e)) /\ exists fs_q_b5ccqc_right_entry. d = fs_q_b5ccqc_right_entry * S ((S (i)) * e) + (q))) /\ (((n + n) = (D) * (q) + (r) /\ (exists bcf_lt_gap_b5ccqc_right_division_bound. bcf_lt_gap_b5ccqc_right_division_bound + S (r) = D)))) - 0022
specialize hright i - 0023
apply hright - 0024
exact hi - 0025
cases hright_data - 0026
cases hright_data_witness - 0027
cases hright_data_witness_witness - 0028
cases hright_data_witness_witness_witness - 0029
cases hright_data_witness_witness_witness_right - 0030
have hpower_eq : x = x3 - 0031
specialize pow_functional p - 0032
specialize pow_functional (S i) - 0033
specialize pow_functional x - 0034
specialize pow_functional x3 - 0035
apply pow_functional - 0036
exact hleft_data_witness_witness_witness_left - 0037
exact hright_data_witness_witness_witness_left - 0038
rewrite <- hpower_eq at hright_data_witness_witness_witness_right_right - 0039
rewrite <- hpower_eq at hright_data_witness_witness_witness_right_right - 0040
have hcarry : x4 = x1 + x1 \/ x4 = S (x1 + x1) - 0041
specialize division_double_quotient_bit x - 0042
specialize division_double_quotient_bit n - 0043
specialize division_double_quotient_bit x1 - 0044
specialize division_double_quotient_bit x2 - 0045
specialize division_double_quotient_bit x4 - 0046
specialize division_double_quotient_bit x5 - 0047
apply division_double_quotient_bit - 0048
exact hleft_data_witness_witness_witness_right_right - 0049
exact hright_data_witness_witness_witness_right_right - 0050
cases hcarry - 0051
exists x1 - 0052
exists x4 - 0053
exists 0 - 0054
split - 0055
exact hleft_data_witness_witness_witness_right_left - 0056
split - 0057
exact hright_data_witness_witness_witness_right_left - 0058
left - 0059
split - 0060
refl - 0061
exact hcarry_left - 0062
exists x1 - 0063
exists x4 - 0064
exists 1 - 0065
split - 0066
exact hleft_data_witness_witness_witness_right_left - 0067
split - 0068
exact hright_data_witness_witness_witness_right_left - 0069
right - 0070
split - 0071
refl - 0072
exact hcarry_right