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
∀ m. ∀ x. ∀ y. (∀ z. ∀ n. ∃ k. Pow(z,n,k)) → Pow(6,10 · m,x) → Pow(4,13 · m,y) → Le(x,y)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
4 occurrences
In local proof propositions
4 occurrences
Exact expanded native-PA statement
forall m x y. (forall bpt_a_hj32_six_block bpt_e_hj32_six_block. exists bpt_x_hj32_six_block. (exists ff_b_bpt_value_hj32_six_block ff_c_bpt_value_hj32_six_block. ((forall ff_i_bpt_value_hj32_six_block_repeat. (exists ff_lt_bpt_value_hj32_six_block_repeat_bound. ff_lt_bpt_value_hj32_six_block_repeat_bound + S ff_i_bpt_value_hj32_six_block_repeat = bpt_e_hj32_six_block) -> (((exists ff_h_bpt_value_hj32_six_block_repeat_decoded. ff_h_bpt_value_hj32_six_block_repeat_decoded + S (bpt_a_hj32_six_block) = S ((S (ff_i_bpt_value_hj32_six_block_repeat)) * ff_c_bpt_value_hj32_six_block)) /\ exists ff_q_bpt_value_hj32_six_block_repeat_decoded. ff_b_bpt_value_hj32_six_block = ff_q_bpt_value_hj32_six_block_repeat_decoded * S ((S (ff_i_bpt_value_hj32_six_block_repeat)) * ff_c_bpt_value_hj32_six_block) + (bpt_a_hj32_six_block)))) /\ (exists ff_u_bpt_value_hj32_six_block_product ff_v_bpt_value_hj32_six_block_product. ((((exists ff_h_bpt_value_hj32_six_block_product_start. ff_h_bpt_value_hj32_six_block_product_start + S (1) = S ((S (0)) * ff_v_bpt_value_hj32_six_block_product)) /\ exists ff_q_bpt_value_hj32_six_block_product_start. ff_u_bpt_value_hj32_six_block_product = ff_q_bpt_value_hj32_six_block_product_start * S ((S (0)) * ff_v_bpt_value_hj32_six_block_product) + (1))) /\ ((((exists ff_h_bpt_value_hj32_six_block_product_terminal. ff_h_bpt_value_hj32_six_block_product_terminal + S (bpt_x_hj32_six_block) = S ((S (bpt_e_hj32_six_block)) * ff_v_bpt_value_hj32_six_block_product)) /\ exists ff_q_bpt_value_hj32_six_block_product_terminal. ff_u_bpt_value_hj32_six_block_product = ff_q_bpt_value_hj32_six_block_product_terminal * S ((S (bpt_e_hj32_six_block)) * ff_v_bpt_value_hj32_six_block_product) + (bpt_x_hj32_six_block))) /\ forall ff_i_bpt_value_hj32_six_block_product. (exists ff_lt_bpt_value_hj32_six_block_product_bound. ff_lt_bpt_value_hj32_six_block_product_bound + S ff_i_bpt_value_hj32_six_block_product = bpt_e_hj32_six_block) -> exists ff_p_bpt_value_hj32_six_block_product ff_r_bpt_value_hj32_six_block_product ff_s_bpt_value_hj32_six_block_product. ((((exists ff_h_bpt_value_hj32_six_block_product_factor. ff_h_bpt_value_hj32_six_block_product_factor + S (ff_p_bpt_value_hj32_six_block_product) = S ((S (ff_i_bpt_value_hj32_six_block_product)) * ff_c_bpt_value_hj32_six_block)) /\ exists ff_q_bpt_value_hj32_six_block_product_factor. ff_b_bpt_value_hj32_six_block = ff_q_bpt_value_hj32_six_block_product_factor * S ((S (ff_i_bpt_value_hj32_six_block_product)) * ff_c_bpt_value_hj32_six_block) + (ff_p_bpt_value_hj32_six_block_product))) /\ ((((exists ff_h_bpt_value_hj32_six_block_product_partial. ff_h_bpt_value_hj32_six_block_product_partial + S (ff_r_bpt_value_hj32_six_block_product) = S ((S (ff_i_bpt_value_hj32_six_block_product)) * ff_v_bpt_value_hj32_six_block_product)) /\ exists ff_q_bpt_value_hj32_six_block_product_partial. ff_u_bpt_value_hj32_six_block_product = ff_q_bpt_value_hj32_six_block_product_partial * S ((S (ff_i_bpt_value_hj32_six_block_product)) * ff_v_bpt_value_hj32_six_block_product) + (ff_r_bpt_value_hj32_six_block_product))) /\ ((((exists ff_h_bpt_value_hj32_six_block_product_successor. ff_h_bpt_value_hj32_six_block_product_successor + S (ff_s_bpt_value_hj32_six_block_product) = S ((S (S ff_i_bpt_value_hj32_six_block_product)) * ff_v_bpt_value_hj32_six_block_product)) /\ exists ff_q_bpt_value_hj32_six_block_product_successor. ff_u_bpt_value_hj32_six_block_product = ff_q_bpt_value_hj32_six_block_product_successor * S ((S (S ff_i_bpt_value_hj32_six_block_product)) * ff_v_bpt_value_hj32_six_block_product) + (ff_s_bpt_value_hj32_six_block_product))) /\ ff_s_bpt_value_hj32_six_block_product = ff_r_bpt_value_hj32_six_block_product * ff_p_bpt_value_hj32_six_block_product))))))))) -> (exists pa_b_hj32_six_block_left pa_c_hj32_six_block_left. ((forall pa_i_hj32_six_block_left_repeat. (exists pa_lt_hj32_six_block_left_repeat_bound. pa_lt_hj32_six_block_left_repeat_bound + S pa_i_hj32_six_block_left_repeat = 10 * m) -> (((exists pa_h_hj32_six_block_left_repeat_decoded. pa_h_hj32_six_block_left_repeat_decoded + S (6) = S ((S (pa_i_hj32_six_block_left_repeat)) * pa_c_hj32_six_block_left)) /\ exists pa_q_hj32_six_block_left_repeat_decoded. pa_b_hj32_six_block_left = pa_q_hj32_six_block_left_repeat_decoded * S ((S (pa_i_hj32_six_block_left_repeat)) * pa_c_hj32_six_block_left) + (6)))) /\ (exists pa_u_hj32_six_block_left_product pa_v_hj32_six_block_left_product. ((((exists pa_h_hj32_six_block_left_product_start. pa_h_hj32_six_block_left_product_start + S (1) = S ((S (0)) * pa_v_hj32_six_block_left_product)) /\ exists pa_q_hj32_six_block_left_product_start. pa_u_hj32_six_block_left_product = pa_q_hj32_six_block_left_product_start * S ((S (0)) * pa_v_hj32_six_block_left_product) + (1))) /\ ((((exists pa_h_hj32_six_block_left_product_terminal. pa_h_hj32_six_block_left_product_terminal + S (x) = S ((S (10 * m)) * pa_v_hj32_six_block_left_product)) /\ exists pa_q_hj32_six_block_left_product_terminal. pa_u_hj32_six_block_left_product = pa_q_hj32_six_block_left_product_terminal * S ((S (10 * m)) * pa_v_hj32_six_block_left_product) + (x))) /\ forall pa_i_hj32_six_block_left_product. (exists pa_lt_hj32_six_block_left_product_bound. pa_lt_hj32_six_block_left_product_bound + S pa_i_hj32_six_block_left_product = 10 * m) -> exists pa_p_hj32_six_block_left_product pa_r_hj32_six_block_left_product pa_s_hj32_six_block_left_product. ((((exists pa_h_hj32_six_block_left_product_factor. pa_h_hj32_six_block_left_product_factor + S (pa_p_hj32_six_block_left_product) = S ((S (pa_i_hj32_six_block_left_product)) * pa_c_hj32_six_block_left)) /\ exists pa_q_hj32_six_block_left_product_factor. pa_b_hj32_six_block_left = pa_q_hj32_six_block_left_product_factor * S ((S (pa_i_hj32_six_block_left_product)) * pa_c_hj32_six_block_left) + (pa_p_hj32_six_block_left_product))) /\ ((((exists pa_h_hj32_six_block_left_product_partial. pa_h_hj32_six_block_left_product_partial + S (pa_r_hj32_six_block_left_product) = S ((S (pa_i_hj32_six_block_left_product)) * pa_v_hj32_six_block_left_product)) /\ exists pa_q_hj32_six_block_left_product_partial. pa_u_hj32_six_block_left_product = pa_q_hj32_six_block_left_product_partial * S ((S (pa_i_hj32_six_block_left_product)) * pa_v_hj32_six_block_left_product) + (pa_r_hj32_six_block_left_product))) /\ ((((exists pa_h_hj32_six_block_left_product_successor. pa_h_hj32_six_block_left_product_successor + S (pa_s_hj32_six_block_left_product) = S ((S (S pa_i_hj32_six_block_left_product)) * pa_v_hj32_six_block_left_product)) /\ exists pa_q_hj32_six_block_left_product_successor. pa_u_hj32_six_block_left_product = pa_q_hj32_six_block_left_product_successor * S ((S (S pa_i_hj32_six_block_left_product)) * pa_v_hj32_six_block_left_product) + (pa_s_hj32_six_block_left_product))) /\ pa_s_hj32_six_block_left_product = pa_r_hj32_six_block_left_product * pa_p_hj32_six_block_left_product)))))))) -> (exists pa_b_hj32_six_block_right pa_c_hj32_six_block_right. ((forall pa_i_hj32_six_block_right_repeat. (exists pa_lt_hj32_six_block_right_repeat_bound. pa_lt_hj32_six_block_right_repeat_bound + S pa_i_hj32_six_block_right_repeat = 13 * m) -> (((exists pa_h_hj32_six_block_right_repeat_decoded. pa_h_hj32_six_block_right_repeat_decoded + S (4) = S ((S (pa_i_hj32_six_block_right_repeat)) * pa_c_hj32_six_block_right)) /\ exists pa_q_hj32_six_block_right_repeat_decoded. pa_b_hj32_six_block_right = pa_q_hj32_six_block_right_repeat_decoded * S ((S (pa_i_hj32_six_block_right_repeat)) * pa_c_hj32_six_block_right) + (4)))) /\ (exists pa_u_hj32_six_block_right_product pa_v_hj32_six_block_right_product. ((((exists pa_h_hj32_six_block_right_product_start. pa_h_hj32_six_block_right_product_start + S (1) = S ((S (0)) * pa_v_hj32_six_block_right_product)) /\ exists pa_q_hj32_six_block_right_product_start. pa_u_hj32_six_block_right_product = pa_q_hj32_six_block_right_product_start * S ((S (0)) * pa_v_hj32_six_block_right_product) + (1))) /\ ((((exists pa_h_hj32_six_block_right_product_terminal. pa_h_hj32_six_block_right_product_terminal + S (y) = S ((S (13 * m)) * pa_v_hj32_six_block_right_product)) /\ exists pa_q_hj32_six_block_right_product_terminal. pa_u_hj32_six_block_right_product = pa_q_hj32_six_block_right_product_terminal * S ((S (13 * m)) * pa_v_hj32_six_block_right_product) + (y))) /\ forall pa_i_hj32_six_block_right_product. (exists pa_lt_hj32_six_block_right_product_bound. pa_lt_hj32_six_block_right_product_bound + S pa_i_hj32_six_block_right_product = 13 * m) -> exists pa_p_hj32_six_block_right_product pa_r_hj32_six_block_right_product pa_s_hj32_six_block_right_product. ((((exists pa_h_hj32_six_block_right_product_factor. pa_h_hj32_six_block_right_product_factor + S (pa_p_hj32_six_block_right_product) = S ((S (pa_i_hj32_six_block_right_product)) * pa_c_hj32_six_block_right)) /\ exists pa_q_hj32_six_block_right_product_factor. pa_b_hj32_six_block_right = pa_q_hj32_six_block_right_product_factor * S ((S (pa_i_hj32_six_block_right_product)) * pa_c_hj32_six_block_right) + (pa_p_hj32_six_block_right_product))) /\ ((((exists pa_h_hj32_six_block_right_product_partial. pa_h_hj32_six_block_right_product_partial + S (pa_r_hj32_six_block_right_product) = S ((S (pa_i_hj32_six_block_right_product)) * pa_v_hj32_six_block_right_product)) /\ exists pa_q_hj32_six_block_right_product_partial. pa_u_hj32_six_block_right_product = pa_q_hj32_six_block_right_product_partial * S ((S (pa_i_hj32_six_block_right_product)) * pa_v_hj32_six_block_right_product) + (pa_r_hj32_six_block_right_product))) /\ ((((exists pa_h_hj32_six_block_right_product_successor. pa_h_hj32_six_block_right_product_successor + S (pa_s_hj32_six_block_right_product) = S ((S (S pa_i_hj32_six_block_right_product)) * pa_v_hj32_six_block_right_product)) /\ exists pa_q_hj32_six_block_right_product_successor. pa_u_hj32_six_block_right_product = pa_q_hj32_six_block_right_product_successor * S ((S (S pa_i_hj32_six_block_right_product)) * pa_v_hj32_six_block_right_product) + (pa_s_hj32_six_block_right_product))) /\ pa_s_hj32_six_block_right_product = pa_r_hj32_six_block_right_product * pa_p_hj32_six_block_right_product)))))))) -> (exists bqb_le_gap_hj32_six_block_result. bqb_le_gap_hj32_six_block_result + (x) = (y))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 assumptionsL1–6
02Establish sb_p6_tenL7–10
Establish this local claim before using it. It is not an additional assumption.
- L7
have sb_p6_ten : ∃ hj32_local_value_sb_p6_ten. Pow(6,10,hj32_local_value_sb_p6_ten)Definitions: Pow(6,10,hj32_local_value_sb_p6_ten)Original native command in the exact edition - L8
specialize htotal 6 - L9
specialize htotal 10 - L10
exact htotal
03Separate the logical casesL11–11
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L11
cases sb_p6_ten
04Establish sb_p4_thirteenL12–15
Establish this local claim before using it. It is not an additional assumption.
- L12
have sb_p4_thirteen : ∃ hj32_local_value_sb_p4_thirteen. Pow(4,13,hj32_local_value_sb_p4_thirteen)Definitions: Pow(4,13,hj32_local_value_sb_p4_thirteen)Original native command in the exact edition - L13
specialize htotal 4 - L14
specialize htotal 13 - L15
exact htotal
05Separate the logical casesL16–16
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L16
cases sb_p4_thirteen
06Establish sb_seedL17–23
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow six ten le pow four thirteen from total.
07Establish sb_boundL24–33
Establish this local claim before using it. It is not an additional assumption.
- L24
- L25
specialize pow_block_bound_from_total 6 - L26
specialize pow_block_bound_from_total 4 - L27
specialize pow_block_bound_from_total 10 - L28
specialize pow_block_bound_from_total 13 - L29
specialize pow_block_bound_from_total m - L30
specialize pow_block_bound_from_total x1 - L31
specialize pow_block_bound_from_total x2 - L32
specialize pow_block_bound_from_total x - L33
specialize pow_block_bound_from_total y
Original defined command ledger · 41 lines
- 0001
intro m - 0002
intro x - 0003
intro y - 0004
intro htotal - 0005
intro hx - 0006
intro hy - 0007
have sb_p6_ten : ∃ hj32_local_value_sb_p6_ten. Pow(6,10,hj32_local_value_sb_p6_ten)Exact native replay line
have sb_p6_ten : exists hj32_local_value_sb_p6_ten. (exists pa_b_hj32_local_total_sb_p6_ten pa_c_hj32_local_total_sb_p6_ten. ((forall pa_i_hj32_local_total_sb_p6_ten_repeat. (exists pa_lt_hj32_local_total_sb_p6_ten_repeat_bound. pa_lt_hj32_local_total_sb_p6_ten_repeat_bound + S pa_i_hj32_local_total_sb_p6_ten_repeat = 10) -> (((exists pa_h_hj32_local_total_sb_p6_ten_repeat_decoded. pa_h_hj32_local_total_sb_p6_ten_repeat_decoded + S (6) = S ((S (pa_i_hj32_local_total_sb_p6_ten_repeat)) * pa_c_hj32_local_total_sb_p6_ten)) /\ exists pa_q_hj32_local_total_sb_p6_ten_repeat_decoded. pa_b_hj32_local_total_sb_p6_ten = pa_q_hj32_local_total_sb_p6_ten_repeat_decoded * S ((S (pa_i_hj32_local_total_sb_p6_ten_repeat)) * pa_c_hj32_local_total_sb_p6_ten) + (6)))) /\ (exists pa_u_hj32_local_total_sb_p6_ten_product pa_v_hj32_local_total_sb_p6_ten_product. ((((exists pa_h_hj32_local_total_sb_p6_ten_product_start. pa_h_hj32_local_total_sb_p6_ten_product_start + S (1) = S ((S (0)) * pa_v_hj32_local_total_sb_p6_ten_product)) /\ exists pa_q_hj32_local_total_sb_p6_ten_product_start. pa_u_hj32_local_total_sb_p6_ten_product = pa_q_hj32_local_total_sb_p6_ten_product_start * S ((S (0)) * pa_v_hj32_local_total_sb_p6_ten_product) + (1))) /\ ((((exists pa_h_hj32_local_total_sb_p6_ten_product_terminal. pa_h_hj32_local_total_sb_p6_ten_product_terminal + S (hj32_local_value_sb_p6_ten) = S ((S (10)) * pa_v_hj32_local_total_sb_p6_ten_product)) /\ exists pa_q_hj32_local_total_sb_p6_ten_product_terminal. pa_u_hj32_local_total_sb_p6_ten_product = pa_q_hj32_local_total_sb_p6_ten_product_terminal * S ((S (10)) * pa_v_hj32_local_total_sb_p6_ten_product) + (hj32_local_value_sb_p6_ten))) /\ forall pa_i_hj32_local_total_sb_p6_ten_product. (exists pa_lt_hj32_local_total_sb_p6_ten_product_bound. pa_lt_hj32_local_total_sb_p6_ten_product_bound + S pa_i_hj32_local_total_sb_p6_ten_product = 10) -> exists pa_p_hj32_local_total_sb_p6_ten_product pa_r_hj32_local_total_sb_p6_ten_product pa_s_hj32_local_total_sb_p6_ten_product. ((((exists pa_h_hj32_local_total_sb_p6_ten_product_factor. pa_h_hj32_local_total_sb_p6_ten_product_factor + S (pa_p_hj32_local_total_sb_p6_ten_product) = S ((S (pa_i_hj32_local_total_sb_p6_ten_product)) * pa_c_hj32_local_total_sb_p6_ten)) /\ exists pa_q_hj32_local_total_sb_p6_ten_product_factor. pa_b_hj32_local_total_sb_p6_ten = pa_q_hj32_local_total_sb_p6_ten_product_factor * S ((S (pa_i_hj32_local_total_sb_p6_ten_product)) * pa_c_hj32_local_total_sb_p6_ten) + (pa_p_hj32_local_total_sb_p6_ten_product))) /\ ((((exists pa_h_hj32_local_total_sb_p6_ten_product_partial. pa_h_hj32_local_total_sb_p6_ten_product_partial + S (pa_r_hj32_local_total_sb_p6_ten_product) = S ((S (pa_i_hj32_local_total_sb_p6_ten_product)) * pa_v_hj32_local_total_sb_p6_ten_product)) /\ exists pa_q_hj32_local_total_sb_p6_ten_product_partial. pa_u_hj32_local_total_sb_p6_ten_product = pa_q_hj32_local_total_sb_p6_ten_product_partial * S ((S (pa_i_hj32_local_total_sb_p6_ten_product)) * pa_v_hj32_local_total_sb_p6_ten_product) + (pa_r_hj32_local_total_sb_p6_ten_product))) /\ ((((exists pa_h_hj32_local_total_sb_p6_ten_product_successor. pa_h_hj32_local_total_sb_p6_ten_product_successor + S (pa_s_hj32_local_total_sb_p6_ten_product) = S ((S (S pa_i_hj32_local_total_sb_p6_ten_product)) * pa_v_hj32_local_total_sb_p6_ten_product)) /\ exists pa_q_hj32_local_total_sb_p6_ten_product_successor. pa_u_hj32_local_total_sb_p6_ten_product = pa_q_hj32_local_total_sb_p6_ten_product_successor * S ((S (S pa_i_hj32_local_total_sb_p6_ten_product)) * pa_v_hj32_local_total_sb_p6_ten_product) + (pa_s_hj32_local_total_sb_p6_ten_product))) /\ pa_s_hj32_local_total_sb_p6_ten_product = pa_r_hj32_local_total_sb_p6_ten_product * pa_p_hj32_local_total_sb_p6_ten_product)))))))) - 0008
specialize htotal 6 - 0009
specialize htotal 10 - 0010
exact htotal - 0011
cases sb_p6_ten - 0012
have sb_p4_thirteen : ∃ hj32_local_value_sb_p4_thirteen. Pow(4,13,hj32_local_value_sb_p4_thirteen)Exact native replay line
have sb_p4_thirteen : exists hj32_local_value_sb_p4_thirteen. (exists pa_b_hj32_local_total_sb_p4_thirteen pa_c_hj32_local_total_sb_p4_thirteen. ((forall pa_i_hj32_local_total_sb_p4_thirteen_repeat. (exists pa_lt_hj32_local_total_sb_p4_thirteen_repeat_bound. pa_lt_hj32_local_total_sb_p4_thirteen_repeat_bound + S pa_i_hj32_local_total_sb_p4_thirteen_repeat = 13) -> (((exists pa_h_hj32_local_total_sb_p4_thirteen_repeat_decoded. pa_h_hj32_local_total_sb_p4_thirteen_repeat_decoded + S (4) = S ((S (pa_i_hj32_local_total_sb_p4_thirteen_repeat)) * pa_c_hj32_local_total_sb_p4_thirteen)) /\ exists pa_q_hj32_local_total_sb_p4_thirteen_repeat_decoded. pa_b_hj32_local_total_sb_p4_thirteen = pa_q_hj32_local_total_sb_p4_thirteen_repeat_decoded * S ((S (pa_i_hj32_local_total_sb_p4_thirteen_repeat)) * pa_c_hj32_local_total_sb_p4_thirteen) + (4)))) /\ (exists pa_u_hj32_local_total_sb_p4_thirteen_product pa_v_hj32_local_total_sb_p4_thirteen_product. ((((exists pa_h_hj32_local_total_sb_p4_thirteen_product_start. pa_h_hj32_local_total_sb_p4_thirteen_product_start + S (1) = S ((S (0)) * pa_v_hj32_local_total_sb_p4_thirteen_product)) /\ exists pa_q_hj32_local_total_sb_p4_thirteen_product_start. pa_u_hj32_local_total_sb_p4_thirteen_product = pa_q_hj32_local_total_sb_p4_thirteen_product_start * S ((S (0)) * pa_v_hj32_local_total_sb_p4_thirteen_product) + (1))) /\ ((((exists pa_h_hj32_local_total_sb_p4_thirteen_product_terminal. pa_h_hj32_local_total_sb_p4_thirteen_product_terminal + S (hj32_local_value_sb_p4_thirteen) = S ((S (13)) * pa_v_hj32_local_total_sb_p4_thirteen_product)) /\ exists pa_q_hj32_local_total_sb_p4_thirteen_product_terminal. pa_u_hj32_local_total_sb_p4_thirteen_product = pa_q_hj32_local_total_sb_p4_thirteen_product_terminal * S ((S (13)) * pa_v_hj32_local_total_sb_p4_thirteen_product) + (hj32_local_value_sb_p4_thirteen))) /\ forall pa_i_hj32_local_total_sb_p4_thirteen_product. (exists pa_lt_hj32_local_total_sb_p4_thirteen_product_bound. pa_lt_hj32_local_total_sb_p4_thirteen_product_bound + S pa_i_hj32_local_total_sb_p4_thirteen_product = 13) -> exists pa_p_hj32_local_total_sb_p4_thirteen_product pa_r_hj32_local_total_sb_p4_thirteen_product pa_s_hj32_local_total_sb_p4_thirteen_product. ((((exists pa_h_hj32_local_total_sb_p4_thirteen_product_factor. pa_h_hj32_local_total_sb_p4_thirteen_product_factor + S (pa_p_hj32_local_total_sb_p4_thirteen_product) = S ((S (pa_i_hj32_local_total_sb_p4_thirteen_product)) * pa_c_hj32_local_total_sb_p4_thirteen)) /\ exists pa_q_hj32_local_total_sb_p4_thirteen_product_factor. pa_b_hj32_local_total_sb_p4_thirteen = pa_q_hj32_local_total_sb_p4_thirteen_product_factor * S ((S (pa_i_hj32_local_total_sb_p4_thirteen_product)) * pa_c_hj32_local_total_sb_p4_thirteen) + (pa_p_hj32_local_total_sb_p4_thirteen_product))) /\ ((((exists pa_h_hj32_local_total_sb_p4_thirteen_product_partial. pa_h_hj32_local_total_sb_p4_thirteen_product_partial + S (pa_r_hj32_local_total_sb_p4_thirteen_product) = S ((S (pa_i_hj32_local_total_sb_p4_thirteen_product)) * pa_v_hj32_local_total_sb_p4_thirteen_product)) /\ exists pa_q_hj32_local_total_sb_p4_thirteen_product_partial. pa_u_hj32_local_total_sb_p4_thirteen_product = pa_q_hj32_local_total_sb_p4_thirteen_product_partial * S ((S (pa_i_hj32_local_total_sb_p4_thirteen_product)) * pa_v_hj32_local_total_sb_p4_thirteen_product) + (pa_r_hj32_local_total_sb_p4_thirteen_product))) /\ ((((exists pa_h_hj32_local_total_sb_p4_thirteen_product_successor. pa_h_hj32_local_total_sb_p4_thirteen_product_successor + S (pa_s_hj32_local_total_sb_p4_thirteen_product) = S ((S (S pa_i_hj32_local_total_sb_p4_thirteen_product)) * pa_v_hj32_local_total_sb_p4_thirteen_product)) /\ exists pa_q_hj32_local_total_sb_p4_thirteen_product_successor. pa_u_hj32_local_total_sb_p4_thirteen_product = pa_q_hj32_local_total_sb_p4_thirteen_product_successor * S ((S (S pa_i_hj32_local_total_sb_p4_thirteen_product)) * pa_v_hj32_local_total_sb_p4_thirteen_product) + (pa_s_hj32_local_total_sb_p4_thirteen_product))) /\ pa_s_hj32_local_total_sb_p4_thirteen_product = pa_r_hj32_local_total_sb_p4_thirteen_product * pa_p_hj32_local_total_sb_p4_thirteen_product)))))))) - 0013
specialize htotal 4 - 0014
specialize htotal 13 - 0015
exact htotal - 0016
cases sb_p4_thirteen - 0017
have sb_seed : Le(x1,x2)Exact native replay line
have sb_seed : exists bqb_le_gap_hj32_sb_seed. bqb_le_gap_hj32_sb_seed + (x1) = (x2) - 0018
specialize pow_six_ten_le_pow_four_thirteen_from_total x1 - 0019
specialize pow_six_ten_le_pow_four_thirteen_from_total x2 - 0020
apply pow_six_ten_le_pow_four_thirteen_from_total - 0021
exact htotal - 0022
exact sb_p6_ten_witness - 0023
exact sb_p4_thirteen_witness - 0024
have sb_bound : Le(x,y)Exact native replay line
have sb_bound : exists bqb_le_gap_hj32_local_block_bound_sb_bound. bqb_le_gap_hj32_local_block_bound_sb_bound + (x) = (y) - 0025
specialize pow_block_bound_from_total 6 - 0026
specialize pow_block_bound_from_total 4 - 0027
specialize pow_block_bound_from_total 10 - 0028
specialize pow_block_bound_from_total 13 - 0029
specialize pow_block_bound_from_total m - 0030
specialize pow_block_bound_from_total x1 - 0031
specialize pow_block_bound_from_total x2 - 0032
specialize pow_block_bound_from_total x - 0033
specialize pow_block_bound_from_total y - 0034
apply pow_block_bound_from_total - 0035
exact htotal - 0036
exact sb_p6_ten_witness - 0037
exact sb_p4_thirteen_witness - 0038
exact sb_seed - 0039
exact hx - 0040
exact hy - 0041
exact sb_bound