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(11,2 · m,x) → Pow(2,7 · 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_eleven_block bpt_e_hj32_eleven_block. exists bpt_x_hj32_eleven_block. (exists ff_b_bpt_value_hj32_eleven_block ff_c_bpt_value_hj32_eleven_block. ((forall ff_i_bpt_value_hj32_eleven_block_repeat. (exists ff_lt_bpt_value_hj32_eleven_block_repeat_bound. ff_lt_bpt_value_hj32_eleven_block_repeat_bound + S ff_i_bpt_value_hj32_eleven_block_repeat = bpt_e_hj32_eleven_block) -> (((exists ff_h_bpt_value_hj32_eleven_block_repeat_decoded. ff_h_bpt_value_hj32_eleven_block_repeat_decoded + S (bpt_a_hj32_eleven_block) = S ((S (ff_i_bpt_value_hj32_eleven_block_repeat)) * ff_c_bpt_value_hj32_eleven_block)) /\ exists ff_q_bpt_value_hj32_eleven_block_repeat_decoded. ff_b_bpt_value_hj32_eleven_block = ff_q_bpt_value_hj32_eleven_block_repeat_decoded * S ((S (ff_i_bpt_value_hj32_eleven_block_repeat)) * ff_c_bpt_value_hj32_eleven_block) + (bpt_a_hj32_eleven_block)))) /\ (exists ff_u_bpt_value_hj32_eleven_block_product ff_v_bpt_value_hj32_eleven_block_product. ((((exists ff_h_bpt_value_hj32_eleven_block_product_start. ff_h_bpt_value_hj32_eleven_block_product_start + S (1) = S ((S (0)) * ff_v_bpt_value_hj32_eleven_block_product)) /\ exists ff_q_bpt_value_hj32_eleven_block_product_start. ff_u_bpt_value_hj32_eleven_block_product = ff_q_bpt_value_hj32_eleven_block_product_start * S ((S (0)) * ff_v_bpt_value_hj32_eleven_block_product) + (1))) /\ ((((exists ff_h_bpt_value_hj32_eleven_block_product_terminal. ff_h_bpt_value_hj32_eleven_block_product_terminal + S (bpt_x_hj32_eleven_block) = S ((S (bpt_e_hj32_eleven_block)) * ff_v_bpt_value_hj32_eleven_block_product)) /\ exists ff_q_bpt_value_hj32_eleven_block_product_terminal. ff_u_bpt_value_hj32_eleven_block_product = ff_q_bpt_value_hj32_eleven_block_product_terminal * S ((S (bpt_e_hj32_eleven_block)) * ff_v_bpt_value_hj32_eleven_block_product) + (bpt_x_hj32_eleven_block))) /\ forall ff_i_bpt_value_hj32_eleven_block_product. (exists ff_lt_bpt_value_hj32_eleven_block_product_bound. ff_lt_bpt_value_hj32_eleven_block_product_bound + S ff_i_bpt_value_hj32_eleven_block_product = bpt_e_hj32_eleven_block) -> exists ff_p_bpt_value_hj32_eleven_block_product ff_r_bpt_value_hj32_eleven_block_product ff_s_bpt_value_hj32_eleven_block_product. ((((exists ff_h_bpt_value_hj32_eleven_block_product_factor. ff_h_bpt_value_hj32_eleven_block_product_factor + S (ff_p_bpt_value_hj32_eleven_block_product) = S ((S (ff_i_bpt_value_hj32_eleven_block_product)) * ff_c_bpt_value_hj32_eleven_block)) /\ exists ff_q_bpt_value_hj32_eleven_block_product_factor. ff_b_bpt_value_hj32_eleven_block = ff_q_bpt_value_hj32_eleven_block_product_factor * S ((S (ff_i_bpt_value_hj32_eleven_block_product)) * ff_c_bpt_value_hj32_eleven_block) + (ff_p_bpt_value_hj32_eleven_block_product))) /\ ((((exists ff_h_bpt_value_hj32_eleven_block_product_partial. ff_h_bpt_value_hj32_eleven_block_product_partial + S (ff_r_bpt_value_hj32_eleven_block_product) = S ((S (ff_i_bpt_value_hj32_eleven_block_product)) * ff_v_bpt_value_hj32_eleven_block_product)) /\ exists ff_q_bpt_value_hj32_eleven_block_product_partial. ff_u_bpt_value_hj32_eleven_block_product = ff_q_bpt_value_hj32_eleven_block_product_partial * S ((S (ff_i_bpt_value_hj32_eleven_block_product)) * ff_v_bpt_value_hj32_eleven_block_product) + (ff_r_bpt_value_hj32_eleven_block_product))) /\ ((((exists ff_h_bpt_value_hj32_eleven_block_product_successor. ff_h_bpt_value_hj32_eleven_block_product_successor + S (ff_s_bpt_value_hj32_eleven_block_product) = S ((S (S ff_i_bpt_value_hj32_eleven_block_product)) * ff_v_bpt_value_hj32_eleven_block_product)) /\ exists ff_q_bpt_value_hj32_eleven_block_product_successor. ff_u_bpt_value_hj32_eleven_block_product = ff_q_bpt_value_hj32_eleven_block_product_successor * S ((S (S ff_i_bpt_value_hj32_eleven_block_product)) * ff_v_bpt_value_hj32_eleven_block_product) + (ff_s_bpt_value_hj32_eleven_block_product))) /\ ff_s_bpt_value_hj32_eleven_block_product = ff_r_bpt_value_hj32_eleven_block_product * ff_p_bpt_value_hj32_eleven_block_product))))))))) -> (exists pa_b_hj32_eleven_block_left pa_c_hj32_eleven_block_left. ((forall pa_i_hj32_eleven_block_left_repeat. (exists pa_lt_hj32_eleven_block_left_repeat_bound. pa_lt_hj32_eleven_block_left_repeat_bound + S pa_i_hj32_eleven_block_left_repeat = 2 * m) -> (((exists pa_h_hj32_eleven_block_left_repeat_decoded. pa_h_hj32_eleven_block_left_repeat_decoded + S (11) = S ((S (pa_i_hj32_eleven_block_left_repeat)) * pa_c_hj32_eleven_block_left)) /\ exists pa_q_hj32_eleven_block_left_repeat_decoded. pa_b_hj32_eleven_block_left = pa_q_hj32_eleven_block_left_repeat_decoded * S ((S (pa_i_hj32_eleven_block_left_repeat)) * pa_c_hj32_eleven_block_left) + (11)))) /\ (exists pa_u_hj32_eleven_block_left_product pa_v_hj32_eleven_block_left_product. ((((exists pa_h_hj32_eleven_block_left_product_start. pa_h_hj32_eleven_block_left_product_start + S (1) = S ((S (0)) * pa_v_hj32_eleven_block_left_product)) /\ exists pa_q_hj32_eleven_block_left_product_start. pa_u_hj32_eleven_block_left_product = pa_q_hj32_eleven_block_left_product_start * S ((S (0)) * pa_v_hj32_eleven_block_left_product) + (1))) /\ ((((exists pa_h_hj32_eleven_block_left_product_terminal. pa_h_hj32_eleven_block_left_product_terminal + S (x) = S ((S (2 * m)) * pa_v_hj32_eleven_block_left_product)) /\ exists pa_q_hj32_eleven_block_left_product_terminal. pa_u_hj32_eleven_block_left_product = pa_q_hj32_eleven_block_left_product_terminal * S ((S (2 * m)) * pa_v_hj32_eleven_block_left_product) + (x))) /\ forall pa_i_hj32_eleven_block_left_product. (exists pa_lt_hj32_eleven_block_left_product_bound. pa_lt_hj32_eleven_block_left_product_bound + S pa_i_hj32_eleven_block_left_product = 2 * m) -> exists pa_p_hj32_eleven_block_left_product pa_r_hj32_eleven_block_left_product pa_s_hj32_eleven_block_left_product. ((((exists pa_h_hj32_eleven_block_left_product_factor. pa_h_hj32_eleven_block_left_product_factor + S (pa_p_hj32_eleven_block_left_product) = S ((S (pa_i_hj32_eleven_block_left_product)) * pa_c_hj32_eleven_block_left)) /\ exists pa_q_hj32_eleven_block_left_product_factor. pa_b_hj32_eleven_block_left = pa_q_hj32_eleven_block_left_product_factor * S ((S (pa_i_hj32_eleven_block_left_product)) * pa_c_hj32_eleven_block_left) + (pa_p_hj32_eleven_block_left_product))) /\ ((((exists pa_h_hj32_eleven_block_left_product_partial. pa_h_hj32_eleven_block_left_product_partial + S (pa_r_hj32_eleven_block_left_product) = S ((S (pa_i_hj32_eleven_block_left_product)) * pa_v_hj32_eleven_block_left_product)) /\ exists pa_q_hj32_eleven_block_left_product_partial. pa_u_hj32_eleven_block_left_product = pa_q_hj32_eleven_block_left_product_partial * S ((S (pa_i_hj32_eleven_block_left_product)) * pa_v_hj32_eleven_block_left_product) + (pa_r_hj32_eleven_block_left_product))) /\ ((((exists pa_h_hj32_eleven_block_left_product_successor. pa_h_hj32_eleven_block_left_product_successor + S (pa_s_hj32_eleven_block_left_product) = S ((S (S pa_i_hj32_eleven_block_left_product)) * pa_v_hj32_eleven_block_left_product)) /\ exists pa_q_hj32_eleven_block_left_product_successor. pa_u_hj32_eleven_block_left_product = pa_q_hj32_eleven_block_left_product_successor * S ((S (S pa_i_hj32_eleven_block_left_product)) * pa_v_hj32_eleven_block_left_product) + (pa_s_hj32_eleven_block_left_product))) /\ pa_s_hj32_eleven_block_left_product = pa_r_hj32_eleven_block_left_product * pa_p_hj32_eleven_block_left_product)))))))) -> (exists pa_b_hj32_eleven_block_right pa_c_hj32_eleven_block_right. ((forall pa_i_hj32_eleven_block_right_repeat. (exists pa_lt_hj32_eleven_block_right_repeat_bound. pa_lt_hj32_eleven_block_right_repeat_bound + S pa_i_hj32_eleven_block_right_repeat = 7 * m) -> (((exists pa_h_hj32_eleven_block_right_repeat_decoded. pa_h_hj32_eleven_block_right_repeat_decoded + S (2) = S ((S (pa_i_hj32_eleven_block_right_repeat)) * pa_c_hj32_eleven_block_right)) /\ exists pa_q_hj32_eleven_block_right_repeat_decoded. pa_b_hj32_eleven_block_right = pa_q_hj32_eleven_block_right_repeat_decoded * S ((S (pa_i_hj32_eleven_block_right_repeat)) * pa_c_hj32_eleven_block_right) + (2)))) /\ (exists pa_u_hj32_eleven_block_right_product pa_v_hj32_eleven_block_right_product. ((((exists pa_h_hj32_eleven_block_right_product_start. pa_h_hj32_eleven_block_right_product_start + S (1) = S ((S (0)) * pa_v_hj32_eleven_block_right_product)) /\ exists pa_q_hj32_eleven_block_right_product_start. pa_u_hj32_eleven_block_right_product = pa_q_hj32_eleven_block_right_product_start * S ((S (0)) * pa_v_hj32_eleven_block_right_product) + (1))) /\ ((((exists pa_h_hj32_eleven_block_right_product_terminal. pa_h_hj32_eleven_block_right_product_terminal + S (y) = S ((S (7 * m)) * pa_v_hj32_eleven_block_right_product)) /\ exists pa_q_hj32_eleven_block_right_product_terminal. pa_u_hj32_eleven_block_right_product = pa_q_hj32_eleven_block_right_product_terminal * S ((S (7 * m)) * pa_v_hj32_eleven_block_right_product) + (y))) /\ forall pa_i_hj32_eleven_block_right_product. (exists pa_lt_hj32_eleven_block_right_product_bound. pa_lt_hj32_eleven_block_right_product_bound + S pa_i_hj32_eleven_block_right_product = 7 * m) -> exists pa_p_hj32_eleven_block_right_product pa_r_hj32_eleven_block_right_product pa_s_hj32_eleven_block_right_product. ((((exists pa_h_hj32_eleven_block_right_product_factor. pa_h_hj32_eleven_block_right_product_factor + S (pa_p_hj32_eleven_block_right_product) = S ((S (pa_i_hj32_eleven_block_right_product)) * pa_c_hj32_eleven_block_right)) /\ exists pa_q_hj32_eleven_block_right_product_factor. pa_b_hj32_eleven_block_right = pa_q_hj32_eleven_block_right_product_factor * S ((S (pa_i_hj32_eleven_block_right_product)) * pa_c_hj32_eleven_block_right) + (pa_p_hj32_eleven_block_right_product))) /\ ((((exists pa_h_hj32_eleven_block_right_product_partial. pa_h_hj32_eleven_block_right_product_partial + S (pa_r_hj32_eleven_block_right_product) = S ((S (pa_i_hj32_eleven_block_right_product)) * pa_v_hj32_eleven_block_right_product)) /\ exists pa_q_hj32_eleven_block_right_product_partial. pa_u_hj32_eleven_block_right_product = pa_q_hj32_eleven_block_right_product_partial * S ((S (pa_i_hj32_eleven_block_right_product)) * pa_v_hj32_eleven_block_right_product) + (pa_r_hj32_eleven_block_right_product))) /\ ((((exists pa_h_hj32_eleven_block_right_product_successor. pa_h_hj32_eleven_block_right_product_successor + S (pa_s_hj32_eleven_block_right_product) = S ((S (S pa_i_hj32_eleven_block_right_product)) * pa_v_hj32_eleven_block_right_product)) /\ exists pa_q_hj32_eleven_block_right_product_successor. pa_u_hj32_eleven_block_right_product = pa_q_hj32_eleven_block_right_product_successor * S ((S (S pa_i_hj32_eleven_block_right_product)) * pa_v_hj32_eleven_block_right_product) + (pa_s_hj32_eleven_block_right_product))) /\ pa_s_hj32_eleven_block_right_product = pa_r_hj32_eleven_block_right_product * pa_p_hj32_eleven_block_right_product)))))))) -> (exists bqb_le_gap_hj32_eleven_block_result. bqb_le_gap_hj32_eleven_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 eb_p11_twoL7–10
Establish this local claim before using it. It is not an additional assumption.
- L7
have eb_p11_two : ∃ hj32_local_value_eb_p11_two. Pow(11,2,hj32_local_value_eb_p11_two)Definitions: Pow(11,2,hj32_local_value_eb_p11_two)Original native command in the exact edition - L8
specialize htotal 11 - L9
specialize htotal 2 - L10
exact htotal
03Separate the logical casesL11–11
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L11
cases eb_p11_two
04Establish eb_p2_sevenL12–15
Establish this local claim before using it. It is not an additional assumption.
- L12
have eb_p2_seven : ∃ hj32_local_value_eb_p2_seven. Pow(2,7,hj32_local_value_eb_p2_seven)Definitions: Pow(2,7,hj32_local_value_eb_p2_seven)Original native command in the exact edition - L13
specialize htotal 2 - L14
specialize htotal 7 - L15
exact htotal
05Separate the logical casesL16–16
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L16
cases eb_p2_seven
06Establish eb_seedL17–23
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow eleven two le pow two seven from total.
07Establish eb_boundL24–33
Establish this local claim before using it. It is not an additional assumption.
- L24
- L25
specialize pow_block_bound_from_total 11 - L26
specialize pow_block_bound_from_total 2 - L27
specialize pow_block_bound_from_total 2 - L28
specialize pow_block_bound_from_total 7 - 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 eb_p11_two : ∃ hj32_local_value_eb_p11_two. Pow(11,2,hj32_local_value_eb_p11_two)Exact native replay line
have eb_p11_two : exists hj32_local_value_eb_p11_two. (exists pa_b_hj32_local_total_eb_p11_two pa_c_hj32_local_total_eb_p11_two. ((forall pa_i_hj32_local_total_eb_p11_two_repeat. (exists pa_lt_hj32_local_total_eb_p11_two_repeat_bound. pa_lt_hj32_local_total_eb_p11_two_repeat_bound + S pa_i_hj32_local_total_eb_p11_two_repeat = 2) -> (((exists pa_h_hj32_local_total_eb_p11_two_repeat_decoded. pa_h_hj32_local_total_eb_p11_two_repeat_decoded + S (11) = S ((S (pa_i_hj32_local_total_eb_p11_two_repeat)) * pa_c_hj32_local_total_eb_p11_two)) /\ exists pa_q_hj32_local_total_eb_p11_two_repeat_decoded. pa_b_hj32_local_total_eb_p11_two = pa_q_hj32_local_total_eb_p11_two_repeat_decoded * S ((S (pa_i_hj32_local_total_eb_p11_two_repeat)) * pa_c_hj32_local_total_eb_p11_two) + (11)))) /\ (exists pa_u_hj32_local_total_eb_p11_two_product pa_v_hj32_local_total_eb_p11_two_product. ((((exists pa_h_hj32_local_total_eb_p11_two_product_start. pa_h_hj32_local_total_eb_p11_two_product_start + S (1) = S ((S (0)) * pa_v_hj32_local_total_eb_p11_two_product)) /\ exists pa_q_hj32_local_total_eb_p11_two_product_start. pa_u_hj32_local_total_eb_p11_two_product = pa_q_hj32_local_total_eb_p11_two_product_start * S ((S (0)) * pa_v_hj32_local_total_eb_p11_two_product) + (1))) /\ ((((exists pa_h_hj32_local_total_eb_p11_two_product_terminal. pa_h_hj32_local_total_eb_p11_two_product_terminal + S (hj32_local_value_eb_p11_two) = S ((S (2)) * pa_v_hj32_local_total_eb_p11_two_product)) /\ exists pa_q_hj32_local_total_eb_p11_two_product_terminal. pa_u_hj32_local_total_eb_p11_two_product = pa_q_hj32_local_total_eb_p11_two_product_terminal * S ((S (2)) * pa_v_hj32_local_total_eb_p11_two_product) + (hj32_local_value_eb_p11_two))) /\ forall pa_i_hj32_local_total_eb_p11_two_product. (exists pa_lt_hj32_local_total_eb_p11_two_product_bound. pa_lt_hj32_local_total_eb_p11_two_product_bound + S pa_i_hj32_local_total_eb_p11_two_product = 2) -> exists pa_p_hj32_local_total_eb_p11_two_product pa_r_hj32_local_total_eb_p11_two_product pa_s_hj32_local_total_eb_p11_two_product. ((((exists pa_h_hj32_local_total_eb_p11_two_product_factor. pa_h_hj32_local_total_eb_p11_two_product_factor + S (pa_p_hj32_local_total_eb_p11_two_product) = S ((S (pa_i_hj32_local_total_eb_p11_two_product)) * pa_c_hj32_local_total_eb_p11_two)) /\ exists pa_q_hj32_local_total_eb_p11_two_product_factor. pa_b_hj32_local_total_eb_p11_two = pa_q_hj32_local_total_eb_p11_two_product_factor * S ((S (pa_i_hj32_local_total_eb_p11_two_product)) * pa_c_hj32_local_total_eb_p11_two) + (pa_p_hj32_local_total_eb_p11_two_product))) /\ ((((exists pa_h_hj32_local_total_eb_p11_two_product_partial. pa_h_hj32_local_total_eb_p11_two_product_partial + S (pa_r_hj32_local_total_eb_p11_two_product) = S ((S (pa_i_hj32_local_total_eb_p11_two_product)) * pa_v_hj32_local_total_eb_p11_two_product)) /\ exists pa_q_hj32_local_total_eb_p11_two_product_partial. pa_u_hj32_local_total_eb_p11_two_product = pa_q_hj32_local_total_eb_p11_two_product_partial * S ((S (pa_i_hj32_local_total_eb_p11_two_product)) * pa_v_hj32_local_total_eb_p11_two_product) + (pa_r_hj32_local_total_eb_p11_two_product))) /\ ((((exists pa_h_hj32_local_total_eb_p11_two_product_successor. pa_h_hj32_local_total_eb_p11_two_product_successor + S (pa_s_hj32_local_total_eb_p11_two_product) = S ((S (S pa_i_hj32_local_total_eb_p11_two_product)) * pa_v_hj32_local_total_eb_p11_two_product)) /\ exists pa_q_hj32_local_total_eb_p11_two_product_successor. pa_u_hj32_local_total_eb_p11_two_product = pa_q_hj32_local_total_eb_p11_two_product_successor * S ((S (S pa_i_hj32_local_total_eb_p11_two_product)) * pa_v_hj32_local_total_eb_p11_two_product) + (pa_s_hj32_local_total_eb_p11_two_product))) /\ pa_s_hj32_local_total_eb_p11_two_product = pa_r_hj32_local_total_eb_p11_two_product * pa_p_hj32_local_total_eb_p11_two_product)))))))) - 0008
specialize htotal 11 - 0009
specialize htotal 2 - 0010
exact htotal - 0011
cases eb_p11_two - 0012
have eb_p2_seven : ∃ hj32_local_value_eb_p2_seven. Pow(2,7,hj32_local_value_eb_p2_seven)Exact native replay line
have eb_p2_seven : exists hj32_local_value_eb_p2_seven. (exists pa_b_hj32_local_total_eb_p2_seven pa_c_hj32_local_total_eb_p2_seven. ((forall pa_i_hj32_local_total_eb_p2_seven_repeat. (exists pa_lt_hj32_local_total_eb_p2_seven_repeat_bound. pa_lt_hj32_local_total_eb_p2_seven_repeat_bound + S pa_i_hj32_local_total_eb_p2_seven_repeat = 7) -> (((exists pa_h_hj32_local_total_eb_p2_seven_repeat_decoded. pa_h_hj32_local_total_eb_p2_seven_repeat_decoded + S (2) = S ((S (pa_i_hj32_local_total_eb_p2_seven_repeat)) * pa_c_hj32_local_total_eb_p2_seven)) /\ exists pa_q_hj32_local_total_eb_p2_seven_repeat_decoded. pa_b_hj32_local_total_eb_p2_seven = pa_q_hj32_local_total_eb_p2_seven_repeat_decoded * S ((S (pa_i_hj32_local_total_eb_p2_seven_repeat)) * pa_c_hj32_local_total_eb_p2_seven) + (2)))) /\ (exists pa_u_hj32_local_total_eb_p2_seven_product pa_v_hj32_local_total_eb_p2_seven_product. ((((exists pa_h_hj32_local_total_eb_p2_seven_product_start. pa_h_hj32_local_total_eb_p2_seven_product_start + S (1) = S ((S (0)) * pa_v_hj32_local_total_eb_p2_seven_product)) /\ exists pa_q_hj32_local_total_eb_p2_seven_product_start. pa_u_hj32_local_total_eb_p2_seven_product = pa_q_hj32_local_total_eb_p2_seven_product_start * S ((S (0)) * pa_v_hj32_local_total_eb_p2_seven_product) + (1))) /\ ((((exists pa_h_hj32_local_total_eb_p2_seven_product_terminal. pa_h_hj32_local_total_eb_p2_seven_product_terminal + S (hj32_local_value_eb_p2_seven) = S ((S (7)) * pa_v_hj32_local_total_eb_p2_seven_product)) /\ exists pa_q_hj32_local_total_eb_p2_seven_product_terminal. pa_u_hj32_local_total_eb_p2_seven_product = pa_q_hj32_local_total_eb_p2_seven_product_terminal * S ((S (7)) * pa_v_hj32_local_total_eb_p2_seven_product) + (hj32_local_value_eb_p2_seven))) /\ forall pa_i_hj32_local_total_eb_p2_seven_product. (exists pa_lt_hj32_local_total_eb_p2_seven_product_bound. pa_lt_hj32_local_total_eb_p2_seven_product_bound + S pa_i_hj32_local_total_eb_p2_seven_product = 7) -> exists pa_p_hj32_local_total_eb_p2_seven_product pa_r_hj32_local_total_eb_p2_seven_product pa_s_hj32_local_total_eb_p2_seven_product. ((((exists pa_h_hj32_local_total_eb_p2_seven_product_factor. pa_h_hj32_local_total_eb_p2_seven_product_factor + S (pa_p_hj32_local_total_eb_p2_seven_product) = S ((S (pa_i_hj32_local_total_eb_p2_seven_product)) * pa_c_hj32_local_total_eb_p2_seven)) /\ exists pa_q_hj32_local_total_eb_p2_seven_product_factor. pa_b_hj32_local_total_eb_p2_seven = pa_q_hj32_local_total_eb_p2_seven_product_factor * S ((S (pa_i_hj32_local_total_eb_p2_seven_product)) * pa_c_hj32_local_total_eb_p2_seven) + (pa_p_hj32_local_total_eb_p2_seven_product))) /\ ((((exists pa_h_hj32_local_total_eb_p2_seven_product_partial. pa_h_hj32_local_total_eb_p2_seven_product_partial + S (pa_r_hj32_local_total_eb_p2_seven_product) = S ((S (pa_i_hj32_local_total_eb_p2_seven_product)) * pa_v_hj32_local_total_eb_p2_seven_product)) /\ exists pa_q_hj32_local_total_eb_p2_seven_product_partial. pa_u_hj32_local_total_eb_p2_seven_product = pa_q_hj32_local_total_eb_p2_seven_product_partial * S ((S (pa_i_hj32_local_total_eb_p2_seven_product)) * pa_v_hj32_local_total_eb_p2_seven_product) + (pa_r_hj32_local_total_eb_p2_seven_product))) /\ ((((exists pa_h_hj32_local_total_eb_p2_seven_product_successor. pa_h_hj32_local_total_eb_p2_seven_product_successor + S (pa_s_hj32_local_total_eb_p2_seven_product) = S ((S (S pa_i_hj32_local_total_eb_p2_seven_product)) * pa_v_hj32_local_total_eb_p2_seven_product)) /\ exists pa_q_hj32_local_total_eb_p2_seven_product_successor. pa_u_hj32_local_total_eb_p2_seven_product = pa_q_hj32_local_total_eb_p2_seven_product_successor * S ((S (S pa_i_hj32_local_total_eb_p2_seven_product)) * pa_v_hj32_local_total_eb_p2_seven_product) + (pa_s_hj32_local_total_eb_p2_seven_product))) /\ pa_s_hj32_local_total_eb_p2_seven_product = pa_r_hj32_local_total_eb_p2_seven_product * pa_p_hj32_local_total_eb_p2_seven_product)))))))) - 0013
specialize htotal 2 - 0014
specialize htotal 7 - 0015
exact htotal - 0016
cases eb_p2_seven - 0017
have eb_seed : Le(x1,x2)Exact native replay line
have eb_seed : exists bqb_le_gap_hj32_eb_seed. bqb_le_gap_hj32_eb_seed + (x1) = (x2) - 0018
specialize pow_eleven_two_le_pow_two_seven_from_total x1 - 0019
specialize pow_eleven_two_le_pow_two_seven_from_total x2 - 0020
apply pow_eleven_two_le_pow_two_seven_from_total - 0021
exact htotal - 0022
exact eb_p11_two_witness - 0023
exact eb_p2_seven_witness - 0024
have eb_bound : Le(x,y)Exact native replay line
have eb_bound : exists bqb_le_gap_hj32_local_block_bound_eb_bound. bqb_le_gap_hj32_local_block_bound_eb_bound + (x) = (y) - 0025
specialize pow_block_bound_from_total 11 - 0026
specialize pow_block_bound_from_total 2 - 0027
specialize pow_block_bound_from_total 2 - 0028
specialize pow_block_bound_from_total 7 - 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 eb_p11_two_witness - 0037
exact eb_p2_seven_witness - 0038
exact eb_seed - 0039
exact hx - 0040
exact hy - 0041
exact eb_bound