BT00X5 · Bertrand theorem

bertrand_four_power_product_le_of_sum_from_total

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Fourth-power factors are bounded by the power at every larger exponent sum.

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

∀ q. ∀ e. ∀ n. ∀ B. ∀ U. ∀ F. (∀ x. ∀ y. ∃ z. Pow(x,y,z)) → Le(q + e,n) → Pow(4,q,B) → Pow(4,e,U) → Pow(4,n,F) → Le(U · B,F)

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

PD0001 Le PD0020 Pow

6 occurrences

In local proof propositions

PD0001 Le PD0002 Lt PD0020 Pow

3 occurrences

Exact expanded native-PA statement
forall q e n B U F. (forall bpt_a_b6_four_product bpt_e_b6_four_product. exists bpt_x_b6_four_product. (exists ff_b_bpt_value_b6_four_product ff_c_bpt_value_b6_four_product. ((forall ff_i_bpt_value_b6_four_product_repeat. (exists ff_lt_bpt_value_b6_four_product_repeat_bound. ff_lt_bpt_value_b6_four_product_repeat_bound + S ff_i_bpt_value_b6_four_product_repeat = bpt_e_b6_four_product) -> (((exists ff_h_bpt_value_b6_four_product_repeat_decoded. ff_h_bpt_value_b6_four_product_repeat_decoded + S (bpt_a_b6_four_product) = S ((S (ff_i_bpt_value_b6_four_product_repeat)) * ff_c_bpt_value_b6_four_product)) /\ exists ff_q_bpt_value_b6_four_product_repeat_decoded. ff_b_bpt_value_b6_four_product = ff_q_bpt_value_b6_four_product_repeat_decoded * S ((S (ff_i_bpt_value_b6_four_product_repeat)) * ff_c_bpt_value_b6_four_product) + (bpt_a_b6_four_product)))) /\ (exists ff_u_bpt_value_b6_four_product_product ff_v_bpt_value_b6_four_product_product. ((((exists ff_h_bpt_value_b6_four_product_product_start. ff_h_bpt_value_b6_four_product_product_start + S (1) = S ((S (0)) * ff_v_bpt_value_b6_four_product_product)) /\ exists ff_q_bpt_value_b6_four_product_product_start. ff_u_bpt_value_b6_four_product_product = ff_q_bpt_value_b6_four_product_product_start * S ((S (0)) * ff_v_bpt_value_b6_four_product_product) + (1))) /\ ((((exists ff_h_bpt_value_b6_four_product_product_terminal. ff_h_bpt_value_b6_four_product_product_terminal + S (bpt_x_b6_four_product) = S ((S (bpt_e_b6_four_product)) * ff_v_bpt_value_b6_four_product_product)) /\ exists ff_q_bpt_value_b6_four_product_product_terminal. ff_u_bpt_value_b6_four_product_product = ff_q_bpt_value_b6_four_product_product_terminal * S ((S (bpt_e_b6_four_product)) * ff_v_bpt_value_b6_four_product_product) + (bpt_x_b6_four_product))) /\ forall ff_i_bpt_value_b6_four_product_product. (exists ff_lt_bpt_value_b6_four_product_product_bound. ff_lt_bpt_value_b6_four_product_product_bound + S ff_i_bpt_value_b6_four_product_product = bpt_e_b6_four_product) -> exists ff_p_bpt_value_b6_four_product_product ff_r_bpt_value_b6_four_product_product ff_s_bpt_value_b6_four_product_product. ((((exists ff_h_bpt_value_b6_four_product_product_factor. ff_h_bpt_value_b6_four_product_product_factor + S (ff_p_bpt_value_b6_four_product_product) = S ((S (ff_i_bpt_value_b6_four_product_product)) * ff_c_bpt_value_b6_four_product)) /\ exists ff_q_bpt_value_b6_four_product_product_factor. ff_b_bpt_value_b6_four_product = ff_q_bpt_value_b6_four_product_product_factor * S ((S (ff_i_bpt_value_b6_four_product_product)) * ff_c_bpt_value_b6_four_product) + (ff_p_bpt_value_b6_four_product_product))) /\ ((((exists ff_h_bpt_value_b6_four_product_product_partial. ff_h_bpt_value_b6_four_product_product_partial + S (ff_r_bpt_value_b6_four_product_product) = S ((S (ff_i_bpt_value_b6_four_product_product)) * ff_v_bpt_value_b6_four_product_product)) /\ exists ff_q_bpt_value_b6_four_product_product_partial. ff_u_bpt_value_b6_four_product_product = ff_q_bpt_value_b6_four_product_product_partial * S ((S (ff_i_bpt_value_b6_four_product_product)) * ff_v_bpt_value_b6_four_product_product) + (ff_r_bpt_value_b6_four_product_product))) /\ ((((exists ff_h_bpt_value_b6_four_product_product_successor. ff_h_bpt_value_b6_four_product_product_successor + S (ff_s_bpt_value_b6_four_product_product) = S ((S (S ff_i_bpt_value_b6_four_product_product)) * ff_v_bpt_value_b6_four_product_product)) /\ exists ff_q_bpt_value_b6_four_product_product_successor. ff_u_bpt_value_b6_four_product_product = ff_q_bpt_value_b6_four_product_product_successor * S ((S (S ff_i_bpt_value_b6_four_product_product)) * ff_v_bpt_value_b6_four_product_product) + (ff_s_bpt_value_b6_four_product_product))) /\ ff_s_bpt_value_b6_four_product_product = ff_r_bpt_value_b6_four_product_product * ff_p_bpt_value_b6_four_product_product))))))))) -> (exists bqb_le_gap_b6_four_product_sum. bqb_le_gap_b6_four_product_sum + (q + e) = (n)) -> (exists pa_b_b6_four_product_q pa_c_b6_four_product_q. ((forall pa_i_b6_four_product_q_repeat. (exists pa_lt_b6_four_product_q_repeat_bound. pa_lt_b6_four_product_q_repeat_bound + S pa_i_b6_four_product_q_repeat = q) -> (((exists pa_h_b6_four_product_q_repeat_decoded. pa_h_b6_four_product_q_repeat_decoded + S (4) = S ((S (pa_i_b6_four_product_q_repeat)) * pa_c_b6_four_product_q)) /\ exists pa_q_b6_four_product_q_repeat_decoded. pa_b_b6_four_product_q = pa_q_b6_four_product_q_repeat_decoded * S ((S (pa_i_b6_four_product_q_repeat)) * pa_c_b6_four_product_q) + (4)))) /\ (exists pa_u_b6_four_product_q_product pa_v_b6_four_product_q_product. ((((exists pa_h_b6_four_product_q_product_start. pa_h_b6_four_product_q_product_start + S (1) = S ((S (0)) * pa_v_b6_four_product_q_product)) /\ exists pa_q_b6_four_product_q_product_start. pa_u_b6_four_product_q_product = pa_q_b6_four_product_q_product_start * S ((S (0)) * pa_v_b6_four_product_q_product) + (1))) /\ ((((exists pa_h_b6_four_product_q_product_terminal. pa_h_b6_four_product_q_product_terminal + S (B) = S ((S (q)) * pa_v_b6_four_product_q_product)) /\ exists pa_q_b6_four_product_q_product_terminal. pa_u_b6_four_product_q_product = pa_q_b6_four_product_q_product_terminal * S ((S (q)) * pa_v_b6_four_product_q_product) + (B))) /\ forall pa_i_b6_four_product_q_product. (exists pa_lt_b6_four_product_q_product_bound. pa_lt_b6_four_product_q_product_bound + S pa_i_b6_four_product_q_product = q) -> exists pa_p_b6_four_product_q_product pa_r_b6_four_product_q_product pa_s_b6_four_product_q_product. ((((exists pa_h_b6_four_product_q_product_factor. pa_h_b6_four_product_q_product_factor + S (pa_p_b6_four_product_q_product) = S ((S (pa_i_b6_four_product_q_product)) * pa_c_b6_four_product_q)) /\ exists pa_q_b6_four_product_q_product_factor. pa_b_b6_four_product_q = pa_q_b6_four_product_q_product_factor * S ((S (pa_i_b6_four_product_q_product)) * pa_c_b6_four_product_q) + (pa_p_b6_four_product_q_product))) /\ ((((exists pa_h_b6_four_product_q_product_partial. pa_h_b6_four_product_q_product_partial + S (pa_r_b6_four_product_q_product) = S ((S (pa_i_b6_four_product_q_product)) * pa_v_b6_four_product_q_product)) /\ exists pa_q_b6_four_product_q_product_partial. pa_u_b6_four_product_q_product = pa_q_b6_four_product_q_product_partial * S ((S (pa_i_b6_four_product_q_product)) * pa_v_b6_four_product_q_product) + (pa_r_b6_four_product_q_product))) /\ ((((exists pa_h_b6_four_product_q_product_successor. pa_h_b6_four_product_q_product_successor + S (pa_s_b6_four_product_q_product) = S ((S (S pa_i_b6_four_product_q_product)) * pa_v_b6_four_product_q_product)) /\ exists pa_q_b6_four_product_q_product_successor. pa_u_b6_four_product_q_product = pa_q_b6_four_product_q_product_successor * S ((S (S pa_i_b6_four_product_q_product)) * pa_v_b6_four_product_q_product) + (pa_s_b6_four_product_q_product))) /\ pa_s_b6_four_product_q_product = pa_r_b6_four_product_q_product * pa_p_b6_four_product_q_product)))))))) -> (exists pa_b_b6_four_product_e pa_c_b6_four_product_e. ((forall pa_i_b6_four_product_e_repeat. (exists pa_lt_b6_four_product_e_repeat_bound. pa_lt_b6_four_product_e_repeat_bound + S pa_i_b6_four_product_e_repeat = e) -> (((exists pa_h_b6_four_product_e_repeat_decoded. pa_h_b6_four_product_e_repeat_decoded + S (4) = S ((S (pa_i_b6_four_product_e_repeat)) * pa_c_b6_four_product_e)) /\ exists pa_q_b6_four_product_e_repeat_decoded. pa_b_b6_four_product_e = pa_q_b6_four_product_e_repeat_decoded * S ((S (pa_i_b6_four_product_e_repeat)) * pa_c_b6_four_product_e) + (4)))) /\ (exists pa_u_b6_four_product_e_product pa_v_b6_four_product_e_product. ((((exists pa_h_b6_four_product_e_product_start. pa_h_b6_four_product_e_product_start + S (1) = S ((S (0)) * pa_v_b6_four_product_e_product)) /\ exists pa_q_b6_four_product_e_product_start. pa_u_b6_four_product_e_product = pa_q_b6_four_product_e_product_start * S ((S (0)) * pa_v_b6_four_product_e_product) + (1))) /\ ((((exists pa_h_b6_four_product_e_product_terminal. pa_h_b6_four_product_e_product_terminal + S (U) = S ((S (e)) * pa_v_b6_four_product_e_product)) /\ exists pa_q_b6_four_product_e_product_terminal. pa_u_b6_four_product_e_product = pa_q_b6_four_product_e_product_terminal * S ((S (e)) * pa_v_b6_four_product_e_product) + (U))) /\ forall pa_i_b6_four_product_e_product. (exists pa_lt_b6_four_product_e_product_bound. pa_lt_b6_four_product_e_product_bound + S pa_i_b6_four_product_e_product = e) -> exists pa_p_b6_four_product_e_product pa_r_b6_four_product_e_product pa_s_b6_four_product_e_product. ((((exists pa_h_b6_four_product_e_product_factor. pa_h_b6_four_product_e_product_factor + S (pa_p_b6_four_product_e_product) = S ((S (pa_i_b6_four_product_e_product)) * pa_c_b6_four_product_e)) /\ exists pa_q_b6_four_product_e_product_factor. pa_b_b6_four_product_e = pa_q_b6_four_product_e_product_factor * S ((S (pa_i_b6_four_product_e_product)) * pa_c_b6_four_product_e) + (pa_p_b6_four_product_e_product))) /\ ((((exists pa_h_b6_four_product_e_product_partial. pa_h_b6_four_product_e_product_partial + S (pa_r_b6_four_product_e_product) = S ((S (pa_i_b6_four_product_e_product)) * pa_v_b6_four_product_e_product)) /\ exists pa_q_b6_four_product_e_product_partial. pa_u_b6_four_product_e_product = pa_q_b6_four_product_e_product_partial * S ((S (pa_i_b6_four_product_e_product)) * pa_v_b6_four_product_e_product) + (pa_r_b6_four_product_e_product))) /\ ((((exists pa_h_b6_four_product_e_product_successor. pa_h_b6_four_product_e_product_successor + S (pa_s_b6_four_product_e_product) = S ((S (S pa_i_b6_four_product_e_product)) * pa_v_b6_four_product_e_product)) /\ exists pa_q_b6_four_product_e_product_successor. pa_u_b6_four_product_e_product = pa_q_b6_four_product_e_product_successor * S ((S (S pa_i_b6_four_product_e_product)) * pa_v_b6_four_product_e_product) + (pa_s_b6_four_product_e_product))) /\ pa_s_b6_four_product_e_product = pa_r_b6_four_product_e_product * pa_p_b6_four_product_e_product)))))))) -> (exists pa_b_b6_four_product_n pa_c_b6_four_product_n. ((forall pa_i_b6_four_product_n_repeat. (exists pa_lt_b6_four_product_n_repeat_bound. pa_lt_b6_four_product_n_repeat_bound + S pa_i_b6_four_product_n_repeat = n) -> (((exists pa_h_b6_four_product_n_repeat_decoded. pa_h_b6_four_product_n_repeat_decoded + S (4) = S ((S (pa_i_b6_four_product_n_repeat)) * pa_c_b6_four_product_n)) /\ exists pa_q_b6_four_product_n_repeat_decoded. pa_b_b6_four_product_n = pa_q_b6_four_product_n_repeat_decoded * S ((S (pa_i_b6_four_product_n_repeat)) * pa_c_b6_four_product_n) + (4)))) /\ (exists pa_u_b6_four_product_n_product pa_v_b6_four_product_n_product. ((((exists pa_h_b6_four_product_n_product_start. pa_h_b6_four_product_n_product_start + S (1) = S ((S (0)) * pa_v_b6_four_product_n_product)) /\ exists pa_q_b6_four_product_n_product_start. pa_u_b6_four_product_n_product = pa_q_b6_four_product_n_product_start * S ((S (0)) * pa_v_b6_four_product_n_product) + (1))) /\ ((((exists pa_h_b6_four_product_n_product_terminal. pa_h_b6_four_product_n_product_terminal + S (F) = S ((S (n)) * pa_v_b6_four_product_n_product)) /\ exists pa_q_b6_four_product_n_product_terminal. pa_u_b6_four_product_n_product = pa_q_b6_four_product_n_product_terminal * S ((S (n)) * pa_v_b6_four_product_n_product) + (F))) /\ forall pa_i_b6_four_product_n_product. (exists pa_lt_b6_four_product_n_product_bound. pa_lt_b6_four_product_n_product_bound + S pa_i_b6_four_product_n_product = n) -> exists pa_p_b6_four_product_n_product pa_r_b6_four_product_n_product pa_s_b6_four_product_n_product. ((((exists pa_h_b6_four_product_n_product_factor. pa_h_b6_four_product_n_product_factor + S (pa_p_b6_four_product_n_product) = S ((S (pa_i_b6_four_product_n_product)) * pa_c_b6_four_product_n)) /\ exists pa_q_b6_four_product_n_product_factor. pa_b_b6_four_product_n = pa_q_b6_four_product_n_product_factor * S ((S (pa_i_b6_four_product_n_product)) * pa_c_b6_four_product_n) + (pa_p_b6_four_product_n_product))) /\ ((((exists pa_h_b6_four_product_n_product_partial. pa_h_b6_four_product_n_product_partial + S (pa_r_b6_four_product_n_product) = S ((S (pa_i_b6_four_product_n_product)) * pa_v_b6_four_product_n_product)) /\ exists pa_q_b6_four_product_n_product_partial. pa_u_b6_four_product_n_product = pa_q_b6_four_product_n_product_partial * S ((S (pa_i_b6_four_product_n_product)) * pa_v_b6_four_product_n_product) + (pa_r_b6_four_product_n_product))) /\ ((((exists pa_h_b6_four_product_n_product_successor. pa_h_b6_four_product_n_product_successor + S (pa_s_b6_four_product_n_product) = S ((S (S pa_i_b6_four_product_n_product)) * pa_v_b6_four_product_n_product)) /\ exists pa_q_b6_four_product_n_product_successor. pa_u_b6_four_product_n_product = pa_q_b6_four_product_n_product_successor * S ((S (S pa_i_b6_four_product_n_product)) * pa_v_b6_four_product_n_product) + (pa_s_b6_four_product_n_product))) /\ pa_s_b6_four_product_n_product = pa_r_b6_four_product_n_product * pa_p_b6_four_product_n_product)))))))) -> (exists bqb_le_gap_b6_four_product_result. bqb_le_gap_b6_four_product_result + (U * B) = (F))

Proof neighborhood

Direct theorem prerequisites

BT009X pow_add BT00SN pow_exponent_monotone_from_total BT0006 mul_comm

Direct theorem dependents

BT00X6 bertrand_main_inequality_factorized_from_total

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

51 script commands · 13 reading checkpoints · 5 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 (3)
pow addpow exponent monotone from totalmul comm
01Fix variables and assumptionsL1–10

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

  1. L1
    intro q
  2. L2
    intro e
  3. L3
    intro n
  4. L4
    intro B
  5. L5
    intro U
  6. L6
    intro F
  7. L7
    intro htotal
  8. L8
    intro hsum
  9. L9
    intro hB
  10. L10
    intro hU
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hF
03Establish hx_existsL12–15

Establish this local claim before using it. It is not an additional assumption.

  1. L12
    have hx_exists : ∃ x. Pow(4,q + e,x)Definitions: Pow(4,q + e,x)Original native command in the exact edition
  2. L13
    specialize htotal 4
  3. L14
    specialize htotal (q + e)
  4. L15
    exact htotal
04Separate the logical casesL16–16

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L16
    cases hx_exists
05Establish hfactorL17–26

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow add.

  1. L17
    have hfactor : x = B * U
  2. L18
    specialize pow_add 4
  3. L19
    specialize pow_add q
  4. L20
    specialize pow_add e
  5. L21
    specialize pow_add (q + e)
  6. L22
    specialize pow_add B
  7. L23
    specialize pow_add U
  8. L24
    specialize pow_add x
  9. L25
    apply pow_add
  10. L26
    refl
06Use earlier factsL27–29

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

  1. L27
    exact hB
  2. L28
    exact hU
  3. L29
    exact hx_exists_witness
07Establish hfourL30–30

Establish this local claim before using it. It is not an additional assumption.

  1. L30
08Construct an explicit witnessL31–31

Supply the displayed value, then prove that it has the required property.

  1. L31
    exists 3
09Calculate and transport equalitiesL32–32

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L32
    norm_num
10Establish hcombinedL33–42

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow exponent monotone from total.

  1. L33
    have hcombined : Le(x,F)Definitions: Le(x,F)Original native command in the exact edition
  2. L34
    specialize pow_exponent_monotone_from_total 4
  3. L35
    specialize pow_exponent_monotone_from_total (q + e)
  4. L36
    specialize pow_exponent_monotone_from_total n
  5. L37
    specialize pow_exponent_monotone_from_total x
  6. L38
    specialize pow_exponent_monotone_from_total F
  7. L39
    apply pow_exponent_monotone_from_total
  8. L40
    exact htotal
  9. L41
    exact hfour
  10. L42
    exact hsum
11Use earlier factsL43–44

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

  1. L43
    exact hx_exists_witness
  2. L44
    exact hF
12Calculate and transport equalitiesL45–45

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L45
    rewrite hfactor at hcombined
13Establish hcommL46–51

Establish this local claim before using it. It is not an additional assumption.

  1. L46
    have hcomm : U * B = B * U
  2. L47
    specialize mul_comm U
  3. L48
    specialize mul_comm B
  4. L49
    exact mul_comm
  5. L50
    rewrite hcomm
  6. L51
    exact hcombined

Library-wide reading audit

Original defined command ledger · 51 lines
  1. 0001intro q
  2. 0002intro e
  3. 0003intro n
  4. 0004intro B
  5. 0005intro U
  6. 0006intro F
  7. 0007intro htotal
  8. 0008intro hsum
  9. 0009intro hB
  10. 0010intro hU
  11. 0011intro hF
  12. 0012have hx_exists : ∃ x. Pow(4,q + e,x)
    Exact native replay linehave hx_exists : exists x. (exists pa_b_b6_four_product_combined pa_c_b6_four_product_combined. ((forall pa_i_b6_four_product_combined_repeat. (exists pa_lt_b6_four_product_combined_repeat_bound. pa_lt_b6_four_product_combined_repeat_bound + S pa_i_b6_four_product_combined_repeat = q + e) -> (((exists pa_h_b6_four_product_combined_repeat_decoded. pa_h_b6_four_product_combined_repeat_decoded + S (4) = S ((S (pa_i_b6_four_product_combined_repeat)) * pa_c_b6_four_product_combined)) /\ exists pa_q_b6_four_product_combined_repeat_decoded. pa_b_b6_four_product_combined = pa_q_b6_four_product_combined_repeat_decoded * S ((S (pa_i_b6_four_product_combined_repeat)) * pa_c_b6_four_product_combined) + (4)))) /\ (exists pa_u_b6_four_product_combined_product pa_v_b6_four_product_combined_product. ((((exists pa_h_b6_four_product_combined_product_start. pa_h_b6_four_product_combined_product_start + S (1) = S ((S (0)) * pa_v_b6_four_product_combined_product)) /\ exists pa_q_b6_four_product_combined_product_start. pa_u_b6_four_product_combined_product = pa_q_b6_four_product_combined_product_start * S ((S (0)) * pa_v_b6_four_product_combined_product) + (1))) /\ ((((exists pa_h_b6_four_product_combined_product_terminal. pa_h_b6_four_product_combined_product_terminal + S (x) = S ((S (q + e)) * pa_v_b6_four_product_combined_product)) /\ exists pa_q_b6_four_product_combined_product_terminal. pa_u_b6_four_product_combined_product = pa_q_b6_four_product_combined_product_terminal * S ((S (q + e)) * pa_v_b6_four_product_combined_product) + (x))) /\ forall pa_i_b6_four_product_combined_product. (exists pa_lt_b6_four_product_combined_product_bound. pa_lt_b6_four_product_combined_product_bound + S pa_i_b6_four_product_combined_product = q + e) -> exists pa_p_b6_four_product_combined_product pa_r_b6_four_product_combined_product pa_s_b6_four_product_combined_product. ((((exists pa_h_b6_four_product_combined_product_factor. pa_h_b6_four_product_combined_product_factor + S (pa_p_b6_four_product_combined_product) = S ((S (pa_i_b6_four_product_combined_product)) * pa_c_b6_four_product_combined)) /\ exists pa_q_b6_four_product_combined_product_factor. pa_b_b6_four_product_combined = pa_q_b6_four_product_combined_product_factor * S ((S (pa_i_b6_four_product_combined_product)) * pa_c_b6_four_product_combined) + (pa_p_b6_four_product_combined_product))) /\ ((((exists pa_h_b6_four_product_combined_product_partial. pa_h_b6_four_product_combined_product_partial + S (pa_r_b6_four_product_combined_product) = S ((S (pa_i_b6_four_product_combined_product)) * pa_v_b6_four_product_combined_product)) /\ exists pa_q_b6_four_product_combined_product_partial. pa_u_b6_four_product_combined_product = pa_q_b6_four_product_combined_product_partial * S ((S (pa_i_b6_four_product_combined_product)) * pa_v_b6_four_product_combined_product) + (pa_r_b6_four_product_combined_product))) /\ ((((exists pa_h_b6_four_product_combined_product_successor. pa_h_b6_four_product_combined_product_successor + S (pa_s_b6_four_product_combined_product) = S ((S (S pa_i_b6_four_product_combined_product)) * pa_v_b6_four_product_combined_product)) /\ exists pa_q_b6_four_product_combined_product_successor. pa_u_b6_four_product_combined_product = pa_q_b6_four_product_combined_product_successor * S ((S (S pa_i_b6_four_product_combined_product)) * pa_v_b6_four_product_combined_product) + (pa_s_b6_four_product_combined_product))) /\ pa_s_b6_four_product_combined_product = pa_r_b6_four_product_combined_product * pa_p_b6_four_product_combined_product))))))))
  13. 0013specialize htotal 4
  14. 0014specialize htotal (q + e)
  15. 0015exact htotal
  16. 0016cases hx_exists
  17. 0017have hfactor : x = B * U
  18. 0018specialize pow_add 4
  19. 0019specialize pow_add q
  20. 0020specialize pow_add e
  21. 0021specialize pow_add (q + e)
  22. 0022specialize pow_add B
  23. 0023specialize pow_add U
  24. 0024specialize pow_add x
  25. 0025apply pow_add
  26. 0026refl
  27. 0027exact hB
  28. 0028exact hU
  29. 0029exact hx_exists_witness
  30. 0030have hfour : Lt(0,4)
    Exact native replay linehave hfour : exists bqb_le_gap_b6_four_product_base_positive. bqb_le_gap_b6_four_product_base_positive + (1) = (4)
  31. 0031exists 3
  32. 0032norm_num
  33. 0033have hcombined : Le(x,F)
    Exact native replay linehave hcombined : exists bqb_le_gap_b6_four_product_combined_order. bqb_le_gap_b6_four_product_combined_order + (x) = (F)
  34. 0034specialize pow_exponent_monotone_from_total 4
  35. 0035specialize pow_exponent_monotone_from_total (q + e)
  36. 0036specialize pow_exponent_monotone_from_total n
  37. 0037specialize pow_exponent_monotone_from_total x
  38. 0038specialize pow_exponent_monotone_from_total F
  39. 0039apply pow_exponent_monotone_from_total
  40. 0040exact htotal
  41. 0041exact hfour
  42. 0042exact hsum
  43. 0043exact hx_exists_witness
  44. 0044exact hF
  45. 0045rewrite hfactor at hcombined
  46. 0046have hcomm : U * B = B * U
  47. 0047specialize mul_comm U
  48. 0048specialize mul_comm B
  49. 0049exact mul_comm
  50. 0050rewrite hcomm
  51. 0051exact hcombined