PA005T

pow_one_from_zero_successor

Stable checked-use theorem · independently closed

A successor of a zero exponent gives the relational first power.

Exact expanded PA statement

forall a z e n. z = 0 -> e = S z -> (exists ff_b_one_carrier ff_c_one_carrier. ((forall ff_i_one_carrier_repeat. (exists ff_lt_one_carrier_repeat_bound. ff_lt_one_carrier_repeat_bound + S ff_i_one_carrier_repeat = e) -> (((exists ff_h_one_carrier_repeat_decoded. ff_h_one_carrier_repeat_decoded + S (a) = S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier)) /\ exists ff_q_one_carrier_repeat_decoded. ff_b_one_carrier = ff_q_one_carrier_repeat_decoded * S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier) + (a)))) /\ (exists ff_u_one_carrier_product ff_v_one_carrier_product. ((((exists ff_h_one_carrier_product_start. ff_h_one_carrier_product_start + S (1) = S ((S (0)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_start. ff_u_one_carrier_product = ff_q_one_carrier_product_start * S ((S (0)) * ff_v_one_carrier_product) + (1))) /\ ((((exists ff_h_one_carrier_product_terminal. ff_h_one_carrier_product_terminal + S (n) = S ((S (e)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_terminal. ff_u_one_carrier_product = ff_q_one_carrier_product_terminal * S ((S (e)) * ff_v_one_carrier_product) + (n))) /\ forall ff_i_one_carrier_product. (exists ff_lt_one_carrier_product_bound. ff_lt_one_carrier_product_bound + S ff_i_one_carrier_product = e) -> exists ff_p_one_carrier_product ff_r_one_carrier_product ff_s_one_carrier_product. ((((exists ff_h_one_carrier_product_factor. ff_h_one_carrier_product_factor + S (ff_p_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier)) /\ exists ff_q_one_carrier_product_factor. ff_b_one_carrier = ff_q_one_carrier_product_factor * S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier) + (ff_p_one_carrier_product))) /\ ((((exists ff_h_one_carrier_product_partial. ff_h_one_carrier_product_partial + S (ff_r_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_partial. ff_u_one_carrier_product = ff_q_one_carrier_product_partial * S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_r_one_carrier_product))) /\ ((((exists ff_h_one_carrier_product_successor. ff_h_one_carrier_product_successor + S (ff_s_one_carrier_product) = S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\ exists ff_q_one_carrier_product_successor. ff_u_one_carrier_product = ff_q_one_carrier_product_successor * S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_s_one_carrier_product))) /\ ff_s_one_carrier_product = ff_r_one_carrier_product * ff_p_one_carrier_product)))))))) -> n = a

Structural proof guide

Generated structural guide

A successor of a zero exponent gives the relational first power.

Use the direct prerequisites pow_successor_decompose, pow_zero, one_mul as previously established PA formulas.

The proof proceeds by case analysis (2), intermediate claims (2), equality transport (1).

Referenced ingredients

PA004D pow_successor_decompose PA004B pow_zero PA000M one_mul

Proof neighborhood

Direct dependencies

PA004D pow_successor_decompose PA004B pow_zero PA000M one_mul

Direct dependents

PA005U pow_one

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Linked names are exact direct references. This Stable checked-use theorem is independently kernel-checked when replayed.

  1. 0001intro a
  2. 0002intro z
  3. 0003intro e
  4. 0004intro n
  5. 0005intro hz
  6. 0006intro he
  7. 0007intro hpow
  8. 0008have hstep : exists r. (exists ff_b_one_predecessor ff_c_one_predecessor. ((forall ff_i_one_predecessor_repeat. (exists ff_lt_one_predecessor_repeat_bound. ff_lt_one_predecessor_repeat_bound + S ff_i_one_predecessor_repeat = z) -> (((exists ff_h_one_predecessor_repeat_decoded. ff_h_one_predecessor_repeat_decoded + S (a) = S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor)) /\ exists ff_q_one_predecessor_repeat_decoded. ff_b_one_predecessor = ff_q_one_predecessor_repeat_decoded * S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor) + (a)))) /\ (exists ff_u_one_predecessor_product ff_v_one_predecessor_product. ((((exists ff_h_one_predecessor_product_start. ff_h_one_predecessor_product_start + S (1) = S ((S (0)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_start. ff_u_one_predecessor_product = ff_q_one_predecessor_product_start * S ((S (0)) * ff_v_one_predecessor_product) + (1))) /\ ((((exists ff_h_one_predecessor_product_terminal. ff_h_one_predecessor_product_terminal + S (r) = S ((S (z)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_terminal. ff_u_one_predecessor_product = ff_q_one_predecessor_product_terminal * S ((S (z)) * ff_v_one_predecessor_product) + (r))) /\ forall ff_i_one_predecessor_product. (exists ff_lt_one_predecessor_product_bound. ff_lt_one_predecessor_product_bound + S ff_i_one_predecessor_product = z) -> exists ff_p_one_predecessor_product ff_r_one_predecessor_product ff_s_one_predecessor_product. ((((exists ff_h_one_predecessor_product_factor. ff_h_one_predecessor_product_factor + S (ff_p_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor)) /\ exists ff_q_one_predecessor_product_factor. ff_b_one_predecessor = ff_q_one_predecessor_product_factor * S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor) + (ff_p_one_predecessor_product))) /\ ((((exists ff_h_one_predecessor_product_partial. ff_h_one_predecessor_product_partial + S (ff_r_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_partial. ff_u_one_predecessor_product = ff_q_one_predecessor_product_partial * S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_r_one_predecessor_product))) /\ ((((exists ff_h_one_predecessor_product_successor. ff_h_one_predecessor_product_successor + S (ff_s_one_predecessor_product) = S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\ exists ff_q_one_predecessor_product_successor. ff_u_one_predecessor_product = ff_q_one_predecessor_product_successor * S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_s_one_predecessor_product))) /\ ff_s_one_predecessor_product = ff_r_one_predecessor_product * ff_p_one_predecessor_product)))))))) /\ n = r * a
  9. 0009specialize pow_successor_decompose a
  10. 0010specialize pow_successor_decompose z
  11. 0011specialize pow_successor_decompose e
  12. 0012specialize pow_successor_decompose n
  13. 0013apply pow_successor_decompose
  14. 0014exact he
  15. 0015exact hpow
  16. 0016cases hstep
  17. 0017cases hstep_witness
  18. 0018have hr : x = 1
  19. 0019specialize pow_zero a
  20. 0020specialize pow_zero z
  21. 0021specialize pow_zero x
  22. 0022apply pow_zero
  23. 0023exact hz
  24. 0024exact hstep_witness_left
  25. 0025trans x * a
  26. 0026exact hstep_witness_right
  27. 0027rewrite hr
  28. 0028specialize one_mul a
  29. 0029exact one_mul