BT00QE

succ_le_mul_of_two_le_right

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

Multiplying a nonzero natural by a factor at least two exceeds it.

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.

Exact expanded PA statement

forall r p. ~(r = 0) -> (exists bpvl_gap_factor_two. bpvl_gap_factor_two + (2) = (p)) -> (exists bpvl_gap_factor_result. bpvl_gap_factor_result + (S r) = (r * p))

Structural proof guide

Multiplying a nonzero natural by a factor at least two exceeds it.

Direct prerequisites: mul_lt_mul_succ_left_nonzero, mul_le_mul_left, mul_one, le_trans. The authored body proceeds by intermediate claims (2), equality transport (1).

Proof neighborhood

Direct dependencies

BT001N mul_lt_mul_succ_left_nonzero BT001L mul_le_mul_left BT000A mul_one BT000F le_trans

Direct dependents

BT00QF prime_power_exponent_le

Formal native tactic body

Dependencies are hypotheses of the historical Alpha-v12 body receipt. The complete historical Alpha-v18 proof bundle independently checks every dependency; current Alpha v25 preserves that checked theorem use without changing Stable membership.

Read the argument

Proof checkpoints

23 script commands · 4 reading checkpoints · 2 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.

Named ingredients (4)
mul lt mul succ left nonzeromul le mul leftmul onele trans
01Fix variables and assumptionsL1–4

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

  1. L1
    intro r
  2. L2
    intro p
  3. L3
    intro hr
  4. L4
    intro hp
02Establish hstepL5–11

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul lt mul succ left nonzero.

  1. L5
    have hstep : exists k. k + S (r * 1) = r * 2
  2. L6
    specialize mul_lt_mul_succ_left_nonzero r
  3. L7
    specialize mul_lt_mul_succ_left_nonzero 1
  4. L8
    apply mul_lt_mul_succ_left_nonzero
  5. L9
    exact hr
  6. L10
    specialize mul_one r
  7. L11
    rewrite mul_one at hstep
03Establish hscaledL12–21

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul le mul left.

  1. L12
    have hscaled : exists k. k + r * 2 = r * p
  2. L13
    specialize mul_le_mul_left 2
  3. L14
    specialize mul_le_mul_left p
  4. L15
    specialize mul_le_mul_left r
  5. L16
    apply mul_le_mul_left
  6. L17
    exact hp
  7. L18
    specialize le_trans (S r)
  8. L19
    specialize le_trans (r * 2)
  9. L20
    specialize le_trans (r * p)
  10. L21
    apply le_trans
04Use earlier factsL22–23

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

  1. L22
    exact hstep
  2. L23
    exact hscaled

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro r
  2. 0002intro p
  3. 0003intro hr
  4. 0004intro hp
  5. 0005have hstep : exists k. k + S (r * 1) = r * 2
  6. 0006specialize mul_lt_mul_succ_left_nonzero r
  7. 0007specialize mul_lt_mul_succ_left_nonzero 1
  8. 0008apply mul_lt_mul_succ_left_nonzero
  9. 0009exact hr
  10. 0010specialize mul_one r
  11. 0011rewrite mul_one at hstep
  12. 0012have hscaled : exists k. k + r * 2 = r * p
  13. 0013specialize mul_le_mul_left 2
  14. 0014specialize mul_le_mul_left p
  15. 0015specialize mul_le_mul_left r
  16. 0016apply mul_le_mul_left
  17. 0017exact hp
  18. 0018specialize le_trans (S r)
  19. 0019specialize le_trans (r * 2)
  20. 0020specialize le_trans (r * p)
  21. 0021apply le_trans
  22. 0022exact hstep
  23. 0023exact hscaled