PA002N

le_scaled_nonzero

Stable checked-use theorem · independently closed

Scaling by a nonzero natural does not decrease a natural.

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 C B. ~(C = 0) -> exists h. h + B = C * B

Structural proof guide

Generated structural guide

Scaling by a nonzero natural does not decrease a natural.

Use the direct prerequisites one_le_of_ne_zero, mul_le_mul_right, one_mul as previously established PA formulas.

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

Referenced ingredients

PA002L one_le_of_ne_zero PA002M mul_le_mul_right PA000M one_mul

Proof neighborhood

Direct dependencies

PA002L one_le_of_ne_zero PA002M mul_le_mul_right PA000M one_mul

Direct dependents

PA002Q new_value_lt_scaled_base PA002T beta_value_lt_scaled_base

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.

Read the argument

Proof checkpoints

16 script commands · 3 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 (3)
one le of ne zeromul le mul rightone mul
01Fix variables and assumptionsL1–3

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

  1. L1
    intro C
  2. L2
    intro B
  3. L3
    intro hC
02Establish h1CL4–7

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply one le of ne zero.

  1. L4
    have h1C : exists h. h + 1 = C
  2. L5
    specialize one_le_of_ne_zero C
  3. L6
    apply one_le_of_ne_zero
  4. L7
    exact hC
03Establish hscaledL8–16

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

  1. L8
    have hscaled : exists h. h + 1 * B = C * B
  2. L9
    specialize mul_le_mul_right 1
  3. L10
    specialize mul_le_mul_right C
  4. L11
    specialize mul_le_mul_right B
  5. L12
    apply mul_le_mul_right
  6. L13
    exact h1C
  7. L14
    specialize one_mul B
  8. L15
    rewrite one_mul at hscaled
  9. L16
    exact hscaled

Library-wide reading audit

Original exact command ledger · 16 lines
  1. 0001intro C
  2. 0002intro B
  3. 0003intro hC
  4. 0004have h1C : exists h. h + 1 = C
  5. 0005specialize one_le_of_ne_zero C
  6. 0006apply one_le_of_ne_zero
  7. 0007exact hC
  8. 0008have hscaled : exists h. h + 1 * B = C * B
  9. 0009specialize mul_le_mul_right 1
  10. 0010specialize mul_le_mul_right C
  11. 0011specialize mul_le_mul_right B
  12. 0012apply mul_le_mul_right
  13. 0013exact h1C
  14. 0014specialize one_mul B
  15. 0015rewrite one_mul at hscaled
  16. 0016exact hscaled