PA002R

beta_value_le_code

Stable checked-use theorem · independently closed

Every decoded beta value is at most its code.

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 b c i x. ((exists h. h + S x = S ((S i) * c)) /\ exists q. b = q * S ((S i) * c) + x) -> exists h. h + x = b

Structural proof guide

Generated structural guide

Every decoded beta value is at most its code.

This root lemma is proved directly from the PA rules and the hypotheses introduced by its statement.

The proof proceeds by case analysis (2).

Referenced ingredients

none

Proof neighborhood

Direct dependencies

none

Direct dependents

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

10 script commands · 5 reading checkpoints · 0 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro i
  4. L4
    intro x
  5. L5
    intro hat
02Separate the logical casesL6–7

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

  1. L6
    cases hat
  2. L7
    cases hat_right
03Construct an explicit witnessL8–8

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

  1. L8
    exists x1 * S ((S i) * c)
04Calculate and transport equalitiesL9–9

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

  1. L9
    symm
05Use earlier factsL10–10

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

  1. L10
    exact hat_right_witness

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro i
  4. 0004intro x
  5. 0005intro hat
  6. 0006cases hat
  7. 0007cases hat_right
  8. 0008exists x1 * S ((S i) * c)
  9. 0009symm
  10. 0010exact hat_right_witness