BT0016

succ_le_succ

Stable checked-use theorem · independently kernel verified

Successor preserves the witness-defined order.

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 a b. (exists k. k + a = b) -> exists r. r + S a = S b

Structural proof guide

Successor preserves the witness-defined order.

Direct prerequisites: none. The authored body proceeds by case analysis (1), equality transport (1).

Proof neighborhood

Direct dependencies

none

Direct dependents

BT0057 beta_value_lt_scaled_base BT0058 new_value_lt_scaled_base BT005E beta_prefix_product_trace_exists BT005K beta_product_succ_append BT0089 beta_prefix_sum_trace_exists BT00QF prime_power_exponent_le BT00TC choose_functional BT00TE choose_zero BT00TI choose_succ_succ_of_lt BT00TJ choose_succ_succ

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

8 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 assumptionsL1–3

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro h
02Separate the logical casesL4–4

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

  1. L4
    cases h
03Construct an explicit witnessL5–5

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

  1. L5
    exists x
04Calculate and transport equalitiesL6–7

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

  1. L6
    rewrite PA4
  2. L7
    congr
05Use earlier factsL8–8

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

  1. L8
    exact h_witness

Library-wide reading audit

Original exact command ledger · 8 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro h
  4. 0004cases h
  5. 0005exists x
  6. 0006rewrite PA4
  7. 0007congr
  8. 0008exact h_witness