PA0010

lt_irrefl_expanded

Stable checked-use theorem · independently closed

No natural is strictly below itself, with strict order fully expanded.

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 n. ~(exists k. k + S n = n)

Structural proof guide

Generated structural guide

No natural is strictly below itself, with strict order fully expanded.

Use the direct prerequisites add_succ_left, no_succ_add_fixed as previously established PA formulas.

The proof proceeds by case analysis (1).

Referenced ingredients

PA000E add_succ_left PA000Y no_succ_add_fixed

Proof neighborhood

Direct dependencies

PA000E add_succ_left PA000Y no_succ_add_fixed

Direct dependents

PA001U beta_exclusive_accumulated_product_step PA004K beta_prefix_swap_last_from_entries PA004R finite_last_is_top_from_prefix_surjective PA004W finite_bounded_injective_surjective PA004X finite_fixed_last_prefix_bounded PA0052 beta_product_replace_balance PA0053 beta_product_swap_last_invariant PA0097 finite_short_cover_impossible PA00C3 odd_half_positive_complement_exists PA00CU beta_sum_replace_balance PA00CV beta_sum_swap_last_invariant

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

12 script commands · 7 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.

Named ingredients (2)
add succ leftno succ add fixed
01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro h
02Separate the logical casesL3–3

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

  1. L3
    cases h
03Use earlier factsL4–6

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

  1. L4
    specialize no_succ_add_fixed x
  2. L5
    specialize no_succ_add_fixed n
  3. L6
    apply no_succ_add_fixed
04Calculate and transport equalitiesL7–8

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

  1. L7
    trans x + S n
  2. L8
    trans S (x + n)
05Use earlier factsL9–9

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

  1. L9
    apply add_succ_left
06Calculate and transport equalitiesL10–10

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

  1. L10
    symm
07Use earlier factsL11–12

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

  1. L11
    apply PA4
  2. L12
    exact h_witness

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro n
  2. 0002intro h
  3. 0003cases h
  4. 0004specialize no_succ_add_fixed x
  5. 0005specialize no_succ_add_fixed n
  6. 0006apply no_succ_add_fixed
  7. 0007trans x + S n
  8. 0008trans S (x + n)
  9. 0009apply add_succ_left
  10. 0010symm
  11. 0011apply PA4
  12. 0012exact h_witness