PA004F · theorem

finite_surjective_zero

Stable checked-use theorem · independently closed

The empty decoded prefix is surjective onto the empty interval.

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.

Statement with defined notation

∀ b. ∀ c. ∀ n. n = 0 → SurjectivePrefix(b,c,n)

Every purple notation token opens its conservative definition. This is a reading surface; the compiler expands the statement before the unchanged kernel checks it.

Definitions used by this theorem

In the theorem statement

PD0026 SurjectivePrefix

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall b c n. n = 0 -> (forall fp_value_zero. (exists fp_gap_zero_value. fp_gap_zero_value + S fp_value_zero = n) -> exists fp_i_zero. ((exists fp_gap_zero_index. fp_gap_zero_index + S fp_i_zero = n) /\ (((exists ff_h_zero_entry. ff_h_zero_entry + S (fp_value_zero) = S ((S (fp_i_zero)) * c)) /\ exists ff_q_zero_entry. b = ff_q_zero_entry * S ((S (fp_i_zero)) * c) + (fp_value_zero)))))

Proof neighborhood

Direct theorem prerequisites

PA0004 add_eq_zero_right PA0005 succ_ne_zero

Direct theorem dependents

PA004W finite_bounded_injective_surjective

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. The untrusted compiler re-expands each one before the original tactic script is replayed; defined notation is never accepted by the kernel. Open the exact replay line beneath every changed command.

Read the argument

Proof checkpoints

17 script commands · 4 reading checkpoints · 1 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
add eq zero rightsucc ne zero
01Fix variables and assumptionsL1–6

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro n
  4. L4
    intro hn
  5. L5
    intro y
  6. L6
    intro hy
02Calculate and transport equalitiesL7–7

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

  1. L7
    rewrite hn at hy
03Separate the logical casesL8–9

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

  1. L8
    exfalso
  2. L9
    cases hy
04Establish hsyL10–17

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

  1. L10
    have hsy : S y = 0
  2. L11
    specialize add_eq_zero_right x
  3. L12
    specialize add_eq_zero_right (S y)
  4. L13
    apply add_eq_zero_right
  5. L14
    exact hy_witness
  6. L15
    specialize succ_ne_zero y
  7. L16
    apply succ_ne_zero
  8. L17
    exact hsy

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro n
  4. 0004intro hn
  5. 0005intro y
  6. 0006intro hy
  7. 0007rewrite hn at hy
  8. 0008exfalso
  9. 0009cases hy
  10. 0010have hsy : S y = 0
  11. 0011specialize add_eq_zero_right x
  12. 0012specialize add_eq_zero_right (S y)
  13. 0013apply add_eq_zero_right
  14. 0014exact hy_witness
  15. 0015specialize succ_ne_zero y
  16. 0016apply succ_ne_zero
  17. 0017exact hsy