FS0000 · theorem body

signed_square_cross_term_zero

dependency-curried kernel-checked candidate body; not enrolled in Alpha or Stable

A canonical signed decoding has a zero positive/negative cross term.

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

forall code pos neg. ((code = 2 * pos /\ neg = 0) \/ exists sd_half_fs_cross. ((code = 2 * sd_half_fs_cross + 1 /\ pos = 0) /\ neg = S sd_half_fs_cross)) -> pos * neg = 0

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order statement
forall code pos neg. ((code = 2 * pos /\ neg = 0) \/ exists sd_half_fs_cross. ((code = 2 * sd_half_fs_cross + 1 /\ pos = 0) /\ neg = S sd_half_fs_cross)) -> pos * neg = 0

Proof neighborhood

Direct theorem prerequisites

signed_decode_normal · Alpha closed mul_zero_left · Stable closed

Direct theorem dependents

none

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

15 script commands · 8 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro code
  2. L2
    intro pos
  3. L3
    intro neg
  4. L4
    intro hdecode
02Use earlier factsL5–7

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

  1. L5
    specialize signed_decode_normal code
  2. L6
    specialize signed_decode_normal pos
  3. L7
    specialize signed_decode_normal neg
03Establish hnormalL8–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed decode normal.

  1. L8
    have hnormal : pos = 0 \/ neg = 0
  2. L9
    apply signed_decode_normal
  3. L10
    exact hdecode
04Separate the logical casesL11–11

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

  1. L11
    cases hnormal
05Calculate and transport equalitiesL12–12

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

  1. L12
    rewrite hnormal_left
06Use earlier factsL13–13

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

  1. L13
    apply mul_zero_left
07Calculate and transport equalitiesL14–14

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

  1. L14
    rewrite hnormal_right
08Use earlier factsL15–15

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

  1. L15
    apply PA5

Library-wide reading audit

Original defined command ledger · 15 lines
  1. 0001intro code
  2. 0002intro pos
  3. 0003intro neg
  4. 0004intro hdecode
  5. 0005specialize signed_decode_normal code
  6. 0006specialize signed_decode_normal pos
  7. 0007specialize signed_decode_normal neg
  8. 0008have hnormal : pos = 0 \/ neg = 0
  9. 0009apply signed_decode_normal
  10. 0010exact hdecode
  11. 0011cases hnormal
  12. 0012rewrite hnormal_left
  13. 0013apply mul_zero_left
  14. 0014rewrite hnormal_right
  15. 0015apply PA5