HL0001

hensel_mod_add_zero_cancel

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

A known zero-congruent summand can be cancelled without subtraction.

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 first-order arithmetic statement

forall m a q. (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. (a + q) + m * hgcrt_mod_left_hpl_mod = 0 + m * hgcrt_mod_right_hpl_mod) -> (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. q + m * hgcrt_mod_left_hpl_mod = 0 + m * hgcrt_mod_right_hpl_mod) -> (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. a + m * hgcrt_mod_left_hpl_mod = 0 + m * hgcrt_mod_right_hpl_mod)

Constructive proof overview

Generated structural guide

A known zero-congruent summand can be cancelled without subtraction.

The unchanged tactic script uses 4 declared prerequisites and contains 24 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mod_eq_add_cancel_right Alpha theorem; checked-use authorized mod_eq_trans Stable theorem; checked-use authorized mod_eq_symm Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

HL0007 hensel_lift_correction_of_root

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

24 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro m
  2. L2
    intro a
  3. L3
    intro q
  4. L4
    intro hsum
  5. L5
    intro hq
02Use earlier factsL6–15

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

  1. L6
    specialize mod_eq_add_cancel_right m
  2. L7
    specialize mod_eq_add_cancel_right a
  3. L8
    specialize mod_eq_add_cancel_right 0
  4. L9
    specialize mod_eq_add_cancel_right q
  5. L10
    apply mod_eq_add_cancel_right
  6. L11
    specialize mod_eq_trans m
  7. L12
    specialize mod_eq_trans (a + q)
  8. L13
    specialize mod_eq_trans 0
  9. L14
    specialize mod_eq_trans (0 + q)
  10. L15
    apply mod_eq_trans
03Use earlier factsL16–16

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

  1. L16
    exact hsum
04Establish heqL17–24

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

  1. L17
    have heq : 0 + q = q
  2. L18
    apply zero_add
  3. L19
    rewrite heq
  4. L20
    specialize mod_eq_symm m
  5. L21
    specialize mod_eq_symm q
  6. L22
    specialize mod_eq_symm 0
  7. L23
    apply mod_eq_symm
  8. L24
    exact hq

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro m
  2. 0002intro a
  3. 0003intro q
  4. 0004intro hsum
  5. 0005intro hq
  6. 0006specialize mod_eq_add_cancel_right m
  7. 0007specialize mod_eq_add_cancel_right a
  8. 0008specialize mod_eq_add_cancel_right 0
  9. 0009specialize mod_eq_add_cancel_right q
  10. 0010apply mod_eq_add_cancel_right
  11. 0011specialize mod_eq_trans m
  12. 0012specialize mod_eq_trans (a + q)
  13. 0013specialize mod_eq_trans 0
  14. 0014specialize mod_eq_trans (0 + q)
  15. 0015apply mod_eq_trans
  16. 0016exact hsum
  17. 0017have heq : 0 + q = q
  18. 0018apply zero_add
  19. 0019rewrite heq
  20. 0020specialize mod_eq_symm m
  21. 0021specialize mod_eq_symm q
  22. 0022specialize mod_eq_symm 0
  23. 0023apply mod_eq_symm
  24. 0024exact hq