JT000E

jordan_tuple_congruence_symm

Pointwise balanced congruence is symmetric on every finite prefix.

Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable

95 new Alpha admissions come from 96 source lemmas: tuple equality reflexivity reuses an already-admitted theorem and is not counted twice. All counts use actual finite beta-coded enumerations. G008 multiplicativity is proved; the general prime-power count and distinct-prime product formula are further goals. General prime-power fields (G091) remain open. Stable is unchanged.

Exact theorem in conservative defined notation

∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. JordanTupleCongruence(n,b,c,d,e,k) → JordanTupleCongruence(n,d,e,b,c,k)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

JordanTupleCongruence(n,b,c,d,e,k) · 2

Actual proof prerequisites

mod_eq_symm · checked external prerequisite
Original expanded first-order statement 
forall n b c d e k. (forall jt_index_msource jt_left_msource jt_right_msource. (exists jt_gap_msourceindex. jt_gap_msourceindex+S (jt_index_msource)=(k)) -> (((exists fs_h_jt_msourceleft. fs_h_jt_msourceleft + S (jt_left_msource) = S ((S (jt_index_msource)) * c)) /\ exists fs_q_jt_msourceleft. b = fs_q_jt_msourceleft * S ((S (jt_index_msource)) * c) + (jt_left_msource))) -> (((exists fs_h_jt_msourceright. fs_h_jt_msourceright + S (jt_right_msource) = S ((S (jt_index_msource)) * e)) /\ exists fs_q_jt_msourceright. d = fs_q_jt_msourceright * S ((S (jt_index_msource)) * e) + (jt_right_msource))) -> (exists jt_left_msourcemod jt_right_msourcemod. (jt_left_msource)+(n)*jt_left_msourcemod=(jt_right_msource)+(n)*jt_right_msourcemod)) -> (forall jt_index_mtarget jt_left_mtarget jt_right_mtarget. (exists jt_gap_mtargetindex. jt_gap_mtargetindex+S (jt_index_mtarget)=(k)) -> (((exists fs_h_jt_mtargetleft. fs_h_jt_mtargetleft + S (jt_left_mtarget) = S ((S (jt_index_mtarget)) * e)) /\ exists fs_q_jt_mtargetleft. d = fs_q_jt_mtargetleft * S ((S (jt_index_mtarget)) * e) + (jt_left_mtarget))) -> (((exists fs_h_jt_mtargetright. fs_h_jt_mtargetright + S (jt_right_mtarget) = S ((S (jt_index_mtarget)) * c)) /\ exists fs_q_jt_mtargetright. b = fs_q_jt_mtargetright * S ((S (jt_index_mtarget)) * c) + (jt_right_mtarget))) -> (exists jt_left_mtargetmod jt_right_mtargetmod. (jt_left_mtarget)+(n)*jt_left_mtargetmod=(jt_right_mtarget)+(n)*jt_right_mtargetmod))

Complete tactic proof in conservative notation

All 24 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

24 script commands · 4 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.

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 assumptionsL1–10

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

  1. L1
    intro n
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro e
  6. L6
    intro k
  7. L7
    intro h
  8. L8
    intro i
  9. L9
    intro a
  10. L10
    intro z
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hi
  2. L12
    intro ha
  3. L13
    intro hz
03Use earlier factsL14–23

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

  1. L14
    specialize mod_eq_symm (n)
  2. L15
    specialize mod_eq_symm (z)
  3. L16
    specialize mod_eq_symm (a)
  4. L17
    apply mod_eq_symm
  5. L18
    specialize h (i)
  6. L19
    specialize h (z)
  7. L20
    specialize h (a)
  8. L21
    apply h
  9. L22
    exact hi
  10. L23
    exact hz
04Use earlier factsL24–24

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

  1. L24
    exact ha

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro n
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro k
  7. 0007intro h
  8. 0008intro i
  9. 0009intro a
  10. 0010intro z
  11. 0011intro hi
  12. 0012intro ha
  13. 0013intro hz
  14. 0014specialize mod_eq_symm (n)
  15. 0015specialize mod_eq_symm (z)
  16. 0016specialize mod_eq_symm (a)
  17. 0017apply mod_eq_symm
  18. 0018specialize h (i)
  19. 0019specialize h (z)
  20. 0020specialize h (a)
  21. 0021apply h
  22. 0022exact hi
  23. 0023exact hz
  24. 0024exact ha