PA007A · theorem

gauss_same_sign_scaled_source_unique

Alpha v34 checked-use theorem · independently closed; not Stable

Equal lower signs or equal reflected signs force equality of the bounded source residues.

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

∀ p. ∀ h. ∀ a. ∀ x. ∀ y. ∀ m. Prime(p) → ¬Dvd(p,a) → Lt(x,p) → Lt(y,p) → ModEq(p,a · x,m) ∧ ModEq(p,a · y,m) ∨ ModEq(p,a · x,2 · h · m) ∧ ModEq(p,a · y,2 · h · m) → x = y

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

PD0002 Lt PD0003 Dvd PD0004 Prime PD0008 ModEq

8 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall p h a x y m. ((~(p = 1) /\ forall gsp_prime_left_collision_prime gsp_prime_right_collision_prime. p = gsp_prime_left_collision_prime * gsp_prime_right_collision_prime -> gsp_prime_left_collision_prime = 1 \/ gsp_prime_right_collision_prime = 1)) -> (~(exists gsp_divisor_factor_collision_multiplier. a = p * gsp_divisor_factor_collision_multiplier)) -> (exists gsp_lt_gap_collision_x_bound. gsp_lt_gap_collision_x_bound + S x = p) -> (exists gsp_lt_gap_collision_y_bound. gsp_lt_gap_collision_y_bound + S y = p) -> ((((exists gmp_mod_left_same_sign_x_lower gmp_mod_right_same_sign_x_lower. (a * x) + p * gmp_mod_left_same_sign_x_lower = (m) + p * gmp_mod_right_same_sign_x_lower) /\ (exists gmp_mod_left_same_sign_y_lower gmp_mod_right_same_sign_y_lower. (a * y) + p * gmp_mod_left_same_sign_y_lower = (m) + p * gmp_mod_right_same_sign_y_lower)) \/ ((exists gmp_mod_left_same_sign_x_reflected gmp_mod_right_same_sign_x_reflected. (a * x) + p * gmp_mod_left_same_sign_x_reflected = ((2 * h) * m) + p * gmp_mod_right_same_sign_x_reflected) /\ (exists gmp_mod_left_same_sign_y_reflected gmp_mod_right_same_sign_y_reflected. (a * y) + p * gmp_mod_left_same_sign_y_reflected = ((2 * h) * m) + p * gmp_mod_right_same_sign_y_reflected)))) -> x = y

Proof neighborhood

Direct theorem prerequisites

PA0079 prime_scaled_same_target_unique

Direct theorem dependents

PA007C gauss_signed_half_magnitude_injective

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

38 script commands · 8 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.

Named ingredients (1)
prime scaled same target unique
01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro h
  3. L3
    intro a
  4. L4
    intro x
  5. L5
    intro y
  6. L6
    intro m
  7. L7
    intro hp
  8. L8
    intro hnotdiv
  9. L9
    intro hxbound
  10. L10
    intro hybound
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hsame
03Separate the logical casesL12–13

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

  1. L12
    cases hsame
  2. L13
    cases hsame_left
04Use earlier factsL14–23

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

  1. L14
    specialize prime_scaled_same_target_unique p
  2. L15
    specialize prime_scaled_same_target_unique a
  3. L16
    specialize prime_scaled_same_target_unique x
  4. L17
    specialize prime_scaled_same_target_unique y
  5. L18
    specialize prime_scaled_same_target_unique m
  6. L19
    apply prime_scaled_same_target_unique
  7. L20
    exact hp
  8. L21
    exact hnotdiv
  9. L22
    exact hxbound
  10. L23
    exact hybound
05Use earlier factsL24–25

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

  1. L24
    exact hsame_left_left
  2. L25
    exact hsame_left_right
06Separate the logical casesL26–26

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

  1. L26
    cases hsame_right
07Use earlier factsL27–36

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

  1. L27
    specialize prime_scaled_same_target_unique p
  2. L28
    specialize prime_scaled_same_target_unique a
  3. L29
    specialize prime_scaled_same_target_unique x
  4. L30
    specialize prime_scaled_same_target_unique y
  5. L31
    specialize prime_scaled_same_target_unique (2 * h) * m
  6. L32
    apply prime_scaled_same_target_unique
  7. L33
    exact hp
  8. L34
    exact hnotdiv
  9. L35
    exact hxbound
  10. L36
    exact hybound
08Use earlier factsL37–38

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

  1. L37
    exact hsame_right_left
  2. L38
    exact hsame_right_right

Library-wide reading audit

Original defined command ledger · 38 lines
  1. 0001intro p
  2. 0002intro h
  3. 0003intro a
  4. 0004intro x
  5. 0005intro y
  6. 0006intro m
  7. 0007intro hp
  8. 0008intro hnotdiv
  9. 0009intro hxbound
  10. 0010intro hybound
  11. 0011intro hsame
  12. 0012cases hsame
  13. 0013cases hsame_left
  14. 0014specialize prime_scaled_same_target_unique p
  15. 0015specialize prime_scaled_same_target_unique a
  16. 0016specialize prime_scaled_same_target_unique x
  17. 0017specialize prime_scaled_same_target_unique y
  18. 0018specialize prime_scaled_same_target_unique m
  19. 0019apply prime_scaled_same_target_unique
  20. 0020exact hp
  21. 0021exact hnotdiv
  22. 0022exact hxbound
  23. 0023exact hybound
  24. 0024exact hsame_left_left
  25. 0025exact hsame_left_right
  26. 0026cases hsame_right
  27. 0027specialize prime_scaled_same_target_unique p
  28. 0028specialize prime_scaled_same_target_unique a
  29. 0029specialize prime_scaled_same_target_unique x
  30. 0030specialize prime_scaled_same_target_unique y
  31. 0031specialize prime_scaled_same_target_unique (2 * h) * m
  32. 0032apply prime_scaled_same_target_unique
  33. 0033exact hp
  34. 0034exact hnotdiv
  35. 0035exact hxbound
  36. 0036exact hybound
  37. 0037exact hsame_right_left
  38. 0038exact hsame_right_right