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refactor: Remove hypothesis of FLT three

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jstoobysmith 2024-11-11 05:54:31 +00:00
parent df210420ba
commit 885c3ae204
2 changed files with 12 additions and 12 deletions
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9
HepLean/AnomalyCancellation/SM/NoGrav/One/Lemmas.lean
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@ -6,6 +6,7 @@ Authors: Joseph Tooby-Smith
import HepLean.AnomalyCancellation.SM.Basic
import HepLean.AnomalyCancellation.SM.NoGrav.Basic
import HepLean.AnomalyCancellation.SM.NoGrav.One.LinearParameterization
import Mathlib.NumberTheory.FLT.Three
/-!
# Lemmas for 1 family SM Accs
@ -47,12 +48,12 @@ lemma accGrav_Q_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) = 0) :
simp_all
linear_combination 3 * h2
lemma accGrav_Q_neq_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) ≠ 0)
(FLTThree : FermatLastTheoremWith ℚ 3) :
lemma accGrav_Q_neq_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) ≠ 0) :
accGrav S.val = 0 := by
have hE := E_zero_iff_Q_zero.mpr.mt hQ
let S' := linearParametersQENeqZero.bijection.symm ⟨S.1.1, And.intro hQ hE⟩
have hC := cubeSol S
have FLTThree := fermatLastTheoremFor_iff_rat.mp fermatLastTheoremThree
have hS' := congrArg (fun S => S.val.val)
(linearParametersQENeqZero.bijection.right_inv ⟨S.1.1, And.intro hQ hE⟩)
change _ = S.val at hS'
@ -60,11 +61,11 @@ lemma accGrav_Q_neq_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) ≠ 0
exact S'.grav_of_cubic hC FLTThree
/-- Any solution to the ACCs without gravity satisfies the gravitational ACC. -/
theorem accGravSatisfied {S : (SMNoGrav 1).Sols} (FLTThree : FermatLastTheoremWith ℚ 3) :
theorem accGravSatisfied {S : (SMNoGrav 1).Sols} :
accGrav S.val = 0 := by
by_cases hQ : Q S.val (0 : Fin 1)= 0
· exact accGrav_Q_zero hQ
· exact accGrav_Q_neq_zero hQ FLTThree
· exact accGrav_Q_neq_zero hQ
end One
end SMNoGrav

15
HepLean/AnomalyCancellation/SM/NoGrav/One/LinearParameterization.lean
View file

@ -9,6 +9,7 @@ import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.NumberTheory.FLT.Three
/-!
# Parameterizations for solutions to the linear ACCs for 1 family
@ -279,14 +280,14 @@ lemma cubic (S : linearParametersQENeqZero) :
simp_all
exact add_eq_zero_iff_eq_neg
lemma cubic_v_or_w_zero (S : linearParametersQENeqZero) (h : accCube (bijection S).1.val = 0)
(FLTThree : FermatLastTheoremWith ℚ 3) :
lemma cubic_v_or_w_zero (S : linearParametersQENeqZero) (h : accCube (bijection S).1.val = 0) :
S.v = 0 ∨ S.w = 0 := by
rw [S.cubic] at h
have h1 : (-1)^3 = (-1 : ℚ) := by rfl
rw [← h1] at h
by_contra hn
rw [not_or] at hn
have FLTThree := fermatLastTheoremFor_iff_rat.mp fermatLastTheoremThree
have h2 := FLTThree S.v S.w (-1) hn.1 hn.2 (Ne.symm (ne_of_beq_false (by rfl)))
exact h2 h
@ -326,10 +327,9 @@ lemma cube_w_zero (S : linearParametersQENeqZero) (h : accCube (bijection S).1.v
simp_all
exact eq_neg_of_add_eq_zero_left h'
lemma cube_w_v (S : linearParametersQENeqZero) (h : accCube (bijection S).1.val = 0)
(FLTThree : FermatLastTheoremWith ℚ 3) :
lemma cube_w_v (S : linearParametersQENeqZero) (h : accCube (bijection S).1.val = 0) :
(S.v = -1 ∧ S.w = 0) ∨ (S.v = 0 ∧ S.w = -1) := by
have h' := cubic_v_or_w_zero S h FLTThree
have h' := cubic_v_or_w_zero S h
cases' h' with hx hx
· simpa [hx] using cubic_v_zero S h hx
· simpa [hx] using cube_w_zero S h hx
@ -345,11 +345,10 @@ lemma grav (S : linearParametersQENeqZero) : accGrav (bijection S).1.val = 0 ↔
· rw [h]
exact Eq.symm (mul_neg_one (6 * S.x))
lemma grav_of_cubic (S : linearParametersQENeqZero) (h : accCube (bijection S).1.val = 0)
(FLTThree : FermatLastTheoremWith ℚ 3) :
lemma grav_of_cubic (S : linearParametersQENeqZero) (h : accCube (bijection S).1.val = 0) :
accGrav (bijection S).1.val = 0 := by
rw [grav]
have h' := cube_w_v S h FLTThree
have h' := cube_w_v S h
cases' h' with h h
· rw [h.1, h.2]
exact Rat.add_zero (-1)