refactor: Last batch of multi-goal proofs
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jstoobysmith
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43 changed files with 910 additions and 955 deletions
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@ -108,19 +108,17 @@ def Hu : MSSMCharges.Charges →ₗ[ℚ] ℚ where
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lemma charges_eq_toSpecies_eq (S T : MSSMCharges.Charges) :
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S = T ↔ (∀ i, toSMSpecies i S = toSMSpecies i T) ∧ Hd S = Hd T ∧ Hu S = Hu T := by
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apply Iff.intro
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intro h
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rw [h]
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simp only [forall_const, Hd_apply, Fin.reduceFinMk, Fin.isValue, Hu_apply, and_self]
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intro h
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apply toSpecies.injective
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apply Prod.ext
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funext i
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exact h.1 i
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funext i
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match i with
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| 0 => exact h.2.1
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| 1 => exact h.2.2
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· rw [h]
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simp only [forall_const, Hd_apply, Fin.reduceFinMk, Fin.isValue, Hu_apply, and_self]
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· apply toSpecies.injective
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apply Prod.ext
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· funext i
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exact h.1 i
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· funext i
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match i with
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| 0 => exact h.2.1
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| 1 => exact h.2.2
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lemma Hd_toSpecies_inv (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) :
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Hd (toSpecies.symm f) = f.2 0 := by
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@ -278,14 +276,14 @@ def quadBiLin : BiLinearSymm MSSMCharges.Charges := BiLinearSymm.mk₂ (
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simp only
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rw [mul_add]
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congr 1
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rw [Finset.mul_sum]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_smul]
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simp only [HSMul.hSMul, SMul.smul, toSMSpecies_apply, Fin.isValue, neg_mul, one_mul]
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ring
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simp only [map_smul, Hd_apply, Fin.reduceFinMk, Fin.isValue, smul_eq_mul, neg_mul, Hu_apply]
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ring)
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· rw [Finset.mul_sum]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_smul]
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simp only [HSMul.hSMul, SMul.smul, toSMSpecies_apply, Fin.isValue, neg_mul, one_mul]
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ring
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· simp only [map_smul, Hd_apply, Fin.reduceFinMk, Fin.isValue, smul_eq_mul, neg_mul, Hu_apply]
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ring)
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(by
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intro S T R
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simp only
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@ -294,24 +292,24 @@ def quadBiLin : BiLinearSymm MSSMCharges.Charges := BiLinearSymm.mk₂ (
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rw [add_assoc]
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rw [← add_assoc _ _ (-Hd S * Hd R + Hu S * Hu R + (-Hd T * Hd R + Hu T * Hu R))]
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congr 1
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rw [← Finset.sum_add_distrib]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_add]
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simp only [ACCSystemCharges.chargesAddCommMonoid_add, toSMSpecies_apply, Fin.isValue, neg_mul,
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one_mul]
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ring
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rw [Hd.map_add, Hu.map_add]
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ring)
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· rw [← Finset.sum_add_distrib]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_add]
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simp only [ACCSystemCharges.chargesAddCommMonoid_add, toSMSpecies_apply, Fin.isValue, neg_mul,
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one_mul]
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ring
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· rw [Hd.map_add, Hu.map_add]
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ring)
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(by
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intro S L
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simp only [MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue, neg_mul, one_mul,
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Hd_apply, Fin.reduceFinMk, Hu_apply]
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congr 1
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rw [Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue]
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ring
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ring)
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· rw [Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue]
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ring
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· ring)
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/-- The quadratic ACC. -/
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@[simp]
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@ -351,14 +349,14 @@ lemma cubeTriLinToFun_map_smul₁ (a : ℚ) (S T R : MSSMCharges.Charges) :
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simp only [cubeTriLinToFun]
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rw [mul_add]
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congr 1
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rw [Finset.mul_sum]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_smul]
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simp only [HSMul.hSMul, SMul.smul, toSMSpecies_apply, Fin.isValue]
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ring
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simp only [map_smul, Hd_apply, Fin.reduceFinMk, Fin.isValue, smul_eq_mul, Hu_apply]
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ring
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· rw [Finset.mul_sum]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_smul]
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simp only [HSMul.hSMul, SMul.smul, toSMSpecies_apply, Fin.isValue]
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ring
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· simp only [map_smul, Hd_apply, Fin.reduceFinMk, Fin.isValue, smul_eq_mul, Hu_apply]
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ring
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lemma cubeTriLinToFun_map_add₁ (S T R L : MSSMCharges.Charges) :
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cubeTriLinToFun (S + T, R, L) = cubeTriLinToFun (S, R, L) + cubeTriLinToFun (T, R, L) := by
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@ -369,34 +367,34 @@ lemma cubeTriLinToFun_map_add₁ (S T R L : MSSMCharges.Charges) :
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rw [← add_assoc _ _ (2 * Hd S * Hd R * Hd L + 2 * Hu S * Hu R * Hu L +
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(2 * Hd T * Hd R * Hd L + 2 * Hu T * Hu R * Hu L))]
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congr 1
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rw [← Finset.sum_add_distrib]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_add]
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simp only [ACCSystemCharges.chargesAddCommMonoid_add, toSMSpecies_apply, Fin.isValue]
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ring
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rw [Hd.map_add, Hu.map_add]
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ring
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· rw [← Finset.sum_add_distrib]
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apply Fintype.sum_congr
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intro i
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repeat erw [map_add]
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simp only [ACCSystemCharges.chargesAddCommMonoid_add, toSMSpecies_apply, Fin.isValue]
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ring
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· rw [Hd.map_add, Hu.map_add]
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ring
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lemma cubeTriLinToFun_swap1 (S T R : MSSMCharges.Charges) :
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cubeTriLinToFun (S, T, R) = cubeTriLinToFun (T, S, R) := by
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simp only [cubeTriLinToFun, MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue, Hd_apply,
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Fin.reduceFinMk, Hu_apply]
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congr 1
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rw [Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue]
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ring
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ring
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· rw [Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue]
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ring
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· ring
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lemma cubeTriLinToFun_swap2 (S T R : MSSMCharges.Charges) :
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cubeTriLinToFun (S, T, R) = cubeTriLinToFun (S, R, T) := by
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simp only [cubeTriLinToFun, MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue, Hd_apply,
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Fin.reduceFinMk, Hu_apply]
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congr 1
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rw [Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue]
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ring
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ring
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· rw [Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue]
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ring
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· ring
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/-- The symmetric trilinear form used to define the cubic ACC. -/
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@[simps!]
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