refactor: More simps
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jstoobysmith
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1651b265e7
commit
4a396783ab
17 changed files with 138 additions and 87 deletions
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@ -73,7 +73,7 @@ lemma ud_us_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
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have h2 := V.fst_row_normalized_abs
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refine Iff.intro (fun h h1 => ?_) (fun h => ?_)
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· rw [h1] at h2
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simp at h2
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simp only [Fin.isValue, one_pow, add_left_eq_self] at h2
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rw [add_eq_zero_iff_of_nonneg (sq_nonneg _) (sq_nonneg _)] at h2
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simp_all
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· by_contra hn
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@ -93,7 +93,7 @@ lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℝ {V : CKMMatrix} (hb : [V]ud ≠ 0
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rw [← Complex.sq_abs] at hn
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have h2 : Complex.abs (V.1 0 2) ^2 = 1 := by
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linear_combination -(1 * hn)
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simp at h2
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simp only [Fin.isValue, sq_eq_one_iff] at h2
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cases' h2 with h2 h2
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· exact hb h2
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· have h3 := Complex.abs.nonneg [V]ub
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@ -121,7 +121,7 @@ lemma VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
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lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
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(normSq [V]ud : ℂ) + normSq [V]us ≠ 0 := by
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have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hb
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simp at h1
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simp only [Fin.isValue, ne_eq] at h1
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rw [← ofReal_inj] at h1
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simp_all
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@ -129,7 +129,7 @@ lemma Vabs_sq_add_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
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((VudAbs ⟦V⟧ : ℂ) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧)) ≠ 0 := by
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have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ hb
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rw [← Complex.sq_abs, ← Complex.sq_abs] at h1
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simp [sq] at h1
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simp only [Fin.isValue, sq, ofReal_mul, ne_eq] at h1
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exact h1
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section orthogonal
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@ -139,7 +139,8 @@ lemma fst_row_orthog_snd_row (V : CKMMatrix) :
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have hV := V.prop
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rw [mem_unitaryGroup_iff] at hV
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have ht := congrFun (congrFun hV 1) 0
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simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
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simp only [Fin.isValue, mul_apply, star_apply, RCLike.star_def, Fin.sum_univ_three, ne_eq,
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one_ne_zero, not_false_eq_true, one_apply_ne] at ht
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exact ht
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lemma fst_row_orthog_thd_row (V : CKMMatrix) :
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@ -147,7 +148,8 @@ lemma fst_row_orthog_thd_row (V : CKMMatrix) :
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have hV := V.prop
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rw [mem_unitaryGroup_iff] at hV
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have ht := congrFun (congrFun hV 2) 0
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simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
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simp only [Fin.isValue, mul_apply, star_apply, RCLike.star_def, Fin.sum_univ_three, ne_eq,
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Fin.reduceEq, not_false_eq_true, one_apply_ne] at ht
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exact ht
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lemma Vcd_mul_conj_Vud (V : CKMMatrix) :
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@ -164,14 +166,14 @@ lemma VAbs_thd_eq_one_fst_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV
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VAbs i 0 V = 0 := by
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have h := VAbs_sum_sq_row_eq_one V i
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rw [hV] at h
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simp at h
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simp only [Fin.isValue, one_pow, add_left_eq_self] at h
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nlinarith
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lemma VAbs_thd_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV : VAbs i 2 V = 1) :
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VAbs i 1 V = 0 := by
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have h := VAbs_sum_sq_row_eq_one V i
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rw [hV] at h
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simp at h
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simp only [Fin.isValue, one_pow, add_left_eq_self] at h
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nlinarith
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section crossProduct
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@ -180,9 +182,12 @@ lemma conj_Vtb_cross_product {V : CKMMatrix} {τ : ℝ}
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(hτ : [V]t = cexp (τ * I) • (conj [V]u ×₃ conj [V]c)) :
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conj [V]tb = cexp (- τ * I) * ([V]cs * [V]ud - [V]us * [V]cd) := by
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have h1 := congrFun hτ 2
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simp [crossProduct, tRow, uRow, cRow] at h1
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simp only [tRow, Fin.isValue, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd, tail_cons,
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head_cons, crossProduct, uRow, cRow, LinearMap.mk₂_apply, Pi.conj_apply, cons_val_one,
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cons_val_zero, Pi.smul_apply, smul_eq_mul] at h1
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apply congrArg conj at h1
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simp at h1
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simp only [Fin.isValue, _root_.map_mul, map_sub, RingHomCompTriple.comp_apply,
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RingHom.id_apply] at h1
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rw [h1]
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simp only [← exp_conj, _root_.map_mul, conj_ofReal, conj_I, mul_neg, Fin.isValue, neg_mul]
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field_simp
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@ -195,7 +200,8 @@ lemma conj_Vtb_mul_Vud {V : CKMMatrix} {τ : ℝ}
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cexp (τ * I) * conj [V]tb * conj [V]ud =
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[V]cs * (normSq [V]ud + normSq [V]us) + [V]cb * conj [V]ub * [V]us := by
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rw [conj_Vtb_cross_product hτ]
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simp [exp_neg]
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simp only [neg_mul, exp_neg, Fin.isValue, ne_eq, exp_ne_zero, not_false_eq_true,
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mul_inv_cancel_left₀]
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have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]ud = [V]cs
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* [V]ud * conj [V]ud - [V]us * ([V]cd * conj [V]ud) := by
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ring
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@ -209,7 +215,8 @@ lemma conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ}
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cexp (τ * I) * conj [V]tb * conj [V]us =
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- ([V]cd * (normSq [V]ud + normSq [V]us) + [V]cb * conj ([V]ub) * [V]ud) := by
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rw [conj_Vtb_cross_product hτ]
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simp [exp_neg]
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simp only [neg_mul, exp_neg, Fin.isValue, ne_eq, exp_ne_zero, not_false_eq_true,
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mul_inv_cancel_left₀, neg_add_rev]
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have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]us = ([V]cs
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* conj [V]us) * [V]ud - [V]us * [V]cd * conj [V]us := by
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ring
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@ -275,7 +282,7 @@ lemma VAbs_sum_sq_col_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) :
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lemma thd_col_normalized_abs (V : CKMMatrix) :
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abs [V]ub ^ 2 + abs [V]cb ^ 2 + abs [V]tb ^ 2 = 1 := by
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have h1 := VAbs_sum_sq_col_eq_one ⟦V⟧ 2
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simp [VAbs] at h1
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simp only [VAbs, VAbs', Fin.isValue, Quotient.lift_mk] at h1
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exact h1
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lemma thd_col_normalized_normSq (V : CKMMatrix) :
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@ -288,11 +295,13 @@ lemma cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h : [V]ud = 0 ∧ [V]us = 0) :
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[V]cb = 0 := by
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have h1 := fst_row_normalized_abs V
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rw [← thd_col_normalized_abs V] at h1
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simp [h] at h1
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simp only [Fin.isValue, h, map_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow,
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add_zero, zero_add] at h1
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rw [add_assoc] at h1
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simp at h1
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simp only [Fin.isValue, self_eq_add_right] at h1
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rw [add_eq_zero_iff_of_nonneg (sq_nonneg _) (sq_nonneg _)] at h1
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simp at h1
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simp only [Fin.isValue, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff,
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map_eq_zero] at h1
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exact h1.1
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lemma cs_of_ud_us_zero {V : CKMMatrix} (ha : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) :
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@ -324,14 +333,14 @@ lemma cb_tb_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
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have h2 := V.thd_col_normalized_abs
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refine Iff.intro (fun h h1 => ?_) (fun h => ?_)
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· rw [h1] at h2
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simp at h2
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simp only [one_pow, Fin.isValue] at h2
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have h2 : Complex.abs (V.1 1 2) ^ 2 + Complex.abs (V.1 2 2) ^ 2 = 0 := by
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linear_combination h2
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rw [add_eq_zero_iff_of_nonneg (sq_nonneg _) (sq_nonneg _)] at h2
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simp_all
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· by_contra hn
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rw [not_or] at hn
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simp at hn
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simp only [Fin.isValue, ne_eq, Decidable.not_not] at hn
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simp_all only [map_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, add_zero,
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sq_eq_one_iff, false_or, one_ne_zero]
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have h1 := Complex.abs.nonneg [V]ub
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@ -343,14 +352,14 @@ lemma VAbs_fst_col_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
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(hV : VAbs 0 i V = 1) : VAbs 1 i V = 0 := by
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have h := VAbs_sum_sq_col_eq_one V i
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rw [hV] at h
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simp at h
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simp only [one_pow, Fin.isValue] at h
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nlinarith
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lemma VAbs_fst_col_eq_one_thd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
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(hV : VAbs 0 i V = 1) : VAbs 2 i V = 0 := by
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have h := VAbs_sum_sq_col_eq_one V i
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rw [hV] at h
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simp at h
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simp only [one_pow, Fin.isValue] at h
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nlinarith
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end columns
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