refactor: Last batch of multi-goal proofs
This commit is contained in:
jstoobysmith
parent
b9479c904d
commit
c0499483a8
43 changed files with 910 additions and 955 deletions
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@ -54,19 +54,19 @@ lemma ext_δ (S T : Fin (2 * n.succ) → ℚ) (h1 : ∀ i, S (δ₁ i) = T (δ
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(h2 : ∀ i, S (δ₂ i) = T (δ₂ i)) : S = T := by
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funext i
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by_cases hi : i.val < n.succ
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let j : Fin n.succ := ⟨i, hi⟩
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have h2 := h1 j
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have h3 : δ₁ j = i := by
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simp [δ₁, Fin.ext_iff]
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rw [h3] at h2
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exact h2
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let j : Fin n.succ := ⟨i - n.succ, by omega⟩
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have h2 := h2 j
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have h3 : δ₂ j = i := by
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simp [δ₂, Fin.ext_iff]
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omega
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rw [h3] at h2
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exact h2
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· let j : Fin n.succ := ⟨i, hi⟩
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have h2 := h1 j
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have h3 : δ₁ j = i := by
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simp [δ₁, Fin.ext_iff]
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rw [h3] at h2
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exact h2
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· let j : Fin n.succ := ⟨i - n.succ, by omega⟩
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have h2 := h2 j
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have h3 : δ₂ j = i := by
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simp [δ₂, Fin.ext_iff]
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omega
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rw [h3] at h2
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exact h2
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lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
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∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
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@ -167,18 +167,18 @@ lemma basis_on_δ₁_other {k j : Fin n.succ} (h : k ≠ j) :
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simp [basisAsCharges]
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simp [δ₁, δ₂]
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split
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rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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split
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rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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rfl
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· rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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· split
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· rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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· rfl
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lemma basis_on_other {k : Fin n.succ} {j : Fin (2 * n.succ)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
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basisAsCharges k j = 0 := by
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@ -195,18 +195,18 @@ lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
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simp [basis!AsCharges]
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simp [δ!₁, δ!₂]
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split
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rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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split
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rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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rfl
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· rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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· split
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· rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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· rfl
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lemma basis_δ₂_eq_minus_δ₁ (j i : Fin n.succ) :
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basisAsCharges j (δ₂ i) = - basisAsCharges j (δ₁ i) := by
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@ -233,8 +233,8 @@ lemma basis!_δ!₂_eq_minus_δ!₁ (j i : Fin n) :
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all_goals rename_i h1 h2
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all_goals rw [Fin.ext_iff] at h1 h2
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all_goals simp_all
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subst h1
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exact Fin.elim0 i
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· subst h1
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exact Fin.elim0 i
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all_goals rename_i h3
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all_goals rw [Fin.ext_iff] at h3
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all_goals simp_all
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@ -256,27 +256,27 @@ lemma basis!_on_δ!₂_other {k j : Fin n} (h : k ≠ j) : basis!AsCharges k (δ
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lemma basis!_on_δ!₃ (j : Fin n) : basis!AsCharges j δ!₃ = 0 := by
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simp [basis!AsCharges]
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split <;> rename_i h
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rw [Fin.ext_iff] at h
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simp [δ!₃, δ!₁] at h
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omega
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split <;> rename_i h2
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rw [Fin.ext_iff] at h2
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simp [δ!₃, δ!₂] at h2
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omega
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rfl
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split<;> rename_i h
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· simp only [δ!₃, succ_eq_add_one, Fin.isValue, δ!₁, Fin.ext_iff, Fin.coe_cast, Fin.coe_castAdd,
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Fin.coe_fin_one, Fin.coe_natAdd] at h
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omega
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· split <;> rename_i h2
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· simp only [δ!₃, succ_eq_add_one, Fin.isValue, δ!₂, Fin.ext_iff, Fin.coe_cast,
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Fin.coe_castAdd, Fin.coe_fin_one, Fin.coe_natAdd] at h2
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omega
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· rfl
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lemma basis!_on_δ!₄ (j : Fin n) : basis!AsCharges j δ!₄ = 0 := by
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simp [basis!AsCharges]
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split <;> rename_i h
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rw [Fin.ext_iff] at h
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simp [δ!₄, δ!₁] at h
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omega
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split <;> rename_i h2
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rw [Fin.ext_iff] at h2
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simp [δ!₄, δ!₂] at h2
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omega
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rfl
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· rw [Fin.ext_iff] at h
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simp [δ!₄, δ!₁] at h
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omega
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· split <;> rename_i h2
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· rw [Fin.ext_iff] at h2
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simp [δ!₄, δ!₂] at h2
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omega
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· rfl
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lemma basis_linearACC (j : Fin n.succ) : (accGrav (2 * n.succ)) (basisAsCharges j) = 0 := by
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rw [accGrav]
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@ -343,16 +343,15 @@ lemma swap!_as_add {S S' : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
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funext i
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rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
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by_cases hi : i = δ!₁ j
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subst hi
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simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
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by_cases hi2 : i = δ!₂ j
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subst hi2
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simp [HSMul.hSMul, basis!_on_δ!₂_self, pairSwap_inv_snd]
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simp [HSMul.hSMul]
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rw [basis!_on_other hi hi2]
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change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
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erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
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simp
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· subst hi
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simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
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· by_cases hi2 : i = δ!₂ j
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· simp [HSMul.hSMul, hi2, basis!_on_δ!₂_self, pairSwap_inv_snd]
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· simp [HSMul.hSMul]
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rw [basis!_on_other hi hi2]
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change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
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erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
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simp
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/-- A point in the span of the first part of the basis as a charge. -/
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def P (f : Fin n.succ → ℚ) : (PureU1 (2 * n.succ)).Charges := ∑ i, f i • basisAsCharges i
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@ -367,23 +366,23 @@ lemma P_δ₁ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₁ j) = f j :=
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rw [P, sum_of_charges]
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simp [HSMul.hSMul, SMul.smul]
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rw [Finset.sum_eq_single j]
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rw [basis_on_δ₁_self]
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simp only [mul_one]
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intro k _ hkj
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rw [basis_on_δ₁_other hkj]
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simp only [mul_zero]
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simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
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· rw [basis_on_δ₁_self]
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simp only [mul_one]
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· intro k _ hkj
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rw [basis_on_δ₁_other hkj]
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simp only [mul_zero]
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· simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
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lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
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rw [P!, sum_of_charges]
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simp [HSMul.hSMul, SMul.smul]
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rw [Finset.sum_eq_single j]
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rw [basis!_on_δ!₁_self]
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simp only [mul_one]
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intro k _ hkj
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rw [basis!_on_δ!₁_other hkj]
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simp only [mul_zero]
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simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
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· rw [basis!_on_δ!₁_self]
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simp only [mul_one]
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· intro k _ hkj
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rw [basis!_on_δ!₁_other hkj]
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simp only [mul_zero]
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· simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
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lemma Pa_δ!₁ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (j : Fin n) :
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Pa f g (δ!₁ j) = f j.succ + g j := by
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@ -395,23 +394,21 @@ lemma P_δ₂ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₂ j) = - f j
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rw [P, sum_of_charges]
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simp [HSMul.hSMul, SMul.smul]
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rw [Finset.sum_eq_single j]
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rw [basis_on_δ₂_self]
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simp only [mul_neg, mul_one]
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intro k _ hkj
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rw [basis_on_δ₂_other hkj]
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simp only [mul_zero]
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simp
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· simp only [basis_on_δ₂_self, mul_neg, mul_one]
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· intro k _ hkj
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simp only [basis_on_δ₂_other hkj, mul_zero]
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· simp
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lemma P!_δ!₂ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₂ j) = - f j := by
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rw [P!, sum_of_charges]
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simp [HSMul.hSMul, SMul.smul]
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rw [Finset.sum_eq_single j]
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rw [basis!_on_δ!₂_self]
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simp only [mul_neg, mul_one]
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intro k _ hkj
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rw [basis!_on_δ!₂_other hkj]
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simp only [mul_zero]
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simp
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· rw [basis!_on_δ!₂_self]
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simp only [mul_neg, mul_one]
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· intro k _ hkj
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rw [basis!_on_δ!₂_other hkj]
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simp only [mul_zero]
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· simp
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lemma Pa_δ!₂ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (j : Fin n) :
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Pa f g (δ!₂ j) = - f j.castSucc - g j := by
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@ -474,13 +471,13 @@ lemma P_P_P!_accCube (g : Fin n.succ → ℚ) (j : Fin n) :
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rw [sum_δ!₁_δ!₂, basis!_on_δ!₃, basis!_on_δ!₄]
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simp only [mul_zero, add_zero, Function.comp_apply, zero_add]
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rw [Finset.sum_eq_single j, basis!_on_δ!₁_self, basis!_on_δ!₂_self]
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simp [δ!₁_δ₁, δ!₂_δ₂]
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rw [P_δ₁, P_δ₂]
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ring
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intro k _ hkj
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erw [basis!_on_δ!₁_other hkj.symm, basis!_on_δ!₂_other hkj.symm]
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simp only [mul_zero, add_zero]
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simp
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· simp [δ!₁_δ₁, δ!₂_δ₂]
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rw [P_δ₁, P_δ₂]
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ring
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· intro k _ hkj
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erw [basis!_on_δ!₁_other hkj.symm, basis!_on_δ!₂_other hkj.symm]
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simp only [mul_zero, add_zero]
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· simp
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lemma P_P!_P!_accCube (g : Fin n → ℚ) (j : Fin n.succ) :
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accCubeTriLinSymm (P! g) (P! g) (basisAsCharges j)
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@ -489,12 +486,12 @@ lemma P_P!_P!_accCube (g : Fin n → ℚ) (j : Fin n.succ) :
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rw [sum_δ₁_δ₂]
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simp only [Function.comp_apply]
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rw [Finset.sum_eq_single j, basis_on_δ₁_self, basis_on_δ₂_self]
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simp [δ!₁_δ₁, δ!₂_δ₂]
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ring
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intro k _ hkj
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erw [basis_on_δ₁_other hkj.symm, basis_on_δ₂_other hkj.symm]
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simp only [mul_zero, add_zero]
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simp
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· simp [δ!₁_δ₁, δ!₂_δ₂]
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ring
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· intro k _ hkj
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erw [basis_on_δ₁_other hkj.symm, basis_on_δ₂_other hkj.symm]
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simp only [mul_zero, add_zero]
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· simp
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lemma P_zero (f : Fin n.succ → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
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intro i
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@ -601,27 +598,24 @@ theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n) := by
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intro i
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simp_all
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cases i
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simp_all
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simp_all
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· simp_all
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· simp_all
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lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
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apply Iff.intro
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intro h
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funext i
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rw [Pa', Pa'] at h
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have h1 : ∑ i : Fin (succ n) ⊕ Fin n, (f i + (- f' i)) • basisa i = 0 := by
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simp only [add_smul, neg_smul]
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rw [Finset.sum_add_distrib]
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rw [h]
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rw [← Finset.sum_add_distrib]
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simp
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have h2 : ∀ i, (f i + (- f' i)) = 0 := by
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exact Fintype.linearIndependent_iff.mp (@basisa_linear_independent n)
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(fun i => f i + -f' i) h1
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have h2i := h2 i
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linarith
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intro h
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rw [h]
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refine Iff.intro (fun h => (funext (fun i => ?_))) (fun h => ?_)
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· rw [Pa', Pa'] at h
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have h1 : ∑ i : Fin (succ n) ⊕ Fin n, (f i + (- f' i)) • basisa i = 0 := by
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simp only [add_smul, neg_smul]
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rw [Finset.sum_add_distrib]
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rw [h]
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rw [← Finset.sum_add_distrib]
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simp
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have h2 : ∀ i, (f i + (- f' i)) = 0 := by
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exact Fintype.linearIndependent_iff.mp (@basisa_linear_independent n)
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(fun i => f i + -f' i) h1
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have h2i := h2 i
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linarith
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· rw [h]
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/-! TODO: Replace the definition of `join` with a Mathlib definition, most likely `Sum.elim`. -/
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/-- A helper function for what follows. -/
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@ -632,33 +626,24 @@ def join (g : Fin n.succ → ℚ) (f : Fin n → ℚ) : (Fin n.succ) ⊕ (Fin n)
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lemma join_ext (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
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join g f = join g' f' ↔ g = g' ∧ f = f' := by
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apply Iff.intro
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intro h
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apply And.intro
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funext i
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exact congr_fun h (Sum.inl i)
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funext i
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exact congr_fun h (Sum.inr i)
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intro h
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rw [h.left, h.right]
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· apply And.intro (funext (fun i => congr_fun h (Sum.inl i)))
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(funext (fun i => congr_fun h (Sum.inr i)))
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· rw [h.left, h.right]
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lemma join_Pa (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
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Pa' (join g f) = Pa' (join g' f') ↔ Pa g f = Pa g' f' := by
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apply Iff.intro
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intro h
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rw [Pa'_eq] at h
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rw [join_ext] at h
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rw [h.left, h.right]
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intro h
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apply ACCSystemLinear.LinSols.ext
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rw [Pa'_P'_P!', Pa'_P'_P!']
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simp [P'_val, P!'_val]
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exact h
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· rw [Pa'_eq, join_ext] at h
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rw [h.left, h.right]
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· apply ACCSystemLinear.LinSols.ext
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rw [Pa'_P'_P!', Pa'_P'_P!']
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simp only [succ_eq_add_one, ACCSystemLinear.linSolsAddCommMonoid_add_val, P'_val, P!'_val]
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exact h
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lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
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Pa g f = Pa g' f' ↔ g = g' ∧ f = f' := by
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rw [← join_Pa]
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rw [← join_ext]
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rw [← join_Pa, ← join_ext]
|
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exact Pa'_eq _ _
|
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|
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lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n)) =
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@ -694,8 +679,8 @@ lemma smul_basis!AsCharges_in_span (S : (PureU1 (2 * n.succ)).LinSols) (j : Fin
|
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Submodule.span ℚ (Set.range basis!AsCharges) := by
|
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apply Submodule.smul_mem
|
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apply SetLike.mem_of_subset
|
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apply Submodule.subset_span
|
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simp_all only [Set.mem_range, exists_apply_eq_apply]
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· exact Submodule.subset_span
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· simp_all only [Set.mem_range, exists_apply_eq_apply]
|
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|
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lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
|
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(hS : ((FamilyPermutations (2 * n.succ)).linSolRep
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|
|
@ -732,19 +717,19 @@ lemma vectorLikeEven_in_span (S : (PureU1 (2 * n.succ)).LinSols)
|
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rw [sortAFL_val]
|
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erw [P'_val]
|
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apply ext_δ
|
||||
intro i
|
||||
rw [P_δ₁]
|
||||
rfl
|
||||
intro i
|
||||
rw [P_δ₂]
|
||||
have ht := hS i
|
||||
change sort S.val (δ₁ i) = - sort S.val (δ₂ i) at ht
|
||||
have h : sort S.val (δ₂ i) = - sort S.val (δ₁ i) := by
|
||||
rw [ht]
|
||||
ring
|
||||
rw [h]
|
||||
simp
|
||||
rfl
|
||||
· intro i
|
||||
rw [P_δ₁]
|
||||
rfl
|
||||
· intro i
|
||||
rw [P_δ₂]
|
||||
have ht := hS i
|
||||
change sort S.val (δ₁ i) = - sort S.val (δ₂ i) at ht
|
||||
have h : sort S.val (δ₂ i) = - sort S.val (δ₁ i) := by
|
||||
rw [ht]
|
||||
ring
|
||||
rw [h]
|
||||
simp
|
||||
rfl
|
||||
|
||||
end VectorLikeEvenPlane
|
||||
|
||||
|
|
|
|||
|
|
@ -135,26 +135,22 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ)).LinSols}
|
|||
rw [P_P_P!_accCube' g f hfg] at h1
|
||||
simp at h1
|
||||
cases h1 <;> rename_i h1
|
||||
apply Or.inl
|
||||
linear_combination h1
|
||||
cases h1 <;> rename_i h1
|
||||
apply Or.inr
|
||||
apply Or.inl
|
||||
linear_combination -(1 * h1)
|
||||
apply Or.inr
|
||||
apply Or.inr
|
||||
exact h1
|
||||
· apply Or.inl
|
||||
linear_combination h1
|
||||
· cases h1 <;> rename_i h1
|
||||
· refine Or.inr (Or.inl ?_)
|
||||
linear_combination -(1 * h1)
|
||||
· exact Or.inr (Or.inr h1)
|
||||
|
||||
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
|
||||
(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
|
||||
refine @Prop_three (2 * n.succ.succ) LineInPlaneProp S (δ!₂ (Fin.last n)) (δ!₁ (Fin.last n))
|
||||
δ!₄ ?_ ?_ ?_ ?_
|
||||
simp [Fin.ext_iff, δ!₂, δ!₁]
|
||||
simp [Fin.ext_iff, δ!₂, δ!₄]
|
||||
simp [Fin.ext_iff, δ!₁, δ!₄]
|
||||
omega
|
||||
intro M
|
||||
exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
|
||||
· simp [Fin.ext_iff, δ!₂, δ!₁]
|
||||
· simp [Fin.ext_iff, δ!₂, δ!₄]
|
||||
· simp [Fin.ext_iff, δ!₁, δ!₄]
|
||||
omega
|
||||
· exact fun M => lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
|
||||
|
||||
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ)).Sols}
|
||||
(LIC : LineInCubicPerm S.1.1) : ConstAbs S.val :=
|
||||
|
|
|
|||
|
|
@ -72,9 +72,7 @@ lemma anomalyFree_param {S : (PureU1 (2 * n.succ)).Sols}
|
|||
have hC := S.cubicSol
|
||||
rw [hS] at hC
|
||||
change (accCube (2 * n.succ)) (P g + P! f) = 0 at hC
|
||||
erw [TriLinearSymm.toCubic_add] at hC
|
||||
erw [P_accCube] at hC
|
||||
erw [P!_accCube] at hC
|
||||
erw [TriLinearSymm.toCubic_add, P_accCube, P!_accCube] at hC
|
||||
linear_combination hC / 3
|
||||
|
||||
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
|
||||
|
|
@ -114,9 +112,9 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ)).Sols) :
|
|||
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
|
||||
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
|
||||
exact ne_or_eq _ _
|
||||
cases h1 <;> rename_i h1
|
||||
exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
|
||||
exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
|
||||
rcases h1 with h1 | h1
|
||||
· exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
|
||||
· exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
|
||||
|
||||
theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
|
||||
∃ g f a, S = parameterization g f a := by
|
||||
|
|
@ -128,11 +126,11 @@ theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
|
|||
change S.val = _ • (_ • P g + _• P! f)
|
||||
rw [anomalyFree_param _ _ hS]
|
||||
rw [neg_neg, ← smul_add, smul_smul, inv_mul_cancel, one_smul]
|
||||
exact hS
|
||||
have h := h g f hS
|
||||
rw [anomalyFree_param _ _ hS] at h
|
||||
simp at h
|
||||
exact h
|
||||
· exact hS
|
||||
· have h := h g f hS
|
||||
rw [anomalyFree_param _ _ hS] at h
|
||||
simp at h
|
||||
exact h
|
||||
|
||||
lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ)).Sols}
|
||||
(h : SpecialCase S) : LineInCubic S.1.1 := by
|
||||
|
|
@ -154,9 +152,8 @@ lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ)).Sols}
|
|||
lemma special_case_lineInCubic_perm {S : (PureU1 (2 * n.succ)).Sols}
|
||||
(h : ∀ (M : (FamilyPermutations (2 * n.succ)).group),
|
||||
SpecialCase ((FamilyPermutations (2 * n.succ)).solAction.toFun S M)) :
|
||||
LineInCubicPerm S.1.1 := by
|
||||
intro M
|
||||
exact special_case_lineInCubic (h M)
|
||||
LineInCubicPerm S.1.1 :=
|
||||
fun M => special_case_lineInCubic (h M)
|
||||
|
||||
theorem special_case {S : (PureU1 (2 * n.succ.succ)).Sols}
|
||||
(h : ∀ (M : (FamilyPermutations (2 * n.succ.succ)).group),
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue