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Complete the proof
Signed-off-by: zeramorphic <zeramorphic@proton.me>
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‎ConNF/Model.lean

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@@ -7,3 +7,4 @@ import ConNF.Model.Predicative
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import ConNF.Model.FOA
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import ConNF.Model.RaiseStrong
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import ConNF.Model.Union
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import ConNF.Model.Result

‎ConNF/Model/Predicative.lean

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@@ -106,6 +106,53 @@ theorem smul_up (t₁ t₂ : TSet β) (ρ : Allowable α) :
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intro s
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simp only [mem_smul_iff, mem_up_iff, inv_smul_eq_iff]
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theorem up_self (t : TSet β) : up hβ t t = .singleton hβ t := by
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refine ext hβ _ _ ?_
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simp only [mem_up_iff, or_self, mem_singleton_iff, implies_true]
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theorem up_injective (t₁ t₂ t₁' t₂' : TSet β) (h : up hβ t₁ t₂ = up hβ t₁' t₂') :
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(t₁ = t₁' ∧ t₂ = t₂') ∨ (t₁ = t₂' ∧ t₂ = t₁') := by
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have h₁ : t₁ ∈[hβ] up hβ t₁ t₂
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· rw [mem_up_iff]
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exact Or.inl rfl
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have h₂ : t₂ ∈[hβ] up hβ t₁ t₂
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· rw [mem_up_iff]
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exact Or.inr rfl
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rw [h] at h₁ h₂
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rw [mem_up_iff] at h₁ h₂
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obtain (h₁ | h₁) := h₁ <;> obtain (h₂ | h₂) := h₂
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· refine Or.inl ⟨h₁, ?_⟩
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cases h₁.trans h₂.symm
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have : t₂' ∈[hβ] up hβ t₁' t₂'
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· rw [mem_up_iff]
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exact Or.inr rfl
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rw [← h, up_self, mem_singleton_iff] at this
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exact this.symm
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· refine Or.inl ⟨h₁, h₂⟩
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· refine Or.inr ⟨h₁, h₂⟩
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· refine Or.inr ⟨h₁, ?_⟩
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cases h₁.trans h₂.symm
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have : t₁' ∈[hβ] up hβ t₁' t₂'
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· rw [mem_up_iff]
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exact Or.inl rfl
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rw [← h, up_self, mem_singleton_iff] at this
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exact this.symm
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theorem up_eq_singleton_iff (t t₁ t₂ : TSet β) :
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up hβ t₁ t₂ = .singleton hβ t ↔ t₁ = t ∧ t₂ = t := by
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constructor
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· intro h
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have h₁ : t₁ ∈[hβ] up hβ t₁ t₂
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· rw [mem_up_iff]
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exact Or.inl rfl
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have h₂ : t₂ ∈[hβ] up hβ t₁ t₂
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· rw [mem_up_iff]
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exact Or.inr rfl
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rw [h, mem_singleton_iff] at h₁ h₂
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exact ⟨h₁, h₂⟩
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· rintro ⟨rfl, rfl⟩
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rw [up_self]
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@[simp]
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theorem TangleData.TSet.mem_op_iff (t₁ t₂ : TSet γ) (s : TSet β) :
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s ∈[hβ] op hβ hγ t₁ t₂ ↔ s = singleton hγ t₁ ∨ s = up hγ t₁ t₂ :=
@@ -119,6 +166,44 @@ theorem smul_op (t₁ t₂ : TSet γ) (ρ : Allowable α) :
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intro s
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simp only [mem_smul_iff, mem_op_iff, inv_smul_eq_iff, smul_singleton, smul_up]
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theorem op_injective (t₁ t₂ t₁' t₂' : TSet γ) (h : op hβ hγ t₁ t₂ = op hβ hγ t₁' t₂') :
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t₁ = t₁' ∧ t₂ = t₂' := by
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have h₁ : .singleton hγ t₁ ∈[hβ] op hβ hγ t₁ t₂
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· rw [mem_op_iff]
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exact Or.inl rfl
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have h₂ : up hγ t₁ t₂ ∈[hβ] op hβ hγ t₁ t₂
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· rw [mem_op_iff]
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exact Or.inr rfl
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rw [h] at h₁ h₂
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rw [mem_op_iff] at h₁ h₂
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obtain (h₁ | h₁) := h₁ <;> obtain (h₂ | h₂) := h₂
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· cases singleton_injective' _ _ _ h₁
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rw [up_eq_singleton_iff] at h₂
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have h' : up hγ t₁ t₂' ∈[hβ] op hβ hγ t₁ t₂'
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· rw [mem_op_iff]
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exact Or.inr rfl
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rw [← h, mem_op_iff] at h'
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obtain (h' | h') := h'
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· rw [up_eq_singleton_iff] at h'
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exact ⟨rfl, h₂.2.trans h'.2.symm⟩
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· obtain (h' | ⟨rfl, rfl⟩) := up_injective _ _ _ _ _ h'
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· exact ⟨rfl, h'.2.symm⟩
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· exact ⟨rfl, rfl⟩
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· cases singleton_injective' _ _ _ h₁
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obtain (h' | ⟨rfl, rfl⟩) := up_injective _ _ _ _ _ h₂
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· exact h'
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· exact ⟨rfl, rfl⟩
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· symm at h₁
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rw [up_eq_singleton_iff] at h₁ h₂
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obtain ⟨rfl, rfl⟩ := h₁
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obtain ⟨_, rfl⟩ := h₂
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exact ⟨rfl, rfl⟩
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· symm at h₁
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rw [up_eq_singleton_iff] at h₁
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obtain ⟨rfl, rfl⟩ := h₁
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rw [up_self, up_eq_singleton_iff] at h₂
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exact h₂
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theorem Symmetric.singletonImage {s : Set (TSet γ)} (h : Symmetric hγ s) :
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Symmetric hβ {p | ∃ a b : TSet ε, op hδ hε a b ∈ s ∧
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p = op hγ hδ (.singleton hε a) (.singleton hε b)} := by

‎ConNF/Model/Result.lean

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import ConNF.Model.Union
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namespace ConNF
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noncomputable local instance : Params := minimalParams
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variable {α β γ δ ε ζ : Λ} (hβ : β < α) (hγ : γ < β) (hδ : δ < γ) (hε : ε < δ) (hζ : ζ < ε)
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theorem ext (t₁ t₂ : TSet α) :
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(∀ s, s ∈[hβ] t₁ ↔ s ∈[hβ] t₂) → t₁ = t₂ :=
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TangleData.TSet.ext hβ t₁ t₂
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theorem inter {t₁ t₂ : TSet α} :
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∃ t, ∀ s, s ∈[hβ] t ↔ s ∈[hβ] t₁ ∧ s ∈[hβ] t₂ := by
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refine ⟨t₁.inter hβ t₂, ?_⟩
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simp only [TangleData.TSet.inter, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq, implies_true]
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theorem compl {t : TSet α} :
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∃ t', ∀ s, s ∈[hβ] t' ↔ ¬s ∈[hβ] t := by
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refine ⟨t.compl hβ, ?_⟩
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simp only [TangleData.TSet.compl, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq, implies_true]
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theorem mem_singleton_iff (t : TSet β) (s : TSet β) :
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s ∈[hβ] .singleton hβ t ↔ s = t :=
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TangleData.TSet.mem_singleton_iff hβ t s
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theorem mem_up_iff (t₁ t₂ : TSet β) (s : TSet β) :
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s ∈[hβ] .up hβ t₁ t₂ ↔ s = t₁ ∨ s = t₂ :=
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TangleData.TSet.mem_up_iff hβ t₁ t₂ s
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theorem mem_op_iff (t₁ t₂ : TSet γ) (s : TSet β) :
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s ∈[hβ] .op hβ hγ t₁ t₂ ↔ s = .singleton hγ t₁ ∨ s = .up hγ t₁ t₂ :=
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TangleData.TSet.mem_op_iff hβ hγ t₁ t₂ s
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theorem singletonImage (t : TSet β) :
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∃ t', ∀ a b,
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.op hγ hδ (.singleton hε a) (.singleton hε b) ∈[hβ] t' ↔ .op hδ hε a b ∈[hγ] t := by
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refine ⟨t.singletonImage hβ hγ hδ hε, ?_⟩
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intro a b
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simp only [TangleData.TSet.singletonImage, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq]
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constructor
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· rintro ⟨a', b', h₁, h₂⟩
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have := op_injective _ _ _ _ _ _ h₂
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cases singleton_injective' _ _ _ this.1
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cases singleton_injective' _ _ _ this.2
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exact h₁
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· intro h
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exact ⟨a, b, h, rfl⟩
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theorem insertion2 (t : TSet γ) :
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∃ t', ∀ a b c, .op hγ hδ (.singleton hε (.singleton hζ a)) (.op hε hζ b c) ∈[hβ] t' ↔
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.op hε hζ a c ∈[hδ] t := by
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refine ⟨t.insertion2 hβ hγ hδ hε hζ, ?_⟩
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intro a b c
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simp only [TangleData.TSet.insertion2, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq]
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constructor
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· rintro ⟨a', b', c', h₁, h₂⟩
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have := op_injective _ _ _ _ _ _ h₂
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cases singleton_injective' _ _ _ (singleton_injective' _ _ _ this.1)
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obtain ⟨rfl, rfl⟩ := op_injective _ _ _ _ _ _ this.2
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exact h₁
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· intro h
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exact ⟨a, b, c, h, rfl⟩
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theorem insertion3 (t : TSet γ) :
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∃ t', ∀ a b c, .op hγ hδ (.singleton hε (.singleton hζ a)) (.op hε hζ b c) ∈[hβ] t' ↔
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.op hε hζ a b ∈[hδ] t := by
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refine ⟨t.insertion3 hβ hγ hδ hε hζ, ?_⟩
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intro a b c
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simp only [TangleData.TSet.insertion3, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq]
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constructor
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· rintro ⟨a', b', c', h₁, h₂⟩
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have := op_injective _ _ _ _ _ _ h₂
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cases singleton_injective' _ _ _ (singleton_injective' _ _ _ this.1)
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obtain ⟨rfl, rfl⟩ := op_injective _ _ _ _ _ _ this.2
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exact h₁
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· intro h
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exact ⟨a, b, c, h, rfl⟩
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theorem cross (t : TSet γ) :
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∃ t', ∀ a, a ∈[hβ] t' ↔ ∃ b c, a = .op hγ hδ b c ∧ c ∈[hδ] t := by
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refine ⟨t.cross hβ hγ hδ, ?_⟩
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intro a
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simp only [TangleData.TSet.cross, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq]
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constructor
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· rintro ⟨a', b', h, rfl⟩
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exact ⟨a', b', rfl, h⟩
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· rintro ⟨b, c, rfl, h⟩
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exact ⟨b, c, h, rfl⟩
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theorem typeLower (t : TSet α) :
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∃ t', ∀ a, a ∈[hε] t' ↔ ∀ b, .op hγ hδ b (.singleton hε a) ∈[hβ] t := by
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refine ⟨.mk hε {a | ∀ b : TSet δ, .op hγ hδ b (.singleton hε a) ∈[hβ] t} ?_, ?_⟩
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· refine symmetric_of_image_singleton_symmetric hδ hε ?_ ?_
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refine symmetric_of_image_singleton_symmetric hγ hδ ?_ ?_
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refine symmetric_of_image_singleton_symmetric hβ hγ ?_ ?_
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convert t.typeLower'_supported hβ hγ hδ hε using 1
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aesop
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· simp only [TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq, implies_true]
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theorem converse (t : TSet α) :
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∃ t', ∀ a b, .op hγ hδ a b ∈[hβ] t' ↔ .op hγ hδ b a ∈[hβ] t := by
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refine ⟨t.converse hβ hγ hδ, ?_⟩
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intro a b
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simp only [TangleData.TSet.converse, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq]
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constructor
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· rintro ⟨a', b', h₁, h₂⟩
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obtain ⟨rfl, rfl⟩ := op_injective _ _ _ _ _ _ h₂
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exact h₁
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· intro h
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exact ⟨b, a, h, rfl⟩
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theorem cardinalOne :
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∃ t, ∀ a, a ∈[hβ] t ↔ ∃ b, ∀ c, c ∈[hγ] a ↔ c = b := by
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refine ⟨TangleData.TSet.cardinalOne hβ hγ, ?_⟩
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intro a
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simp only [TangleData.TSet.cardinalOne, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq]
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constructor
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· rintro ⟨a, rfl⟩
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refine ⟨a, ?_⟩
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intro b
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rw [TangleData.TSet.mem_singleton_iff]
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· rintro ⟨b, hb⟩
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refine ⟨b, ?_⟩
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refine ext hγ _ _ ?_
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intro c
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rw [hb, TangleData.TSet.mem_singleton_iff]
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theorem subset :
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∃ t, ∀ a b, .op hγ hδ a b ∈[hβ] t ↔ ∀ c, c ∈[hε] a → c ∈[hε] b := by
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refine ⟨TangleData.TSet.subset hβ hγ hδ hε, ?_⟩
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intro a b
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simp only [TangleData.TSet.subset, TangleData.TSet.mem_mk_iff, Set.mem_setOf_eq]
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constructor
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· rintro ⟨a', b', h₁, h₂⟩
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obtain ⟨rfl, rfl⟩ := op_injective _ _ _ _ _ _ h₂
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exact h₁
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· intro h
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exact ⟨a, b, h, rfl⟩
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end ConNF

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