Homework 8

Homework 8.md

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+**Exercise 26.3.5**:
+```
+SBool  ≜ ∀X <: ⊤. ∀T <: X. ∀F <: X. T → F → X
+STrue  ≜ ∀X <: ⊤. ∀T <: X. ∀F <: X. T → F → T
+SFalse ≜ ∀X <: ⊤. ∀T <: X. ∀F <: X. T → F → F
+
+notft : SFalse → STrue
+notft = λsfalse : SFalse. ΛX <: ⊤. ΛT <: X. ΛF <: X. λt : T. λf : F. t
+
+nottf : STrue → SFalse
+nottf = λstrue : STrue. ΛX <: ⊤. ΛT <: X. ΛF <: X. λt : T. λf : F. f
+```
+**Exercise 26.4.11**:
+1. If Γ ⊢ S₁ → S₂ <: T, then T = ⊤ or T = T₁ → T₂ with Γ ⊢ T₁ <: S₁ and Γ ⊢ S₂ <: T₂.
+2. If Γ ⊢ ∀X <: U. S₂ <: T, then T = ⊤ or T = ∀X <: U. T₂ with Γ, X <: U ⊢ S₂ <: T₂.
+3. If Γ ⊢ X <: T, then T = ⊤ or T = X or Γ ⊢ S <: T with X <: S ∈ Γ.
+4. If Γ ⊢ ⊤ <: T, then T = ⊤.
+
+**Proof**:
+4. By induction on the subtyping derivation. The cases not listed are impossible.
+   * S-Refl: T = ⊤.
+   * S-Trans: The premises are Γ ⊢ ⊤ <: U and Γ ⊢ U <: T.
+     By the induction hypothesis, U = ⊤ and T = ⊤.
+   * S-Top: Trivial.
+1. By induction on the subtyping derivation. The cases not listed are impossible.
+   * S-Refl: T = S₁ → S₂ with Γ ⊢ S₁ <: S₁ and Γ ⊢ S₂ <: S₂ by S-Refl.
+   * S-Trans: The premises are Γ ⊢ S₁ → S₂ <: U and Γ ⊢ U <: T.
+     By the induction hypothesis on the first premise, either U = ⊤, so T = ⊤ by (4),
+     or U = U₁ → U₂ with Γ ⊢ U₁ <: S₁ and Γ ⊢ S₂ <: U₂.
+     By the induction hypothesis on the second premise, either T = ⊤,
+     or T = T₁ → T₂ with Γ ⊢ T₁ <: U₁ and Γ ⊢ U₂ <: T₂.
+     Then by S-Trans, Γ ⊢ T₁ <: S₁ and Γ ⊢ S₂ <: T₂.
+   * S-Arrow: Trivially the second case by the premises.
+2. By induction on the subtyping derivation. The cases not listed are impossible.
+   * S-Refl: T = ∀X <: U. S₂ with Γ, X <: U ⊢ S₂ <: S₂ by S-Refl.
+   * S-Trans: The premises are Γ ∀X <: U. S₂ <: V and Γ ⊢ V <: T.
+     By the induction hypothesis on the first premise, either V = ⊤, so T = ⊤ by (4),
+     or V = ∀X <: U. V₂ with Γ, X <: U ⊢ S₂ <: V₂.
+     By the induction hypothesis on the second premise, either T = ⊤,
+     or T = ∀X <: U. T₂ with Γ, X <: U ⊢ V₂ <: T₂.
+     Then by S-Trans, Γ, X <: U ⊢ S₂ <: T₂.
+   * S-All: Trivially the second case by the premise.
+3. By induction on the subtyping derivation. The cases not listed are impossible.
+   * S-Refl: Trivially the second case.
+   * S-Trans: The premises are Γ ⊢ X <: U and Γ ⊢ U <: T.
+     By the induction hypothesis on the first premise, there are three possibilities:
+     - U = ⊤, so T = ⊤ by (4).
+     - U = X. By the induction hypothesis on the second premise,
+       either T = ⊤ by (4), or T = X, or Γ ⊢ S <: T with X <: S ∈ Γ, as desired.
+     - Γ ⊢ S <: U with X <: S ∈ Γ. Then by S-Trans, Γ ⊢ S <: T.
+   * S-Var: Trivially the last case with Γ ⊢ T <: T by S-Refl.
+
+**Exercise 28.2.3**:
+2. If Γ ⊢ t : T, then Γ ⊢> t : M with Γ ⊢ M <: T.
+
+**Proof**:
+1. Case T-TApp: The premises are Γ ⊢ t₁ : ∀X <: T₁₁. T₁₂ and Γ ⊢ T₂ <: T₁₁.
+   We wish to show that Γ ⊢> t₁ [T₂] : M₂ for some M₂ with Γ ⊢ M₂ <: T₁₂[X ↦ T₂].
+   By the induction hypothesis, we have Γ ⊢> t₁ : M₁ with Γ ⊢ M₁ <: ∀X <: T₁₁. T₁₂.
+   Let M₁ ⇑ N₁, knowing that N₁ is not a variable.
+   By the exposure lemma, we have Γ ⊢ N₁ <: ∀X <: T₁₁. T₁₂.
+   By inversion, N₁ = ∀X <: S₁₁. S₁₂ with Γ ⊢ T₁₁ <: S₁₁ and Γ, X <: S₁₁ ⊢ S₁₂ <: T₁₂.
+   By S-Trans, Γ ⊢ T₂ <: S₁₁.
+   By TA-TApp, we have Γ ⊢> t₁ [T₂] : S₁₂[X ↦ T₂].
+   Finally, we have Γ ⊢ S₁₂[X ↦ T₂] <: T₁₂[X ↦ T₂] by preservation of subtyping under substitution.
+
+**Exercise 28.7.1**:
+It seems to me that you would also need to compute a *maximal X-free subtype*,
+which I'll call Q{X,Γ}(T), defined mutually with R{X,Γ}(T)
+both by recursion on the type.
+
+```
+R{X,Γ} : Type → Type
+Q{X,Γ} : Option Type → Type
+
+R{X,Γ}(Y) ≜ if X == Y then ⊤ else Y
+R{X,Γ}(S → T) ≜
+  case Q{X,Γ}(S) of
+    some S' ⇒ S' → R{X,Γ}(T)
+    none ⇒ ⊤
+R{X,Γ}(∀Y <: U. T) ≜ ∀Y <: U. R{X,Γ}(T)
+R{X,Γ}(∃Y <: S. T) ≜ ∃Y <: R{X,Γ}(S). R{X,Γ}(T)
+
+Q{X,Γ}(Y) ≜ if X == Y then none else some Y
+Q{X,Γ}(S → T) ≜
+  case Q{X,Γ}(T) of
+    some T' ⇒ R{X,Γ}(S) → T'
+    none ⇒ none
+Q{X,Γ}(∀Y <: U. T) ≜ ∀Y <: U. Q{X,Γ}(T)
+Q{X,Γ}(∃Y <: S. T) ≜ ∃Y <: Q{X,Γ}(S). Q{X,Γ}(T)
+```