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From stdpp Require Import gmap base relations.
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From iris Require Import prelude.
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From semantics.ts.systemf Require Import lang notation parallel_subst types logrel tactics.
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(** Exercise 1 (LN Exercise 19): De Bruijn Terms *)
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Module dbterm.
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(** Your type of expressions only needs to encompass the operations of our base lambda calculus. *)
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Inductive expr :=
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| Lit (l : base_lit)
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| Var (n : nat)
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| Lam (e : expr)
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| Plus (e1 e2 : expr)
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| App (e1 e2 : expr)
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.
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(** Formalize substitutions and renamings as functions. *)
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Definition subt := nat → expr.
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Definition rent := nat → nat.
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Implicit Types
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(σ : subt)
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(δ : rent)
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(n x : nat)
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(e : expr).
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Fixpoint subst σ e :=
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(* FIXME *) e.
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Goal (subst
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(λ n, match n with
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| 0 => Lit (LitInt 42)
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| 1 => Var 0
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| _ => Var n
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end)
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(Lam (Plus (Plus (Var 2) (Var 1)) (Var 0)))) =
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Lam (Plus (Plus (Var 1) (Lit 42%Z)) (Var 0)).
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Proof.
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cbn.
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(* Should be by reflexivity. *)
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(* TODO: exercise *)
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Admitted.
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End dbterm.
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Section church_encodings.
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(** Exercise 2 (LN Exercise 24): Church encoding, sum types *)
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(* a) Define your encoding *)
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Definition sum_type (A B : type) : type := #0 (* FIXME *).
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(* b) Implement inj1, inj2, case *)
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Definition injl_val (v : val) : val := #0 (* FIXME *).
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Definition injl_expr (e : expr) : expr := #0 (* FIXME *).
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Definition injr_val (v : val) : val := #0 (* FIXME *).
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Definition injr_expr (e : expr) : expr := #0 (* FIXME *).
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(* You may want to use the variables x1, x2 for the match arms to fit the typing statements below. *)
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Definition match_expr (e : expr) (e1 e2 : expr) : expr := #0. (* FIXME *)
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(* c) Reduction behavior *)
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(* Some lemmas about substitutions might be useful. Look near the end of the lang.v file! *)
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Lemma match_expr_red_injl e e1 e2 (vl v' : val) :
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is_closed [] vl →
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is_closed ["x1"] e1 →
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big_step e (injl_val vl) →
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big_step (subst' "x1" vl e1) v' →
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big_step (match_expr e e1 e2) v'.
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Proof.
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(* TODO: exercise *)
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Admitted.
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Lemma match_expr_red_injr e e1 e2 (vl v' : val) :
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is_closed [] vl →
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big_step e (injr_val vl) →
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big_step (subst' "x2" vl e2) v' →
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big_step (match_expr e e1 e2) v'.
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Proof.
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intros. bs_step_det.
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(* TODO: exercise *)
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Admitted.
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Lemma injr_expr_red e v :
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big_step e v →
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big_step (injr_expr e) (injr_val v).
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Proof.
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intros. bs_step_det.
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(* TODO: exercise *)
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Admitted.
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Lemma injl_expr_red e v :
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big_step e v →
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big_step (injl_expr e) (injl_val v).
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Proof.
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intros. bs_step_det.
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(* TODO: exercise *)
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Admitted.
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(* d) Typing rules *)
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Lemma sum_injl_typed n Γ (e : expr) A B :
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type_wf n B →
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type_wf n A →
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TY n; Γ ⊢ e : A →
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TY n; Γ ⊢ injl_expr e : sum_type A B.
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Proof.
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intros. solve_typing.
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(* TODO: exercise *)
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Admitted.
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Lemma sum_injr_typed n Γ e A B :
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type_wf n B →
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type_wf n A →
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TY n; Γ ⊢ e : B →
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TY n; Γ ⊢ injr_expr e : sum_type A B.
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Proof.
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intros. solve_typing.
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(* TODO: exercise *)
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Admitted.
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Lemma sum_match_typed n Γ A B C e e1 e2 :
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type_wf n A →
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type_wf n B →
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type_wf n C →
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TY n; Γ ⊢ e : sum_type A B →
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TY n; <["x1" := A]> Γ ⊢ e1 : C →
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TY n; <["x2" := B]> Γ ⊢ e2 : C →
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TY n; Γ ⊢ match_expr e e1 e2 : C.
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Proof.
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intros. solve_typing.
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(* TODO: exercise *)
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Admitted.
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(** Exercise 3 (LN Exercise 25): church encoding, list types *)
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(* a) translate the type of lists into De Bruijn. *)
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Definition list_type (A : type) : type := #0 (* FIXME *).
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(* b) Implement nil and cons. *)
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Definition nil_val : val := #0 (* FIXME *).
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Definition cons_val (v1 v2 : val) : val := #0 (* FIXME *).
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Definition cons_expr (e1 e2 : expr) : expr := #0 (* FIXME *).
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(* c) Define typing rules and prove them *)
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Lemma nil_typed n Γ A :
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type_wf n A →
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TY n; Γ ⊢ nil_val : list_type A.
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Proof.
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intros. solve_typing.
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(* TODO: exercise *)
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Admitted.
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Lemma cons_typed n Γ (e1 e2 : expr) A :
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type_wf n A →
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TY n; Γ ⊢ e2 : list_type A →
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TY n; Γ ⊢ e1 : A →
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TY n; Γ ⊢ cons_expr e1 e2 : list_type A.
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Proof.
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intros. repeat solve_typing.
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(* TODO: exercise *)
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Admitted.
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(* d) Define a function head of type list A → A + 1 *)
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Definition head : val := #0 (* FIXME *).
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Lemma head_typed n Γ A :
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type_wf n A →
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TY n; Γ ⊢ head: (list_type A → (A + Unit)).
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Proof.
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intros. solve_typing.
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(* TODO: exercise *)
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Admitted.
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(* e) Define a function [tail] of type list A → list A *)
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Definition tail : val := #0 (* FIXME *).
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Lemma tail_typed n Γ A :
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type_wf n A →
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TY n; Γ ⊢ tail: (list_type A → list_type A).
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Proof.
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intros. repeat solve_typing.
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(* TODO: exercise *)
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Admitted.
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End church_encodings.
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Section free_theorems.
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(** Exercise 4 (LN Exercise 27): Free Theorems I *)
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(* a) State a free theorem for the type ∀ α, β. α → β → α × β *)
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Lemma free_thm_1 :
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∀ f : val,
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TY 0; ∅ ⊢ f : (∀: ∀: #1 → #0 → #1 × #0) →
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True (* FIXME state your theorem *).
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Proof.
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(* TODO: exercise *)
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Admitted.
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(* b) State a free theorem for the type ∀ α, β. α × β → α *)
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Lemma free_thm_2 :
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∀ f : val,
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TY 0; ∅ ⊢ f : (∀: ∀: #1 × #0 → #1) →
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True (* FIXME state your theorem *).
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Proof.
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(* TODO: exercise *)
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Admitted.
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(* c) State a free theorem for the type ∀ α, β. α → β *)
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Lemma free_thm_3 :
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∀ f : val,
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TY 0; ∅ ⊢ f : (∀: ∀: #1 → #0) →
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True (* FIXME state your theorem *).
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Proof.
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(* TODO: exercise *)
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Admitted.
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(** Exercise 5 (LN Exercise 28): Free Theorems II *)
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Lemma free_theorem_either :
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∀ f : val,
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TY 0; ∅ ⊢ f : (∀: #0 → #0 → #0) →
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∀ (v1 v2 : val), is_closed [] v1 → is_closed [] v2 →
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big_step (f <> v1 v2) v1 ∨ big_step (f <> v1 v2) v2.
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Proof.
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(* TODO: exercise *)
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Admitted.
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(** Exercise 6 (LN Exercise 29): Free Theorems III *)
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(* Hint: you might want to use the fact that our reduction is deterministic. *)
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Lemma big_step_det e v1 v2 :
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big_step e v1 → big_step e v2 → v1 = v2.
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Proof.
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induction 1 in v2 |-*; inversion 1; subst; eauto 2.
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all: naive_solver.
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Qed.
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Lemma free_theorems_magic :
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∀ (A A1 A2 : type) (f g : val),
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type_wf 0 A → type_wf 0 A1 → type_wf 0 A2 →
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is_closed [] f → is_closed [] g →
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TY 0; ∅ ⊢ f : (∀: (A1 → A2 → #0) → #0) →
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TY 0; ∅ ⊢ g : (A1 → A2 → A) →
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∀ v, big_step (f <> g) v →
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∃ (v1 v2 : val), big_step (g v1 v2) v.
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Proof.
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(* Hint: you may find the following lemmas useful:
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- [sem_val_rel_cons]
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- [type_wf_closed]
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- [val_rel_is_closed]
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- [big_step_preserve_closed]
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*)
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(* TODO: exercise *)
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Admitted.
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End free_theorems.
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