Revert "ex5 template"

This reverts commit 86353384.

_CoqProject

@@ -65,4 +65,3 @@ theories/systemf/existential_invariants.v
 #theories/stlc/cbn_logrel.v
 #theories/systemf/exercices04.v
 #theories/systemf/exercises04_sol.v
-#theories/systemf/exercises05.v

theories/systemf/exercises05.v

-From stdpp Require Import gmap base relations.
-From iris Require Import prelude.
-From semantics.systemf Require Import lang notation parallel_subst types logrel tactics.
-From semantics.systemf Require Import existential_invariants.
-
-(** * Exercise Sheet 5 *)
-
-Implicit Types
-  (e : expr)
-  (v : val)
-  (A B : type)
-.
-
-(** ** Exercise 3: Existential Fun *)
-Section existential.
-  (** Since extending our language with records would be tedious,
-    we encode records using nested pairs.
-
-   For instance, we would represent the record type
-   { add : Int → Int → Int; sub : Int → Int → Int; neg : Int → Int }
-   as (Int → Int → Int) × (Int → Int → Int) × (Int → Int).
-
-   Similarly, we would represent the record value
-   { add := λ: "x" "y", "x" + "y";
-     sub := λ: "x" "y", "x" - "y";
-     neg := λ: "x", #0 - "x"
-   }
-  as the nested pair
-  ((λ: "x" "y", "x" + "y",    (* add *)
-    λ: "x" "y", "x" - "y"),   (* sub *)
-    λ: "x", #0 - "x").        (* neg *)
-
-  *)
-
-  (** We will also assume a recursion combinator. We have not formally added it to our language, but we could do so. *)
-  Context (Fix : string → string → expr → val).
-  Notation "'fix:' f x := e" := (Fix f x e)%E
-    (at level 200, f, x at level 1, e at level 200,
-     format "'[' 'fix:'  f  x   :=  '/  ' e ']'") : val_scope.
-  Notation "'fix:' f x := e" := (Fix f x e)%E
-    (at level 200, f, x at level 1, e at level 200,
-     format "'[' 'fix:'  f  x   :=  '/  ' e ']'") : expr_scope.
-  Context (fix_typing : ∀ n Γ (f x: string) (A B: type) (e: expr),
-    type_wf n A →
-    type_wf n B →
-    f ≠ x →
-    TY n; <[x := A]> (<[f := (A → B)%ty]> Γ) ⊢ e : B →
-    TY n; Γ ⊢ (fix: f x := e) : (A → B)).
-
-  Definition ISET : type :=
-    #0. (* FIXME: your definition *)
-
-  (* We represent sets as functions of type ((Int → Bool) × Int × Int),
-    storing the mapping, the minimum value, and the maximum value. *)
-
-  Definition iset : val :=
-    #0. (* FIXME: your definition *)
-
-
-  Lemma iset_typed n Γ : TY n; Γ ⊢ iset : ISET.
-  Proof.
-    (* FIXME *)
-  (*Qed.*)
-  Admitted.
-
-  Definition ISETE : type :=
-    #0 (* FIXME *).
-
-  Definition add_equality : val :=
-    #0. (* FIXME *)
-
-  Lemma add_equality_typed n Γ : TY n; Γ ⊢ add_equality : (ISET → ISETE)%ty.
-  Proof.
-    repeat solve_typing.
-  (*Qed.*)
-  Admitted.
-
-End existential.
-
-Section ex4.
-(** ** Exercise 4: Evenness *)
-(* Consider the following existential type: *)
-Definition even_type : type :=
-  ∃: (#0 ×              (* zero *)
-      (#0 → #0) ×       (* add2 *)
-      (#0 → Int)        (* toint *)
-  )%ty.
-
-(* and consider the following implementation of [even_type]: *)
-Definition even_impl : val :=
-  pack (#0,
-        λ: "z", #2 + "z",
-        λ: "z", "z"
-       ).
-(* We want to prove that [toint] will only every yield even numbers. *)
-(* For that purpose, assume that we have a function [even] that decides even parity available: *)
-Context (even_dec : val).
-Context (even_dec_typed : ∀ n Γ, TY n; Γ ⊢ even_dec : (Int → Bool)).
-
-(* a) Change [even_impl] to [even_impl_instrumented] such that [toint] asserts evenness of the argument before returned.
-You may use the [assert] expression.
-*)
-
-Definition even_impl_instrumented : val :=
-  #0. (* FIXME *)
-
-(* b) Prove that [even_impl_instrumented] is safe. You may assume that even works as intended,
-  but be sure to state this here. *)
-
-Lemma even_impl_instrumented_safe δ:
-  𝒱 even_type δ even_impl_instrumented.
-Proof.
-  (* FIXME *)
-(*Qed.*)
-Admitted.
-End ex4.
-
-(** ** Exercise 5: Abstract sums *)
-
-Definition sum_ex_type (A B : type) : type :=
-  ∃: ((A → #0) ×
-      (B → #0) ×
-      (∀: #1 → (A → #0) → (B → #0) → #0)
-     )%ty.
-
-Definition sum_ex_impl : val :=
-  pack (λ: "x", (#1, "x"),
-        λ: "x", (#2, "x"),
-        Λ, λ: "x" "f1" "f2", if: Fst "x" = #1 then "f1" (Snd "x") else "f2" (Snd "x")
-       ).
-
-Lemma sum_ex_safe A B δ:
-  type_wf 0 A →
-  type_wf 0 B →
-  𝒱 (sum_ex_type A B) δ sum_ex_impl.
-Proof.
-  (* FIXME *)
-Admitted.