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@ -2,6 +2,19 @@ From stdpp Require Import gmap base relations.
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From iris Require Import prelude.
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From iris Require Import prelude.
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From semantics.ts.systemf Require Import lang notation types tactics.
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From semantics.ts.systemf Require Import lang notation types tactics.
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(*
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Exercise 3.1:
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With V[A->B] = {v | ∀v', v' ∈ V[A] -> v v' ∈ E[B]}
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The set V[A->B] is equal to:
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V[A->B] = {v | ∀v', v' ∈ V[A], ∃w, v v' | w, w ∈ V[B]}
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According to big step semantics, `v v' | w` inverts to `v | λx.e` and `e[v'/x] | w`.
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Assuming that `v` and `v'` are closed, we have `e[v'/x]` closed, and thus `e[v'/x] ∈ E[B]`.
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Since `v` is a value and `v | λx.e`, we have `v = λx.e`.
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We thus have `{v | ∀v', v' ∈ V[A] -> v v' ∈ E[B]} = {λx.e | ∀v', v' ∈ V[A] -> e[v'/x] ∈ E[B]}`,
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after which we can carry out our proof as before.
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*)
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(** Exercise 3 (LN Exercise 22): Universal Fun *)
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(** Exercise 3 (LN Exercise 22): Universal Fun *)
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Definition fun_comp : val :=
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Definition fun_comp : val :=
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@ -96,5 +109,3 @@ Proof.
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(* Should be solved by reflexivity. *)
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(* Should be solved by reflexivity. *)
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(* TODO: exercise *)
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(* TODO: exercise *)
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Admitted.
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Admitted.
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