84 lines
1.4 KiB
Coq
84 lines
1.4 KiB
Coq
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From Coq Require Import Strings.String.
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Require Import terms.
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Require Import subst.
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Require Import equiv.
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Require Import subtype.
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Require Import smallstep.
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Require Import typing.
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Include Terms.
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Include Subst.
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Include Equiv.
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Include Subtype.
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Include Smallstep.
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Include Typing.
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Module Soundness.
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(* e is stuck when it is neither a value, nor can it be reduced *)
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Definition is_stuck (e:expr_term) : Prop :=
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~(is_value e) ->
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~(exists e', e -->β e')
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.
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(* every exactly typed term is not stuck *)
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Lemma exact_progress :
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forall (e:expr_term),
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(is_exactly_typed e) -> ~(is_stuck e)
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.
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Proof.
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Admitted.
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(* every well typed term is not stuck *)
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Lemma progress :
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forall (e:expr_term),
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(is_well_typed e) -> ~(is_stuck e)
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.
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Proof.
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Admitted.
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(* reduction step preserves the type *)
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Lemma exact_preservation :
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forall Γ e e' τ,
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(Γ |- e \is τ) ->
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(e -->β e') ->
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(Γ |- e' \is τ)
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.
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Proof.
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(*
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intros.
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generalize dependent e'.
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induction H.
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intros e' I.
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inversion I.
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*)
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Admitted.
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(* reduction step preserves well-typedness *)
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Lemma preservation :
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forall Γ e e' τ,
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(Γ |- e \compatible τ) ->
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(e -->β e') ->
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(Γ |- e' \compatible τ)
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.
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Proof.
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Admitted.
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(* every well-typed expression can be reduced to a value *)
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Theorem soundness :
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forall (e:expr_term),
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(is_well_typed e) ->
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(exists e', e -->β* e' /\ (is_value e'))
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.
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Proof.
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intros.
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Admitted.
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End Soundness.
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