102 lines
3 KiB
Coq
102 lines
3 KiB
Coq
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 context.
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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 Context.
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Module Morph.
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(* Given a context, there is a morphism path from τ to τ' *)
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Reserved Notation "Γ '|-' σ '~>' τ" (at level 101, σ at next level, τ at next level).
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Inductive morphism_path : context -> type_term -> type_term -> Prop :=
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| M_Sub : forall Γ τ τ',
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(τ :<= τ') ->
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(Γ |- τ ~> τ')
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| M_Single : forall Γ h τ τ',
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(context_contains Γ h (type_morph τ τ')) ->
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(Γ |- τ ~> τ')
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| M_Chain : forall Γ τ τ' τ'',
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(Γ |- τ ~> τ') ->
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(Γ |- τ' ~> τ'') ->
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(Γ |- τ ~> τ'')
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| M_Lift : forall Γ σ τ τ',
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(Γ |- τ ~> τ') ->
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(Γ |- (type_ladder σ τ) ~> (type_ladder σ τ'))
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| M_MapSeq : forall Γ τ τ',
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(Γ |- τ ~> τ') ->
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(Γ |- (type_spec (type_id "Seq") τ) ~> (type_spec (type_id "Seq") τ'))
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where "Γ '|-' s '~>' t" := (morphism_path Γ s t).
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Inductive translate_morphism_path : context -> type_term -> type_term -> expr_term -> Prop :=
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| Translate_Subtype : forall Γ τ τ',
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(τ :<= τ') ->
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(translate_morphism_path Γ τ τ'
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(expr_morph "x" τ (expr_var "x")))
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| Translate_Lift : forall Γ σ τ τ' m,
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(Γ |- τ ~> τ') ->
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(translate_morphism_path Γ τ τ' m) ->
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(translate_morphism_path Γ (type_ladder σ τ) (type_ladder σ τ')
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(expr_morph "x" (type_ladder σ τ)
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(expr_ascend σ (expr_app m (expr_descend τ (expr_var "x"))))))
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| Translate_Single : forall Γ h τ τ',
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(context_contains Γ h (type_morph τ τ')) ->
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(translate_morphism_path Γ τ τ' (expr_var h))
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| Translate_Chain : forall Γ τ τ' τ'' m1 m2,
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(translate_morphism_path Γ τ τ' m1) ->
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(translate_morphism_path Γ τ' τ'' m2) ->
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(translate_morphism_path Γ τ τ''
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(expr_morph "x" τ (expr_app m2 (expr_app m1 (expr_var "x")))))
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| Translate_MapSeq : forall Γ τ τ' m,
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(translate_morphism_path Γ τ τ' m) ->
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(translate_morphism_path Γ
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(type_spec (type_id "Seq") τ)
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(type_spec (type_id "Seq") τ')
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(expr_morph "xs"
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(type_spec (type_id "Seq") τ)
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(expr_app (expr_app (expr_ty_app (expr_ty_app
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(expr_var "map") τ) τ') m)
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(expr_var "xs"))))
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.
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Example morphism_paths :
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(ctx_assign "degrees-to-turns" [< $"Angle"$~$"Degrees"$~$"ℝ"$ ->morph $"Angle"$~$"Turns"$~$"ℝ"$ >]
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(ctx_assign "turns-to-radians" [< $"Angle"$~$"Turns"$~$"ℝ"$ ->morph $"Angle"$~$"Radians"$~$"ℝ"$ >]
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ctx_empty))
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|- [< <$"Seq"$ $"Hue"$~$"Angle"$~$"Degrees"$~$"ℝ"$> >]
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~> [< <$"Seq"$ $"Hue"$~$"Angle"$~$"Radians"$~$"ℝ"$> >]
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.
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Proof.
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intros.
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apply M_MapSeq.
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apply M_Lift.
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apply M_Chain with (τ':=[<$"Angle"$~$"Turns"$~$"ℝ"$>]).
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apply M_Single with (h:="degrees-to-turns"%string).
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apply C_take.
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apply M_Single with (h:="turns-to-radians"%string).
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apply C_shuffle.
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apply C_take.
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Qed.
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End Morph.
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