-(*********************************************************************************************************************************)
-(* HaskProofCategory: *)
-(* *)
-(* Natural Deduction proofs of the well-typedness of a Haskell term form a category *)
-(* *)
-(*********************************************************************************************************************************)
-
-Generalizable All Variables.
-Require Import Preamble.
-Require Import General.
-Require Import NaturalDeduction.
-Require Import Coq.Strings.String.
-Require Import Coq.Lists.List.
-
-Require Import HaskKinds.
-Require Import HaskCoreTypes.
-Require Import HaskLiteralsAndTyCons.
-Require Import HaskStrongTypes.
-Require Import HaskProof.
-Require Import NaturalDeduction.
-Require Import NaturalDeductionCategory.
-
-Require Import Algebras_ch4.
-Require Import Categories_ch1_3.
-Require Import Functors_ch1_4.
-Require Import Isomorphisms_ch1_5.
-Require Import ProductCategories_ch1_6_1.
-Require Import OppositeCategories_ch1_6_2.
-Require Import Enrichment_ch2_8.
-Require Import Subcategories_ch7_1.
-Require Import NaturalTransformations_ch7_4.
-Require Import NaturalIsomorphisms_ch7_5.
-Require Import MonoidalCategories_ch7_8.
-Require Import Coherence_ch7_8.
-
-Require Import HaskStrongTypes.
-Require Import HaskStrong.
-Require Import HaskProof.
-Require Import HaskStrongToProof.
-Require Import HaskProofToStrong.
-Require Import ProgrammingLanguage.
-
-Open Scope nd_scope.
-
-
-(*
- * The flattening transformation. Currently only TWO-level languages are
- * supported, and the level-1 sublanguage is rather limited.
- *
- *
- * This file abuses terminology pretty badly. For purposes of this file,
- * "PCF" means "the level-1 sublanguage" and "FC" (aka System FC) means
- * the whole language (level-0 language including bracketed level-1 terms)
- *)
-Section HaskProofCategory.
-
- Context (ndr_systemfc:@ND_Relation _ Rule).
-
- Inductive PCFJudg Γ (Δ:CoercionEnv Γ) (ec:HaskTyVar Γ ★) :=
- pcfjudg : Tree ??(HaskType Γ ★) -> Tree ??(HaskType Γ ★) -> PCFJudg Γ Δ ec.
- Implicit Arguments pcfjudg [ [Γ] [Δ] [ec] ].
-
- (* given an PCFJudg at depth (ec::depth) we can turn it into an PCFJudg
- * from depth (depth) by wrapping brackets around everything in the
- * succedent and repopulating *)
- Definition brakify {Γ}{Δ}{ec} (j:PCFJudg Γ Δ ec) : Judg :=
- match j with
- pcfjudg Σ τ => Γ > Δ > (Σ@@@(ec::nil)) |- (mapOptionTree (fun t => HaskBrak ec t) τ @@@ nil)
- end.
-
- Definition pcf_vars {Γ}(ec:HaskTyVar Γ ★)(t:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★)
- := mapOptionTreeAndFlatten (fun lt =>
- match lt with t @@ l => match l with
- | ec'::nil => if eqd_dec ec ec' then [t] else []
- | _ => []
- end
- end) t.
-
- Inductive MatchingJudgments {Γ}{Δ}{ec} : Tree ??(PCFJudg Γ Δ ec) -> Tree ??Judg -> Type :=
- | match_nil : MatchingJudgments [] []
- | match_branch : forall a b c d, MatchingJudgments a b -> MatchingJudgments c d -> MatchingJudgments (a,,c) (b,,d)
- | match_leaf :
- forall Σ τ lev,
- MatchingJudgments
- [pcfjudg (pcf_vars ec Σ) τ ]
- [Γ > Δ > Σ |- (mapOptionTree (HaskBrak ec) τ @@@ lev)].
-
- Definition fc_vars {Γ}(ec:HaskTyVar Γ ★)(t:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★)
- := mapOptionTreeAndFlatten (fun lt =>
- match lt with t @@ l => match l with
- | ec'::nil => if eqd_dec ec ec' then [] else [t]
- | _ => []
- end
- end) t.
-
- Definition pcfjudg2judg {Γ}{Δ:CoercionEnv Γ} ec (cj:PCFJudg Γ Δ ec) :=
- match cj with pcfjudg Σ τ => Γ > Δ > (Σ @@@ (ec::nil)) |- (τ @@@ (ec::nil)) end.
-
- (* Rules allowed in PCF; i.e. rules we know how to turn into GArrows *)
- (* Rule_PCF consists of the rules allowed in flat PCF: everything except *)
- (* AppT, AbsT, AppC, AbsC, Cast, Global, and some Case statements *)
- Inductive Rule_PCF {Γ}{Δ:CoercionEnv Γ} (ec:HaskTyVar Γ ★)
- : forall (h c:Tree ??(PCFJudg Γ Δ ec)), Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c) -> Type :=
- | PCF_RArrange : ∀ x y t a, Rule_PCF ec [pcfjudg _ _ ] [ pcfjudg _ _ ] (RArrange Γ Δ (x@@@(ec::nil)) (y@@@(ec::nil)) (t@@@(ec::nil)) a)
- | PCF_RLit : ∀ lit , Rule_PCF ec [ ] [ pcfjudg [] [_] ] (RLit Γ Δ lit (ec::nil))
- | PCF_RNote : ∀ Σ τ n , Rule_PCF ec [pcfjudg _ [_]] [ pcfjudg _ [_] ] (RNote Γ Δ (Σ@@@(ec::nil)) τ (ec::nil) n)
- | PCF_RVar : ∀ σ , Rule_PCF ec [ ] [ pcfjudg [_] [_] ] (RVar Γ Δ σ (ec::nil) )
- | PCF_RLam : ∀ Σ tx te , Rule_PCF ec [pcfjudg (_,,[_]) [_] ] [ pcfjudg _ [_] ] (RLam Γ Δ (Σ@@@(ec::nil)) tx te (ec::nil) )
-
- | PCF_RApp : ∀ Σ Σ' tx te ,
- Rule_PCF ec ([pcfjudg _ [_]],,[pcfjudg _ [_]]) [pcfjudg (_,,_) [_]]
- (RApp Γ Δ (Σ@@@(ec::nil))(Σ'@@@(ec::nil)) tx te (ec::nil))
-
- | PCF_RLet : ∀ Σ Σ' σ₂ p,
- Rule_PCF ec ([pcfjudg _ [_]],,[pcfjudg (_,,[_]) [_]]) [pcfjudg (_,,_) [_]]
- (RLet Γ Δ (Σ@@@(ec::nil)) (Σ'@@@(ec::nil)) σ₂ p (ec::nil))
-
- | PCF_RVoid : Rule_PCF ec [ ] [ pcfjudg [] [] ] (RVoid Γ Δ )
-(*| PCF_RLetRec : ∀ Σ₁ τ₁ τ₂ , Rule_PCF (ec::nil) _ _ (RLetRec Γ Δ Σ₁ τ₁ τ₂ (ec::nil) )*)
- | PCF_RJoin : ∀ Σ₁ Σ₂ τ₁ τ₂, Rule_PCF ec ([pcfjudg _ _],,[pcfjudg _ _]) [pcfjudg (_,,_) (_,,_)]
- (RJoin Γ Δ (Σ₁@@@(ec::nil)) (Σ₂@@@(ec::nil)) (τ₁@@@(ec::nil)) (τ₂@@@(ec::nil))).
- (* need int/boolean case *)
- Implicit Arguments Rule_PCF [ ].
-
- Definition PCFRule Γ Δ lev h c := { r:_ & @Rule_PCF Γ Δ lev h c r }.
-
- (* An organized deduction has been reorganized into contiguous blocks whose
- * hypotheses (if any) and conclusion have the same Γ and Δ and a fixed nesting depth. The boolean
- * indicates if non-PCF rules have been used *)
- Inductive OrgR : Tree ??Judg -> Tree ??Judg -> Type :=
-
- | org_fc : forall h c (r:Rule h c),
- Rule_Flat r ->
- OrgR h c
-
- | org_pcf : forall Γ Δ ec h h' c c',
- MatchingJudgments h h' ->
- MatchingJudgments c c' ->
- ND (PCFRule Γ Δ ec) h c ->
- OrgR h' c'.
-
- Definition mkEsc {Γ}{Δ}{ec}(h:Tree ??(PCFJudg Γ Δ ec))
- : ND Rule
- (mapOptionTree brakify h)
- (mapOptionTree (pcfjudg2judg ec) h).
- apply nd_replicate; intros.
- destruct o; simpl in *.
- induction t0.
- destruct a; simpl.
- apply nd_rule.
- apply REsc.
- apply nd_id.
- apply (Prelude_error "mkEsc got multi-leaf succedent").
- Defined.
-
- Definition mkBrak {Γ}{Δ}{ec}(h:Tree ??(PCFJudg Γ Δ ec))
- : ND Rule
- (mapOptionTree (pcfjudg2judg ec) h)
- (mapOptionTree brakify h).
- apply nd_replicate; intros.
- destruct o; simpl in *.
- induction t0.
- destruct a; simpl.
- apply nd_rule.
- apply RBrak.
- apply nd_id.
- apply (Prelude_error "mkBrak got multi-leaf succedent").
- Defined.
-
- (*
- Definition Partition {Γ} ec (Σ:Tree ??(LeveledHaskType Γ ★)) :=
- { vars:(_ * _) |
- fc_vars ec Σ = fst vars /\
- pcf_vars ec Σ = snd vars }.
- *)
-
- Definition pcfToND : forall Γ Δ ec h c,
- ND (PCFRule Γ Δ ec) h c -> ND Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c).
- intros.
- eapply (fun q => nd_map' _ q X).
- intros.
- destruct X0.
- apply nd_rule.
- apply x.
- Defined.
-
- Instance OrgPCF Γ Δ lev : @ND_Relation _ (PCFRule Γ Δ lev) :=
- { ndr_eqv := fun a b f g => (pcfToND _ _ _ _ _ f) === (pcfToND _ _ _ _ _ g) }.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- Defined.
-
- (*
- * An intermediate representation necessitated by Coq's termination
- * conditions. This is basically a tree where each node is a
- * subproof which is either entirely level-1 or entirely level-0
- *)
- Inductive Alternating : Tree ??Judg -> Type :=
-
- | alt_nil : Alternating []
-
- | alt_branch : forall a b,
- Alternating a -> Alternating b -> Alternating (a,,b)
-
- | alt_fc : forall h c,
- Alternating h ->
- ND Rule h c ->
- Alternating c
-
- | alt_pcf : forall Γ Δ ec h c h' c',
- MatchingJudgments h h' ->
- MatchingJudgments c c' ->
- Alternating h' ->
- ND (PCFRule Γ Δ ec) h c ->
- Alternating c'.
-
- Require Import Coq.Logic.Eqdep.
-
- Lemma magic a b c d ec e :
- ClosedSIND(Rule:=Rule) [a > b > c |- [d @@ (ec :: e)]] ->
- ClosedSIND(Rule:=Rule) [a > b > pcf_vars ec c @@@ (ec :: nil) |- [d @@ (ec :: nil)]].
- admit.
- Defined.
-
- Definition orgify : forall Γ Δ Σ τ (pf:ClosedSIND(Rule:=Rule) [Γ > Δ > Σ |- τ]), Alternating [Γ > Δ > Σ |- τ].
-
- refine (
- fix orgify_fc' Γ Δ Σ τ (pf:ClosedSIND [Γ > Δ > Σ |- τ]) {struct pf} : Alternating [Γ > Δ > Σ |- τ] :=
- let case_main := tt in _
- with orgify_fc c (pf:ClosedSIND c) {struct pf} : Alternating c :=
- (match c as C return C=c -> Alternating C with
- | T_Leaf None => fun _ => alt_nil
- | T_Leaf (Some (Γ > Δ > Σ |- τ)) => let case_leaf := tt in fun eqpf => _
- | T_Branch b1 b2 => let case_branch := tt in fun eqpf => _
- end (refl_equal _))
- with orgify_pcf Γ Δ ec pcfj j (m:MatchingJudgments pcfj j)
- (pf:ClosedSIND (mapOptionTree (pcfjudg2judg ec) pcfj)) {struct pf} : Alternating j :=
- let case_pcf := tt in _
- for orgify_fc').
-
- destruct case_main.
- inversion pf; subst.
- set (alt_fc _ _ (orgify_fc _ X) (nd_rule X0)) as backup.
- refine (match X0 as R in Rule H C return
- match C with
- | T_Leaf (Some (Γ > Δ > Σ |- τ)) =>
- h=H -> Alternating [Γ > Δ > Σ |- τ] -> Alternating [Γ > Δ > Σ |- τ]
- | _ => True
- end
- with
- | RBrak Σ a b c n m => let case_RBrak := tt in fun pf' backup => _
- | REsc Σ a b c n m => let case_REsc := tt in fun pf' backup => _
- | _ => fun pf' x => x
- end (refl_equal _) backup).
- clear backup0 backup.
-
- destruct case_RBrak.
- rename c into ec.
- set (@match_leaf Σ0 a ec n [b] m) as q.
- set (orgify_pcf Σ0 a ec _ _ q) as q'.
- apply q'.
- simpl.
- rewrite pf' in X.
- apply magic in X.
- apply X.
-
- destruct case_REsc.
- apply (Prelude_error "encountered Esc in wrong side of mkalt").
-
- destruct case_leaf.
- apply orgify_fc'.
- rewrite eqpf.
- apply pf.
-
- destruct case_branch.
- rewrite <- eqpf in pf.
- inversion pf; subst.
- apply no_rules_with_multiple_conclusions in X0.
- inversion X0.
- exists b1. exists b2.
- auto.
- apply (alt_branch _ _ (orgify_fc _ X) (orgify_fc _ X0)).
-
- destruct case_pcf.
- Admitted.
-
- Definition pcfify Γ Δ ec : forall Σ τ,
- ClosedSIND(Rule:=Rule) [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]
- -> ND (PCFRule Γ Δ ec) [] [pcfjudg Σ τ].
-
- refine ((
- fix pcfify Σ τ (pn:@ClosedSIND _ Rule [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]) {struct pn}
- : ND (PCFRule Γ Δ ec) [] [pcfjudg Σ τ] :=
- (match pn in @ClosedSIND _ _ J return J=[Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)] -> _ with
- | cnd_weak => let case_nil := tt in _
- | cnd_rule h c cnd' r => let case_rule := tt in _
- | cnd_branch _ _ c1 c2 => let case_branch := tt in _
- end (refl_equal _)))).
- intros.
- inversion H.
- intros.
- destruct c; try destruct o; inversion H.
- destruct j.
- Admitted.
-
- (* any proof in organized form can be "dis-organized" *)
- Definition unOrgR : forall h c, OrgR h c -> ND Rule h c.
- intros.
-
- induction X.
- apply nd_rule.
- apply r.
-
- eapply nd_comp.
- (*
- apply (mkEsc h).
- eapply nd_comp; [ idtac | apply (mkBrak c) ].
- apply pcfToND.
- apply n.
- *)
- Admitted.
-
- Definition unOrgND h c : ND OrgR h c -> ND Rule h c := nd_map unOrgR.
-
- Instance OrgNDR : @ND_Relation _ OrgR :=
- { ndr_eqv := fun a b f g => (unOrgND _ _ f) === (unOrgND _ _ g) }.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- Defined.
-
- Hint Constructors Rule_Flat.
-
- Instance PCF_sequents Γ Δ lev : @SequentCalculus _ (PCFRule Γ Δ lev) _ pcfjudg.
- apply Build_SequentCalculus.
- intros.
- induction a.
- destruct a; simpl.
- apply nd_rule.
- exists (RVar _ _ _ _).
- apply PCF_RVar.
- apply nd_rule.
- exists (RVoid _ _ ).
- apply PCF_RVoid.
- eapply nd_comp.
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply (nd_prod IHa1 IHa2).
- apply nd_rule.
- exists (RJoin _ _ _ _ _ _).
- apply PCF_RJoin.
- Defined.
-
- Definition PCF_Arrange {Γ}{Δ}{lev} : forall x y z, Arrange x y -> ND (PCFRule Γ Δ lev) [pcfjudg x z] [pcfjudg y z].
- admit.
- Defined.
-
- Definition PCF_cut Γ Δ lev : forall a b c, ND (PCFRule Γ Δ lev) ([ pcfjudg a b ],,[ pcfjudg b c ]) [ pcfjudg a c ].
- intros.
- destruct b.
- destruct o.
- destruct c.
- destruct o.
-
- (* when the cut is a single leaf and the RHS is a single leaf: *)
- eapply nd_comp.
- eapply nd_prod.
- apply nd_id.
- apply (PCF_Arrange [h] ([],,[h]) [h0]).
- apply RuCanL.
- eapply nd_comp; [ idtac | apply (PCF_Arrange ([],,a) a [h0]); apply RCanL ].
- apply nd_rule.
-(*
- set (@RLet Γ Δ [] (a@@@(ec::nil)) h0 h (ec::nil)) as q.
- exists q.
- apply (PCF_RLet _ [] a h0 h).
- apply (Prelude_error "cut rule invoked with [a|=[b]] [[b]|=[]]").
- apply (Prelude_error "cut rule invoked with [a|=[b]] [[b]|=[x,,y]]").
- apply (Prelude_error "cut rule invoked with [a|=[]] [[]|=c]").
- apply (Prelude_error "cut rule invoked with [a|=[b,,c]] [[b,,c]|=z]").
-*)
- admit.
- admit.
- admit.
- admit.
- admit.
- Defined.
-
- Instance PCF_cutrule Γ Δ lev : CutRule (PCF_sequents Γ Δ lev) :=
- { nd_cut := PCF_cut Γ Δ lev }.
- admit.
- admit.
- admit.
- Defined.
-
- Definition PCF_left Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (a,,b) (a,,c)].
- eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
- eapply nd_prod; [ apply nd_seq_reflexive | apply nd_id ].
- apply nd_rule.
- set (@PCF_RJoin Γ Δ lev a b a c) as q'.
- refine (existT _ _ _).
- apply q'.
- Defined.
-
- Definition PCF_right Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (b,,a) (c,,a)].
- eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
- eapply nd_prod; [ apply nd_id | apply nd_seq_reflexive ].
- apply nd_rule.
- set (@PCF_RJoin Γ Δ lev b a c a) as q'.
- refine (existT _ _ _).
- apply q'.
- Defined.
-
- Instance PCF_sequent_join Γ Δ lev : @SequentExpansion _ _ _ _ _ (PCF_sequents Γ Δ lev) (PCF_cutrule Γ Δ lev) :=
- { se_expand_left := PCF_left Γ Δ lev
- ; se_expand_right := PCF_right Γ Δ lev }.
- admit.
- admit.
- admit.
- admit.
- Defined.
-
- (* 5.1.3 *)
- Instance PCF Γ Δ lev : @ProgrammingLanguage _ _ pcfjudg (PCFRule Γ Δ lev) :=
- { pl_eqv := OrgPCF Γ Δ lev
- ; pl_sc := PCF_sequents Γ Δ lev
- ; pl_subst := PCF_cutrule Γ Δ lev
- ; pl_sequent_join := PCF_sequent_join Γ Δ lev
- }.
- apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
-
- apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RCossa _ _ _)).
- apply (PCF_RArrange lev ((a,,b),,c) (a,,(b,,c)) x).
-
- apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RAssoc _ _ _)).
- apply (PCF_RArrange lev (a,,(b,,c)) ((a,,b),,c) x).
-
- apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RCanL _)).
- apply (PCF_RArrange lev ([],,a) _ _).
-
- apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RCanR _)).
- apply (PCF_RArrange lev (a,,[]) _ _).
-
- apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RuCanL _)).
- apply (PCF_RArrange lev _ ([],,a) _).
-
- apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RuCanR _)).
- apply (PCF_RArrange lev _ (a,,[]) _).
- Defined.
-
- Instance SystemFCa_sequents Γ Δ : @SequentCalculus _ OrgR _ (mkJudg Γ Δ).
- apply Build_SequentCalculus.
- intros.
- induction a.
- destruct a; simpl.
- apply nd_rule.
- destruct l.
- apply org_fc with (r:=RVar _ _ _ _).
- auto.
- apply nd_rule.
- apply org_fc with (r:=RVoid _ _ ).
- auto.
- eapply nd_comp.
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply (nd_prod IHa1 IHa2).
- apply nd_rule.
- apply org_fc with (r:=RJoin _ _ _ _ _ _).
- auto.
- Defined.
-
- Definition SystemFCa_cut Γ Δ : forall a b c, ND OrgR ([ Γ > Δ > a |- b ],,[ Γ > Δ > b |- c ]) [ Γ > Δ > a |- c ].
- intros.
- destruct b.
- destruct o.
- destruct c.
- destruct o.
-
- (* when the cut is a single leaf and the RHS is a single leaf: *)
- eapply nd_comp.
- eapply nd_prod.
- apply nd_id.
- eapply nd_rule.
- apply org_fc with (r:=RArrange _ _ _ _ _ (RuCanL [l])).
- auto.
- eapply nd_comp; [ idtac | eapply nd_rule; apply org_fc with (r:=RArrange _ _ _ _ _ (RCanL _)) ].
- apply nd_rule.
- destruct l.
- destruct l0.
- assert (h0=h2). admit.
- subst.
- apply org_fc with (r:=@RLet Γ Δ [] a h1 h h2).
- auto.
- auto.
- apply (Prelude_error "systemfc cut rule invoked with [a|=[b]] [[b]|=[]]").
- apply (Prelude_error "systemfc cut rule invoked with [a|=[b]] [[b]|=[x,,y]]").
- apply (Prelude_error "systemfc rule invoked with [a|=[]] [[]|=c]").
- apply (Prelude_error "systemfc rule invoked with [a|=[b,,c]] [[b,,c]|=z]").
- Defined.
-
- Instance SystemFCa_cutrule Γ Δ : CutRule (SystemFCa_sequents Γ Δ) :=
- { nd_cut := SystemFCa_cut Γ Δ }.
- admit.
- admit.
- admit.
- Defined.
-
- Definition SystemFCa_left Γ Δ a b c : ND OrgR [Γ > Δ > b |- c] [Γ > Δ > (a,,b) |- (a,,c)].
- eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
- eapply nd_prod; [ apply nd_seq_reflexive | apply nd_id ].
- apply nd_rule.
- apply org_fc with (r:=RJoin Γ Δ a b a c).
- auto.
- Defined.
-
- Definition SystemFCa_right Γ Δ a b c : ND OrgR [Γ > Δ > b |- c] [Γ > Δ > (b,,a) |- (c,,a)].
- eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
- eapply nd_prod; [ apply nd_id | apply nd_seq_reflexive ].
- apply nd_rule.
- apply org_fc with (r:=RJoin Γ Δ b a c a).
- auto.
- Defined.
-
- Instance SystemFCa_sequent_join Γ Δ : @SequentExpansion _ _ _ _ _ (SystemFCa_sequents Γ Δ) (SystemFCa_cutrule Γ Δ) :=
- { se_expand_left := SystemFCa_left Γ Δ
- ; se_expand_right := SystemFCa_right Γ Δ }.
- admit.
- admit.
- admit.
- admit.
- Defined.
-
- (* 5.1.2 *)
- Instance SystemFCa Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) OrgR :=
- { pl_eqv := OrgNDR
- ; pl_sc := SystemFCa_sequents Γ Δ
- ; pl_subst := SystemFCa_cutrule Γ Δ
- ; pl_sequent_join := SystemFCa_sequent_join Γ Δ
- }.
- apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
- apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCossa a b c))). apply Flat_RArrange.
- apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RAssoc a b c))). apply Flat_RArrange.
- apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanL a ))). apply Flat_RArrange.
- apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanR a ))). apply Flat_RArrange.
- apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanL a ))). apply Flat_RArrange.
- apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanR a ))). apply Flat_RArrange.
- Defined.
-
-
-(*
- Definition code2garrow Γ (ec t:RawHaskType Γ ★) :=
- match t with
-(* | TApp ★ ★ (TApp _ ★ TArrow tx) t' => code2garrow0 ec tx t'*)
- | _ => code2garrow0 ec unitType t
- end.
- Opaque code2garrow.
- Fixpoint typeMap {TV}{κ}(ty:@RawHaskType TV κ) : @RawHaskType TV κ :=
- match ty as TY in RawHaskType _ K return RawHaskType TV K with
- | TCode ec t => code2garrow _ ec t
- | TApp _ _ t1 t2 => TApp (typeMap t1) (typeMap t2)
- | TAll _ f => TAll _ (fun tv => typeMap (f tv))
- | TCoerc _ t1 t2 t3 => TCoerc (typeMap t1) (typeMap t2) (typeMap t3)
- | TVar _ v => TVar v
- | TArrow => TArrow
- | TCon tc => TCon tc
- | TyFunApp tf rhtl => (* FIXME *) TyFunApp tf rhtl
- end.
-
- Definition typeMapL {Γ}(lht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
- match lht with
-(* | t @@ nil => (fun TV ite => typeMap (t TV ite)) @@ lev*)
- | t @@ lev => (fun TV ite => typeMap (t TV ite)) @@ lev
- end.
-*)
-
- (* gathers a tree of guest-language types into a single host-language types via the tensor *)
- Definition tensorizeType {Γ} (lt:Tree ??(HaskType Γ ★)) : HaskType Γ ★.
- admit.
- Defined.
-
- Definition mkGA {Γ} : HaskType Γ ★ -> HaskType Γ ★ -> HaskType Γ ★.
- admit.
- Defined.
-
- Definition guestJudgmentAsGArrowType {Γ}{Δ}{ec}(lt:PCFJudg Γ Δ ec) : HaskType Γ ★ :=
- match lt with
- pcfjudg x y =>
- (mkGA (tensorizeType x) (tensorizeType y))
- end.
-
- Definition obact {Γ}{Δ} ec (X:Tree ??(PCFJudg Γ Δ ec)) : Tree ??(LeveledHaskType Γ ★) :=
- mapOptionTree guestJudgmentAsGArrowType X @@@ nil.
-
- (*
- * Here it is, what you've all been waiting for! When reading this,
- * it might help to have the definition for "Inductive ND" (see
- * NaturalDeduction.v) handy as a cross-reference.
- *)
- Definition FlatteningFunctor_fmor {Γ}{Δ}{ec}
- : forall h c,
- (h~~{JudgmentsL _ _ (PCF Γ Δ ec)}~~>c) ->
- ((obact ec h)~~{TypesL _ _ (SystemFCa Γ Δ)}~~>(obact ec c)).
-
- set (@nil (HaskTyVar Γ ★)) as lev.
-
- unfold hom; unfold ob; unfold ehom; simpl; unfold mon_i; unfold obact; intros.
-
- induction X; simpl.
-
- (* the proof from no hypotheses of no conclusions (nd_id0) becomes RVoid *)
- apply nd_rule; apply (org_fc _ _ (RVoid _ _ )). auto.
-
- (* the proof from hypothesis X of conclusion X (nd_id1) becomes RVar *)
- apply nd_rule; apply (org_fc _ _ (RVar _ _ _ _)). auto.
-
- (* the proof from hypothesis X of no conclusions (nd_weak) becomes RWeak;;RVoid *)
- eapply nd_comp;
- [ idtac
- | eapply nd_rule
- ; eapply (org_fc _ _ (RArrange _ _ _ _ _ (RWeak _)))
- ; auto ].
- eapply nd_rule.
- eapply (org_fc _ _ (RVoid _ _)); auto.
-
- (* the proof from hypothesis X of two identical conclusions X,,X (nd_copy) becomes RVar;;RJoin;;RCont *)
- eapply nd_comp; [ idtac | eapply nd_rule; eapply (org_fc _ _ (RArrange _ _ _ _ _ (RCont _))) ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- set (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa Γ Δ))
- (mapOptionTree guestJudgmentAsGArrowType h @@@ lev)) as q.
- eapply nd_comp.
- eapply nd_prod.
- apply q.
- apply q.
- apply nd_rule.
- eapply (org_fc _ _ (RJoin _ _ _ _ _ _ )).
- auto.
- auto.
-
- (* nd_prod becomes nd_llecnac;;nd_prod;;RJoin *)
- eapply nd_comp.
- apply (nd_llecnac ;; nd_prod IHX1 IHX2).
- apply nd_rule.
- eapply (org_fc _ _ (RJoin _ _ _ _ _ _ )).
- auto.
-
- (* nd_comp becomes pl_subst (aka nd_cut) *)
- eapply nd_comp.
- apply (nd_llecnac ;; nd_prod IHX1 IHX2).
- clear IHX1 IHX2 X1 X2.
- apply (@nd_cut _ _ _ _ _ _ (@pl_subst _ _ _ _ (SystemFCa Γ Δ))).
-
- (* nd_cancell becomes RVar;;RuCanL *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanL _))) ].
- apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa Γ Δ))).
- auto.
-
- (* nd_cancelr becomes RVar;;RuCanR *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanR _))) ].
- apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa Γ Δ))).
- auto.
-
- (* nd_llecnac becomes RVar;;RCanL *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanL _))) ].
- apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa Γ Δ))).
- auto.
-
- (* nd_rlecnac becomes RVar;;RCanR *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanR _))) ].
- apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa Γ Δ))).
- auto.
-
- (* nd_assoc becomes RVar;;RAssoc *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RAssoc _ _ _))) ].
- apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa Γ Δ))).
- auto.
-
- (* nd_cossa becomes RVar;;RCossa *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RCossa _ _ _))) ].
- apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa Γ Δ))).
- auto.
-
- destruct r as [r rp].
- refine (match rp as R in Rule_PCF _ _ _ H C _ with
- | PCF_RArrange h c r q => let case_RURule := tt in _
- | PCF_RLit lit => let case_RLit := tt in _
- | PCF_RNote Σ τ n => let case_RNote := tt in _
- | PCF_RVar σ => let case_RVar := tt in _
- | PCF_RLam Σ tx te => let case_RLam := tt in _
- | PCF_RApp Σ tx te p => let case_RApp := tt in _
- | PCF_RLet Σ σ₁ σ₂ p => let case_RLet := tt in _
- | PCF_RJoin b c d e => let case_RJoin := tt in _
- | PCF_RVoid => let case_RVoid := tt in _
- (*| PCF_RCase T κlen κ θ l x => let case_RCase := tt in _*)
- (*| PCF_RLetRec Σ₁ τ₁ τ₂ lev => let case_RLetRec := tt in _*)
- end); simpl in *.
- clear rp.
- clear r h c.
- rename r0 into r; rename h0 into h; rename c0 into c.
-
- destruct case_RURule.
- refine (match q with
- | RLeft a b c r => let case_RLeft := tt in _
- | RRight a b c r => let case_RRight := tt in _
- | RCanL b => let case_RCanL := tt in _
- | RCanR b => let case_RCanR := tt in _
- | RuCanL b => let case_RuCanL := tt in _
- | RuCanR b => let case_RuCanR := tt in _
- | RAssoc b c d => let case_RAssoc := tt in _
- | RCossa b c d => let case_RCossa := tt in _
- | RExch b c => let case_RExch := tt in _
- | RWeak b => let case_RWeak := tt in _
- | RCont b => let case_RCont := tt in _
- | RComp a b c f g => let case_RComp := tt in _
- end).
-
- destruct case_RCanL.
- (* ga_cancell *)
- admit.
-
- destruct case_RCanR.
- (* ga_cancelr *)
- admit.
-
- destruct case_RuCanL.
- (* ga_uncancell *)
- admit.
-
- destruct case_RuCanR.
- (* ga_uncancelr *)
- admit.
-
- destruct case_RAssoc.
- (* ga_assoc *)
- admit.
-
- destruct case_RCossa.
- (* ga_unassoc *)
- admit.
-
- destruct case_RExch.
- (* ga_swap *)
- admit.
-
- destruct case_RWeak.
- (* ga_drop *)
- admit.
-
- destruct case_RCont.
- (* ga_copy *)
- admit.
-
- destruct case_RLeft.
- (* ga_second *)
- admit.
-
- destruct case_RRight.
- (* ga_first *)
- admit.
-
- destruct case_RComp.
- (* ga_comp *)
- admit.
-
- destruct case_RLit.
- (* ga_literal *)
- admit.
-
- (* hey cool, I figured out how to pass CoreNote's through... *)
- destruct case_RNote.
- eapply nd_comp.
- eapply nd_rule.
- eapply (org_fc _ _ (RVar _ _ _ _)) . auto.
- apply nd_rule.
- apply (org_fc _ _ (RNote _ _ _ _ _ n)). auto.
-
- destruct case_RVar.
- (* ga_id *)
- admit.
-
- destruct case_RLam.
- (* ga_curry, but try to avoid this someday in the future if the argument type isn't a function *)
- admit.
-
- destruct case_RApp.
- (* ga_apply *)
- admit.
-
- destruct case_RLet.
- (* ga_comp! perhaps this means the ga_curry avoidance can be done by turning lambdas into lets? *)
- admit.
-
- destruct case_RVoid.
- (* ga_id u *)
- admit.
-
- destruct case_RJoin.
- (* ga_first+ga_second; technically this assumes a specific evaluation order, which is bad *)
- admit.
-
- Defined.
-
- Instance FlatteningFunctor {Γ}{Δ}{ec} : Functor (JudgmentsL _ _ (PCF Γ Δ ec)) (TypesL _ _ (SystemFCa Γ Δ)) (obact ec) :=
- { fmor := FlatteningFunctor_fmor }.
- admit.
- admit.
- admit.
- Defined.
-(*
- Definition ReificationFunctor Γ Δ : Functor (JudgmentsL _ _ (PCF n Γ Δ)) SystemFCa' (mapOptionTree brakifyJudg).
- refine {| fmor := ReificationFunctor_fmor Γ Δ |}; unfold hom; unfold ob; simpl ; intros.
- unfold ReificationFunctor_fmor; simpl.
- admit.
- unfold ReificationFunctor_fmor; simpl.
- admit.
- unfold ReificationFunctor_fmor; simpl.
- admit.
- Defined.
-
-
- Definition PCF_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
- refine {| plsmme_pl := PCF n Γ Δ |}.
- admit.
- Defined.
-
- Definition SystemFCa_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
- refine {| plsmme_pl := SystemFCa n Γ Δ |}.
- admit.
- Defined.
-
- Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n).
- admit.
- Defined.
-
- (* 5.1.4 *)
- Definition PCF_SystemFCa_two_level n Γ Δ : TwoLevelLanguage (PCF_SMME n Γ Δ) (SystemFCa_SMME (S n) Γ Δ).
- admit.
- (* ... and the retraction exists *)
- Defined.
-*)
- (* Any particular proof in HaskProof is only finitely large, so it uses only finitely many levels of nesting, so
- * it falls within (SystemFCa n) for some n. This function calculates that "n" and performs the translation *)
- (*
- Definition HaskProof_to_SystemFCa :
- forall h c (pf:ND Rule h c),
- { n:nat & h ~~{JudgmentsL (SystemFCa_SMME n)}~~> c }.
- *)
-
- (* for every n we have a functor from the category of (n+1)-bounded proofs to the category of n-bounded proofs *)
-
-
-
-End HaskProofCategory.
-
-(*
- Definition code2garrow0 {Γ}(ec t1 t2:RawHaskType Γ ★) : RawHaskType Γ ★.
- admit.
- Defined.
- Definition code2garrow Γ (ec t:RawHaskType Γ ★) :=
- match t with
-(* | TApp ★ ★ (TApp _ ★ TArrow tx) t' => code2garrow0 ec tx t'*)
- | _ => code2garrow0 ec unitType t
- end.
- Opaque code2garrow.
- Fixpoint typeMap {TV}{κ}(ty:@RawHaskType TV κ) : @RawHaskType TV κ :=
- match ty as TY in RawHaskType _ K return RawHaskType TV K with
- | TCode ec t => code2garrow _ ec t
- | TApp _ _ t1 t2 => TApp (typeMap t1) (typeMap t2)
- | TAll _ f => TAll _ (fun tv => typeMap (f tv))
- | TCoerc _ t1 t2 t3 => TCoerc (typeMap t1) (typeMap t2) (typeMap t3)
- | TVar _ v => TVar v
- | TArrow => TArrow
- | TCon tc => TCon tc
- | TyFunApp tf rhtl => (* FIXME *) TyFunApp tf rhtl
- end.
-*)
-
-
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