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.
- Definition unitType {Γ} : RawHaskType Γ ★.
- admit.
- Defined.
+ Context (ndr_systemfc:@ND_Relation _ Rule).
- Definition brakifyType {Γ} (lt:LeveledHaskType Γ ★) : LeveledHaskType (★ ::Γ) ★ :=
- match lt with
- t @@ l => HaskBrak (FreshHaskTyVar ★) (weakT t) @@ weakL l
- end.
+ 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] ].
- Definition brakify (j:Judg) : Judg :=
+ (* 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
- Γ > Δ > Σ |- τ =>
- (★ ::Γ) > weakCE Δ > mapOptionTree brakifyType Σ |- mapOptionTree brakifyType τ
- end.
+ pcfjudg Σ τ => Γ > Δ > (Σ@@@(ec::nil)) |- (mapOptionTree (fun t => HaskBrak ec t) τ @@@ nil)
+ end.
- (* Rules allowed in PCF; i.e. rules we know how to turn into GArrows *)
- Section RulePCF.
-
- Context {Γ:TypeEnv}{Δ:CoercionEnv Γ}.
-
- (* 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 : forall {h}{c}, Rule h c -> Prop :=
- | PCF_RURule : ∀ h c r , Rule_PCF (RURule Γ Δ h c r)
- | PCF_RLit : ∀ Γ Δ Σ τ , Rule_PCF (RLit Γ Δ Σ τ )
- | PCF_RNote : ∀ Σ τ l n , Rule_PCF (RNote Γ Δ Σ τ l n)
- | PCF_RVar : ∀ σ l, Rule_PCF (RVar Γ Δ σ l)
- | PCF_RLam : ∀ Σ tx te q , Rule_PCF (RLam Γ Δ Σ tx te q )
- | PCF_RApp : ∀ Σ tx te p l, Rule_PCF (RApp Γ Δ Σ tx te p l)
- | PCF_RLet : ∀ Σ σ₁ σ₂ p l, Rule_PCF (RLet Γ Δ Σ σ₁ σ₂ p l)
- | PCF_RBindingGroup : ∀ q a b c d e , Rule_PCF (RBindingGroup q a b c d e)
- | PCF_RCase : ∀ T κlen κ θ l x , Rule_PCF (RCase Γ Δ T κlen κ θ l x) (* FIXME: only for boolean and int *)
- | Flat_REmptyGroup : ∀ q a , Rule_PCF (REmptyGroup q a)
- | Flat_RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂ lev , Rule_PCF (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
-
- (* "RulePCF n" is the type of rules permitted in PCF with n-level deep nesting (so, "RulePCF 0" is flat PCF) *)
- Inductive RulePCF : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
-
- (* any proof using only PCF rules is an n-bounded proof for any n>0 *)
- | pcf_flat : forall n h c (r:Rule h c), Rule_PCF r -> RulePCF n h c
-
- (* any n-bounded proof may be used as an (n+1)-bounded proof by prepending Esc and appending Brak *)
- | pfc_nest : forall n h c, ND (RulePCF n) h c -> RulePCF (S n) (mapOptionTree brakify h) (mapOptionTree brakify c)
- .
- End RulePCF.
-
- (* "RuleSystemFCa n" is the type of rules permitted in SystemFC^\alpha with n-level-deep nesting
- * (so, "RuleSystemFCa 0" is damn close to System FC) *)
- Inductive RuleSystemFCa : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
- | sfc_flat : forall n h c (r:Rule h c), Rule_Flat r -> RuleSystemFCa n h c
- | sfc_nest : forall n h c Γ Δ, ND (@RulePCF Γ Δ n) h c -> RuleSystemFCa (S n) h c
- .
-
- Context (ndr_pcf :forall n Γ Δ, @ND_Relation _ (@RulePCF Γ Δ n)).
- Context (ndr_systemfca:forall n, @ND_Relation _ (RuleSystemFCa n)).
+ 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.
- (* 5.1.3 *)
- Instance PCF n Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) (@RulePCF Γ Δ n) :=
- { pl_eqv := _ (*@ND_Relation Judg Rule where "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2)*)
- ; pl_tsr := _ (*@TreeStructuralRules Judg Rule T sequent*)
- ; pl_sc := _ (*@SequentCalculus Judg Rule _ sequent*)
- ; pl_subst := _ (*@CutRule Judg Rule _ sequent pl_eqv pl_sc*)
- ; pl_sequent_join := _ (*@SequentExpansion Judg Rule T sequent pl_eqv pl_sc pl_subst*)
- }.
- Admitted.
-
- (* 5.1.2 *)
- Instance SystemFCa n Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) (RuleSystemFCa n) :=
- { pl_eqv := _ (*@ND_Relation Judg Rule where "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2)*)
- ; pl_tsr := _ (*@TreeStructuralRules Judg Rule T sequent*)
- ; pl_sc := _ (*@SequentCalculus Judg Rule _ sequent*)
- ; pl_subst := _ (*@CutRule Judg Rule _ sequent pl_eqv pl_sc*)
- ; pl_sequent_join := _ (*@SequentExpansion Judg Rule T sequent pl_eqv pl_sc pl_subst*)
- }.
- Admitted.
+ 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 PCF_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
- refine {| plsmme_pl := PCF n Γ Δ |}.
- admit.
+ (*
+ 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 SystemFCa_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
- refine {| plsmme_pl := SystemFCa n Γ Δ |}.
+ 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.
- (* 5.1.4 *)
- Definition PCF_SystemFCa_two_level n Γ Δ : TwoLevelLanguage (PCF_SMME n Γ Δ) (SystemFCa_SMME (S n) Γ Δ).
+ 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.
- (* ... and the retraction exists *)
-
- (* 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 Judg h c),
- { n:nat & h ~~{JudgmentsL SystemFCa_SMME n *)
- (* for every n we have a functor from the category of (n+1)-bounded proofs to the category of n-bounded proofs *)
- (*
- Definition ReificationFunctor n : Functor (JudgmentsN n) (JudgmentsN (S n)) (mapOptionTree brakify).
- refine {| fmor := fun h c (f:h~~{JudgmentsN n}~~>c) => nd_rule (br_nest _ _ _ f) |}; intros; simpl.
+ Instance PCF_cutrule Γ Δ lev : CutRule (PCF_sequents Γ Δ lev) :=
+ { nd_cut := PCF_cut Γ Δ lev }.
admit.
admit.
admit.
Defined.
- Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n).
- admit.
+ 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 makeTree : Tree ??(LeveledHaskType Γ ★) -> HaskType Γ ★.
- admit.
+ 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.
- Definition flattenType n (j:JudgmentsN n) : TypesN n.
- unfold eob_eob.
- unfold ob in j.
- refine (mapOptionTree _ j).
- clear j; intro j.
- destruct j as [ Γ' Δ' Σ τ ].
- assert (Γ'=Γ). admit.
- rewrite H in *.
- clear H Γ'.
- refine (_ @@ nil).
- refine (HaskBrak _ ( (makeTree Σ) ---> (makeTree τ) )); intros.
+ 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.
- Definition FlattenFunctor_fmor n :
- forall h c,
- (h~~{JudgmentsN n}~~>c) ->
- ((flattenType n h)~~{TypesN n}~~>(flattenType n c)).
+ (* 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.
- unfold hom in *; simpl.
- unfold mon_i.
- unfold ehom.
- unfold TypesNmor.
+ 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.
- admit.
+ 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.
- Definition FlattenFunctor n : Functor (JudgmentsN n) (TypesN n) (flattenType n).
- refine {| fmor := FlattenFunctor_fmor n |}; intros.
+ Instance SystemFCa_cutrule Γ Δ : CutRule (SystemFCa_sequents Γ Δ) :=
+ { nd_cut := SystemFCa_cut Γ Δ }.
admit.
admit.
admit.
Defined.
-
- End RulePCF.
- Implicit Arguments Rule_PCF [ [h] [c] ].
- Implicit Arguments BoundedRule [ ].
-*)
-(*
- Definition code2garrow0 {Γ}(ec t1 t2:RawHaskType Γ ★) : RawHaskType Γ ★.
+ 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'*)
(* | t @@ nil => (fun TV ite => typeMap (t TV ite)) @@ lev*)
| t @@ lev => (fun TV ite => typeMap (t TV ite)) @@ lev
end.
+*)
- Definition coMap {Γ}(ck:HaskCoercionKind Γ) :=
- fun TV ite => match ck TV ite with
- | mkRawCoercionKind _ t1 t2 => mkRawCoercionKind _ (typeMap t1) (typeMap t2)
- end.
-
- Definition flattenCoercion {Γ}{Δ}{ck}(hk:HaskCoercion Γ Δ ck) : HaskCoercion Γ (map coMap Δ) (coMap ck).
+ (* 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.
- Lemma update_typeMap Γ (lev:HaskLevel Γ) ξ v t
- : (typeMap ○ (update_ξ ξ lev ((⟨v, t ⟩) :: nil)))
- = ( update_ξ (typeMap ○ ξ) lev ((⟨v, typeMap_ t ⟩) :: nil)).
+ Definition mkGA {Γ} : HaskType Γ ★ -> HaskType Γ ★ -> HaskType Γ ★.
admit.
- Qed.
+ Defined.
- Lemma foo κ Γ σ τ : typeMap_ (substT σ τ) = substT(Γ:=Γ)(κ₁:=κ) (fun TV ite => typeMap ○ σ TV ite) τ.
- admit.
- Qed.
+ Definition guestJudgmentAsGArrowType {Γ}{Δ}{ec}(lt:PCFJudg Γ Δ ec) : HaskType Γ ★ :=
+ match lt with
+ pcfjudg x y =>
+ (mkGA (tensorizeType x) (tensorizeType y))
+ end.
- Lemma lit_lemma lit Γ : typeMap_ (literalType lit) = literalType(Γ:=Γ) lit.
- admit.
- Qed.
-*)
-(*
- Definition flatten : forall h c, Rule h c -> @ND Judg Rule (mapOptionTree flattenJudgment h) (mapOptionTree flattenJudgment c).
- intros h c r.
- refine (match r as R in Rule H C return ND Rule (mapOptionTree flattenJudgment H) (mapOptionTree flattenJudgment C) with
- | RURule a b c d e => let case_RURule := tt in _
- | RNote Γ Δ Σ τ l n => let case_RNote := tt in _
- | RLit Γ Δ l _ => let case_RLit := tt in _
- | RVar Γ Δ σ p => let case_RVar := tt in _
- | RGlobal Γ Δ σ l wev => let case_RGlobal := tt in _
- | RLam Γ Δ Σ tx te x => let case_RLam := tt in _
- | RCast Γ Δ Σ σ τ γ x => let case_RCast := tt in _
- | RAbsT Γ Δ Σ κ σ a => let case_RAbsT := tt in _
- | RAppT Γ Δ Σ κ σ τ y => let case_RAppT := tt in _
- | RAppCo Γ Δ Σ κ σ₁ σ₂ γ σ l => let case_RAppCo := tt in _
- | RAbsCo Γ Δ Σ κ σ σ₁ σ₂ y => let case_RAbsCo := tt in _
- | RApp Γ Δ Σ₁ Σ₂ tx te p => let case_RApp := tt in _
- | RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p => let case_RLet := tt in _
- | RBindingGroup Γ p lri m x q => let case_RBindingGroup := tt in _
- | REmptyGroup _ _ => let case_REmptyGroup := tt in _
- | RBrak Σ a b c n m => let case_RBrak := tt in _
- | REsc Σ a b c n m => let case_REsc := tt in _
- | RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _
- | RLetRec Γ Δ lri x y t => let case_RLetRec := tt in _
- 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_RBrak.
- simpl.
+ 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_REsc.
- simpl.
+ 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_RNote.
- eapply nd_rule. simpl. apply RNote; auto.
+ destruct case_RComp.
+ (* ga_comp *)
+ admit.
destruct case_RLit.
- simpl.
-
- set (@RNote Γ Δ Σ τ l) as q.
- Defined.
-
- Definition flatten' {h}{c} (pf:ND Rule h c) := nd_map' flattenJudgment flatten pf.
-
+ (* ga_literal *)
+ admit.
- @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c).
+ (* 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.
- refine (fix flatten : forall Γ Δ Σ τ
- (pf:SCND Rule [] [Γ > Δ > Σ |- τ ]) :
- SCND Rule [] [Γ > Δ > mapOptionTree typeMap Σ |- mapOptionTree typeMap τ ] :=
- match pf as SCND _ _
- | scnd_comp : forall ht ct c , SCND ht ct -> Rule ct [c] -> SCND ht [c]
- | scnd_weak : forall c , SCND c []
- | scnd_leaf : forall ht c , SCND ht [c] -> SCND ht [c]
- | scnd_branch : forall ht c1 c2, SCND ht c1 -> SCND ht c2 -> SCND ht (c1,,c2)
- Expr Γ Δ ξ τ -> Expr Γ (map coMap Δ) (typeMap ○ ξ) (typeMap τ).
-*)
+ destruct case_RLam.
+ (* ga_curry, but try to avoid this someday in the future if the argument type isn't a function *)
+ admit.
-(*
- Lemma all_lemma Γ κ σ l :
-(@typeMap (κ :: Γ)
- (@HaskTApp (κ :: Γ) κ (@weakF Γ κ ★ σ) (@FreshHaskTyVar Γ κ) @@
- @weakL Γ κ l)) = (@typeMap Γ (@HaskTAll Γ κ σ @@ l)).
-*)
+ destruct case_RApp.
+ (* ga_apply *)
+ admit.
-(*
- Definition flatten : forall Γ Δ ξ τ, Expr Γ Δ ξ τ -> Expr Γ (map coMap Δ) (typeMap ○ ξ) (typeMap τ).
- refine (fix flatten Γ' Δ' ξ' τ' (exp:Expr Γ' Δ' ξ' τ') : Expr Γ' (map coMap Δ') (typeMap ○ ξ') (typeMap τ') :=
- match exp as E in Expr G D X T return Expr G (map coMap D) (typeMap ○ X) (typeMap T) with
- | EGlobal Γ Δ ξ t wev => EGlobal _ _ _ _ wev
- | EVar Γ Δ ξ ev => EVar _ _ _ ev
- | ELit Γ Δ ξ lit lev => let case_ELit := tt in _
- | EApp Γ Δ ξ t1 t2 lev e1 e2 => EApp _ _ _ _ _ _ (flatten _ _ _ _ e1) (flatten _ _ _ _ e2)
- | ELam Γ Δ ξ t1 t2 lev v e => let case_ELam := tt in _
- | ELet Γ Δ ξ tv t l ev elet ebody => let case_ELet := tt in _
- | ELetRec Γ Δ ξ lev t tree branches ebody => let case_ELetRec := tt in _
- | ECast Γ Δ ξ t1 t2 γ lev e => let case_ECast := tt in _
- | ENote Γ Δ ξ t n e => ENote _ _ _ _ n (flatten _ _ _ _ e)
- | ETyLam Γ Δ ξ κ σ l e => let case_ETyLam := tt in _
- | ECoLam Γ Δ κ σ σ₁ σ₂ ξ l e => let case_ECoLam := tt in _
- | ECoApp Γ Δ κ σ₁ σ₂ γ σ ξ l e => let case_ECoApp := tt in _
- | ETyApp Γ Δ κ σ τ ξ l e => let case_ETyApp := tt in _
- | ECase Γ Δ ξ l tc tbranches atypes e alts' => let case_ECase := tt in _
-
- | EEsc Γ Δ ξ ec t lev e => let case_EEsc := tt in _
- | EBrak Γ Δ ξ ec t lev e => let case_EBrak := tt in _
- end); clear exp ξ' τ' Γ' Δ'.
-
- destruct case_ELit.
- simpl.
- rewrite lit_lemma.
- apply ELit.
-
- destruct case_ELam.
- set (flatten _ _ _ _ e) as q.
- rewrite update_typeMap in q.
- apply (@ELam _ _ _ _ _ _ _ _ v q).
-
- destruct case_ELet.
- set (flatten _ _ _ _ ebody) as ebody'.
- set (flatten _ _ _ _ elet) as elet'.
- rewrite update_typeMap in ebody'.
- apply (@ELet _ _ _ _ _ _ _ _ _ elet' ebody').
-
- destruct case_EEsc.
- admit.
- destruct case_EBrak.
- admit.
+ destruct case_RLet.
+ (* ga_comp! perhaps this means the ga_curry avoidance can be done by turning lambdas into lets? *)
+ admit.
- destruct case_ECast.
- apply flatten in e.
- eapply ECast in e.
- apply e.
- apply flattenCoercion in γ.
- apply γ.
-
- destruct case_ETyApp.
- apply flatten in e.
- simpl in e.
- unfold HaskTAll in e.
- unfold typeMap_ in e.
- simpl in e.
- eapply ETyApp in e.
- rewrite <- foo in e.
- apply e.
-
- destruct case_ECoLam.
- apply flatten in e.
- simpl in e.
- set (@ECoLam _ _ _ _ _ _ _ _ _ _ e) as x.
- simpl in x.
- simpl.
- unfold typeMap_.
- simpl.
- apply x.
+ destruct case_RVoid.
+ (* ga_id u *)
+ admit.
- destruct case_ECoApp.
- simpl.
- apply flatten in e.
- eapply ECoApp.
- unfold mkHaskCoercionKind in *.
- simpl in γ.
- apply flattenCoercion in γ.
- unfold coMap in γ at 2.
- apply γ.
- apply e.
-
- destruct case_ETyLam.
- apply flatten in e.
- set (@ETyLam Unique _ Γ (map coMap Δ) (typeMap ○ ξ) κ (fun ite x => typeMap (σ x ite))) as e'.
- unfold HaskTAll in *.
- unfold typeMap_ in *.
- rewrite <- foo in e'.
- unfold typeMap in e'.
- simpl in e'.
- apply ETyLam.
-
-Set Printing Implicit.
-idtac.
-idtac.
+ 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.
-
- destruct case_ECase.
admit.
-
- destruct case_ELetRec.
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.
- (* This proof will work for any dynamic semantics you like, so
- * long as those semantics are an ND_Relation (associativity,
- * neutrality, etc) *)
- Context (dynamic_semantics : @ND_Relation _ Rule).
-
- Section SystemFC_Category.
-
- Context {Γ:TypeEnv}
- {Δ:CoercionEnv Γ}.
-
- Definition Context := Tree ??(LeveledHaskType Γ ★).
-
- Notation "a |= b" := (Γ >> Δ > a |- b).
- (*
- SystemFCa
- PCF
- SystemFCa_two_level
- SystemFCa_initial_GArrow
- *)
-
- Context (nd_eqv:@ND_Relation _ (@URule Γ Δ)).
- Check (@ProgrammingLanguage).
- Context (PL:@ProgrammingLanguage (LeveledHaskType Γ ★)
- (fun x y => match x with x1|=x2 => match y with y1|=y2 => @URule Γ Δ)).
- Definition JudgmentsFC := @Judgments_Category_CartesianCat _ (@URule Γ Δ) nd_eqv.
- Definition TypesFC := @TypesL _ (@URule Γ Δ) nd_eqv.
-
- (* The full subcategory of SystemFC(Γ,Δ) consisting only of judgments involving types at a fixed level. Note that
- * code types are still permitted! *)
- Section SingleLevel.
- Context (lev:HaskLevel Γ).
-
- Inductive ContextAtLevel : Context -> Prop :=
- | contextAtLevel_nil : ContextAtLevel []
- | contextAtLevel_leaf : forall τ, ContextAtLevel [τ @@ lev]
- | contextAtLevel_branch : forall b1 b2, ContextAtLevel b1 -> ContextAtLevel b2 -> ContextAtLevel (b1,,b2).
-
- Inductive JudgmentsAtLevel : JudgmentsFC -> Prop :=
- | judgmentsAtLevel_nil : JudgmentsAtLevel []
- | judgmentsAtLevel_leaf : forall c1 c2, ContextAtLevel c1 -> ContextAtLevel c2 -> JudgmentsAtLevel [c1 |= c2]
- | judgmentsAtLevel_branch : forall j1 j2, JudgmentsAtLevel j1 -> JudgmentsAtLevel j2 -> JudgmentsAtLevel (j1,,j2).
-
- Definition JudgmentsFCAtLevel := FullSubcategory JudgmentsFC JudgmentsAtLevel.
- Definition TypesFCAtLevel := FullSubcategory TypesFC ContextAtLevel.
- End SingleLevel.
-
- End SystemFC_Category.
-
- Implicit Arguments TypesFC [ ].
-
-(*
- Section EscBrak_Functor.
- Context
- (past:@Past V)
- (n:V)
- (Σ₁:Tree ??(@LeveledHaskType V)).
-
- Definition EscBrak_Functor_Fobj
- : SystemFC_Cat_Flat ((Σ₁,n)::past) -> SystemFC_Cat past
- := mapOptionTree (fun q:Tree ??(@CoreType V) * Tree ??(@CoreType V) =>
- let (a,s):=q in (Σ₁,,(``a)^^^n,[`<[ n |- encodeTypeTree_flat s ]>])).
-
- Definition append_brak
- : forall {c}, ND_ML
- (mapOptionTree (ob2judgment_flat ((⟨Σ₁,n⟩) :: past)) c )
- (mapOptionTree (ob2judgment past ) (EscBrak_Functor_Fobj c)).
- intros.
- unfold ND_ML.
- unfold EscBrak_Functor_Fobj.
- rewrite mapOptionTree_comp.
- simpl in *.
- apply nd_replicate.
- intro o; destruct o.
- apply nd_rule.
- apply MLRBrak.
- Defined.
-
- Definition prepend_esc
- : forall {h}, ND_ML
- (mapOptionTree (ob2judgment past ) (EscBrak_Functor_Fobj h))
- (mapOptionTree (ob2judgment_flat ((⟨Σ₁,n⟩) :: past)) h ).
- intros.
- unfold ND_ML.
- unfold EscBrak_Functor_Fobj.
- rewrite mapOptionTree_comp.
- simpl in *.
- apply nd_replicate.
- intro o; destruct o.
- apply nd_rule.
- apply MLREsc.
- Defined.
-
- Definition EscBrak_Functor_Fmor
- : forall a b (f:a~~{SystemFC_Cat_Flat ((Σ₁,n)::past)}~~>b),
- (EscBrak_Functor_Fobj a)~~{SystemFC_Cat past}~~>(EscBrak_Functor_Fobj b).
- intros.
- eapply nd_comp.
- apply prepend_esc.
- eapply nd_comp.
- eapply Flat_to_ML.
- apply f.
- apply append_brak.
- Defined.
-
- Lemma esc_then_brak_is_id : forall a,
- ndr_eqv(ND_Relation:=ml_dynamic_semantics V) (nd_comp prepend_esc append_brak)
- (nd_id (mapOptionTree (ob2judgment past) (EscBrak_Functor_Fobj a))).
- admit.
- Qed.
-
- Lemma brak_then_esc_is_id : forall a,
- ndr_eqv(ND_Relation:=ml_dynamic_semantics V) (nd_comp append_brak prepend_esc)
- (nd_id (mapOptionTree (ob2judgment_flat (((Σ₁,n)::past))) a)).
- admit.
- Qed.
-
- Instance EscBrak_Functor
- : Functor (SystemFC_Cat_Flat ((Σ₁,n)::past)) (SystemFC_Cat past) EscBrak_Functor_Fobj :=
- { fmor := fun a b f => EscBrak_Functor_Fmor a b f }.
- intros; unfold EscBrak_Functor_Fmor; simpl in *.
- apply ndr_comp_respects; try reflexivity.
- apply ndr_comp_respects; try reflexivity.
- auto.
- intros; unfold EscBrak_Functor_Fmor; simpl in *.
- set (@ndr_comp_left_identity _ _ (ml_dynamic_semantics V)) as q.
- setoid_rewrite q.
- apply esc_then_brak_is_id.
- intros; unfold EscBrak_Functor_Fmor; simpl in *.
- set (@ndr_comp_associativity _ _ (ml_dynamic_semantics V)) as q.
- repeat setoid_rewrite q.
- apply ndr_comp_respects; try reflexivity.
- apply ndr_comp_respects; try reflexivity.
- repeat setoid_rewrite <- q.
- apply ndr_comp_respects; try reflexivity.
- setoid_rewrite brak_then_esc_is_id.
- clear q.
- set (@ndr_comp_left_identity _ _ (fc_dynamic_semantics V)) as q.
- setoid_rewrite q.
- reflexivity.
- Defined.
-
- End EscBrak_Functor.
-
-
-
- Ltac rule_helper_tactic' :=
- match goal with
- | [ H : ?A = ?A |- _ ] => clear H
- | [ H : [?A] = [] |- _ ] => inversion H; clear H
- | [ H : [] = [?A] |- _ ] => inversion H; clear H
- | [ H : ?A,,?B = [] |- _ ] => inversion H; clear H
- | [ H : ?A,,?B = [?Y] |- _ ] => inversion H; clear H
- | [ H: ?A :: ?B = ?B |- _ ] => apply symmetry in H; apply list_cannot_be_longer_than_itself in H; destruct H
- | [ H: ?B = ?A :: ?B |- _ ] => apply list_cannot_be_longer_than_itself in H; destruct H
- | [ H: ?A :: ?C :: ?B = ?B |- _ ] => apply symmetry in H; apply list_cannot_be_longer_than_itself' in H; destruct H
- | [ H: ?B = ?A :: ?C :: ?B |- _ ] => apply list_cannot_be_longer_than_itself' in H; destruct H
-(* | [ H : Sequent T |- _ ] => destruct H *)
-(* | [ H : ?D = levelize ?C (?A |= ?B) |- _ ] => inversion H; clear H*)
- | [ H : [?A] = [?B] |- _ ] => inversion H; clear H
- | [ H : [] = mapOptionTree ?B ?C |- _ ] => apply mapOptionTree_on_nil in H; subst
- | [ H : [?A] = mapOptionTree ?B ?C |- _ ] => destruct C as [C|]; simpl in H; [ | inversion H ]; destruct C; simpl in H; simpl
- | [ H : ?A,,?B = mapOptionTree ?C ?D |- _ ] => destruct D as [D|] ; [destruct D|idtac]; simpl in H; inversion H
- end.
+ Definition PCF_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
+ refine {| plsmme_pl := PCF n Γ Δ |}.
+ admit.
+ Defined.
- Lemma fixit : forall a b f c1 c2, (@mapOptionTree a b f c1),,(mapOptionTree f c2) = mapOptionTree f (c1,,c2).
+ Definition SystemFCa_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
+ refine {| plsmme_pl := SystemFCa n Γ Δ |}.
admit.
Defined.
- Lemma grak a b f c : @mapOptionTree a b f c = [] -> [] = c.
+ Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n).
admit.
- Qed.
+ Defined.
- Definition append_brak_to_id : forall {c},
- @ND_FC V
- (mapOptionTree (ob2judgment ((⟨Σ₁,n⟩) :: past)) c )
- (mapOptionTree (ob2judgment past) (EscBrak_Functor_Fobj c)).
- 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 }.
+ *)
- Definition append_brak : forall {h c}
- (pf:@ND_FC V
- h
- (mapOptionTree (ob2judgment ((⟨Σ₁,n⟩) :: past)) c )),
- @ND_FC V
- h
- (mapOptionTree (ob2judgment past) (EscBrak_Functor_Fobj c)).
-
- refine (fix append_brak h c nd {struct nd} :=
- ((match nd
- as nd'
- in @ND _ _ H C
- return
- (H=h) ->
- (C=mapOptionTree (ob2judgment ((⟨Σ₁,n⟩) :: past)) c) ->
- ND_FC h (mapOptionTree (ob2judgment past) (EscBrak_Functor_Fobj c))
- with
- | nd_id0 => let case_nd_id0 := tt in _
- | nd_id1 h => let case_nd_id1 := tt in _
- | nd_weak h => let case_nd_weak := tt in _
- | nd_copy h => let case_nd_copy := tt in _
- | nd_prod _ _ _ _ lpf rpf => let case_nd_prod := tt in _
- | nd_comp _ _ _ top bot => let case_nd_comp := tt in _
- | nd_rule _ _ rule => let case_nd_rule := tt in _
- | nd_cancell _ => let case_nd_cancell := tt in _
- | nd_cancelr _ => let case_nd_cancelr := tt in _
- | nd_llecnac _ => let case_nd_llecnac := tt in _
- | nd_rlecnac _ => let case_nd_rlecnac := tt in _
- | nd_assoc _ _ _ => let case_nd_assoc := tt in _
- | nd_cossa _ _ _ => let case_nd_cossa := tt in _
- end) (refl_equal _) (refl_equal _)
- ));
- simpl in *; intros; subst; simpl in *; try (repeat (rule_helper_tactic' || subst)); subst; simpl in *.
- destruct case_nd_id0. apply nd_id0.
- destruct case_nd_id1. apply nd_rule. destruct p. apply RBrak.
- destruct case_nd_weak. apply nd_weak.
-
- destruct case_nd_copy.
- (*
- destruct c; try destruct o; simpl in *; try rule_helper_tactic'; try destruct p; try rule_helper_tactic'.
- inversion H0; subst.
- simpl.*)
- idtac.
- clear H0.
- admit.
-
- destruct case_nd_prod.
- eapply nd_prod.
- apply (append_brak _ _ lpf).
- apply (append_brak _ _ rpf).
-
- destruct case_nd_comp.
- apply append_brak in bot.
- apply (nd_comp top bot).
-
- destruct case_nd_cancell.
- eapply nd_comp; [ apply nd_cancell | idtac ].
- apply append_brak_to_id.
-
- destruct case_nd_cancelr.
- eapply nd_comp; [ apply nd_cancelr | idtac ].
- apply append_brak_to_id.
-
- destruct case_nd_llecnac.
- eapply nd_comp; [ idtac | apply nd_llecnac ].
- apply append_brak_to_id.
-
- destruct case_nd_rlecnac.
- eapply nd_comp; [ idtac | apply nd_rlecnac ].
- apply append_brak_to_id.
-
- destruct case_nd_assoc.
- eapply nd_comp; [ apply nd_assoc | idtac ].
- repeat rewrite fixit.
- apply append_brak_to_id.
-
- destruct case_nd_cossa.
- eapply nd_comp; [ idtac | apply nd_cossa ].
- repeat rewrite fixit.
- apply append_brak_to_id.
-
- destruct case_nd_rule
-
+ (* for every n we have a functor from the category of (n+1)-bounded proofs to the category of n-bounded proofs *)
- Defined.
- Definition append_brak {h c} : forall
- pf:@ND_FC V
- (fixify Γ ((⟨n, Σ₁ ⟩) :: past) h )
- c,
- @ND_FC V
- (fixify Γ past (EscBrak_Functor_Fobj h))
- c.
+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.
*)
-*)
-End HaskProofCategory.
+
+
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