Notation "Γ > Δ > a '|-' s '@' l" := (mkJudg Γ Δ a s l) (at level 52, Δ at level 50, a at level 52, s at level 50, l at level 50).
(* information needed to define a case branch in a HaskProof *)
-Record ProofCaseBranch {tc:TyCon}{Γ}{Δ}{lev}{branchtype : HaskType Γ ★}{avars}{sac:@StrongAltCon tc} :=
-{ pcb_freevars : Tree ??(LeveledHaskType Γ ★)
-; pcb_judg := sac_gamma sac Γ > sac_delta sac Γ avars (map weakCK' Δ)
+Definition pcb_judg
+ {tc:TyCon}{Γ}{Δ}{lev}{branchtype : HaskType Γ ★}{avars}{sac:@StrongAltCon tc}
+ (pcb_freevars : Tree ??(LeveledHaskType Γ ★)) :=
+ sac_gamma sac Γ > sac_delta sac Γ avars (map weakCK' Δ)
> (mapOptionTree weakLT' pcb_freevars),,(unleaves (map (fun t => t@@weakL' lev)
(vec2list (sac_types sac Γ avars))))
- |- [weakT' branchtype ] @ weakL' lev
-}.
-Implicit Arguments ProofCaseBranch [ ].
+ |- [weakT' branchtype ] @ weakL' lev.
(* Figure 3, production $\vdash_E$, all rules *)
Inductive Rule : Tree ??Judg -> Tree ??Judg -> Type :=
(* order is important here; we want to be able to skolemize without introducing new AExch'es *)
| RApp : ∀ Γ Δ Σ₁ Σ₂ tx te l, Rule ([Γ>Δ> Σ₁ |- [tx--->te]@l],,[Γ>Δ> Σ₂ |- [tx]@l]) [Γ>Δ> Σ₁,,Σ₂ |- [te]@l]
-| RLet : ∀ Γ Δ Σ₁ Σ₂ σ₁ σ₂ l, Rule ([Γ>Δ> Σ₁ |- [σ₁]@l],,[Γ>Δ> [σ₁@@l],,Σ₂ |- [σ₂]@l ]) [Γ>Δ> Σ₁,,Σ₂ |- [σ₂ ]@l]
-| RWhere : ∀ Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ l, Rule ([Γ>Δ> Σ₁,,([σ₁@@l],,Σ₃) |- [σ₂]@l ],,[Γ>Δ> Σ₂ |- [σ₁]@l]) [Γ>Δ> Σ₁,,(Σ₂,,Σ₃) |- [σ₂ ]@l]
-
| RCut : ∀ Γ Δ Σ Σ₁ Σ₁₂ Σ₂ Σ₃ l, Rule ([Γ>Δ> Σ₁ |- Σ₁₂ @l],,[Γ>Δ> Σ,,((Σ₁₂@@@l),,Σ₂) |- Σ₃@l ]) [Γ>Δ> Σ,,(Σ₁,,Σ₂) |- Σ₃@l]
| RLeft : ∀ Γ Δ Σ₁ Σ₂ Σ l, Rule [Γ>Δ> Σ₁ |- Σ₂ @l] [Γ>Δ> (Σ@@@l),,Σ₁ |- Σ,,Σ₂@l]
| RRight : ∀ Γ Δ Σ₁ Σ₂ Σ l, Rule [Γ>Δ> Σ₁ |- Σ₂ @l] [Γ>Δ> Σ₁,,(Σ@@@l) |- Σ₂,,Σ@l]
Rule [Γ > ((σ₁∼∼∼σ₂)::Δ) > Σ |- [σ ]@l]
[Γ > Δ > Σ |- [σ₁∼∼σ₂⇒ σ ]@l]
-| RLetRec : forall Γ Δ Σ₁ τ₁ τ₂ lev, Rule [Γ > Δ > Σ₁,,(τ₂@@@lev) |- (τ₂,,[τ₁]) @lev ] [Γ > Δ > Σ₁ |- [τ₁] @lev]
+| RLetRec : forall Γ Δ Σ₁ τ₁ τ₂ lev, Rule [Γ > Δ > (τ₂@@@lev),,Σ₁ |- (τ₂,,[τ₁]) @lev ] [Γ > Δ > Σ₁ |- [τ₁] @lev]
| RCase : forall Γ Δ lev tc Σ avars tbranches
- (alts:Tree ??{ sac : @StrongAltCon tc & @ProofCaseBranch tc Γ Δ lev tbranches avars sac }),
+ (alts:Tree ??( (@StrongAltCon tc) * (Tree ??(LeveledHaskType Γ ★)) )),
Rule
- ((mapOptionTree (fun x => pcb_judg (projT2 x)) alts),,
+ ((mapOptionTree (fun x => @pcb_judg tc Γ Δ lev tbranches avars (fst x) (snd x)) alts),,
[Γ > Δ > Σ |- [ caseType tc avars ] @lev])
- [Γ > Δ > (mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x)) alts),,Σ |- [ tbranches ] @ lev]
+ [Γ > Δ > (mapOptionTreeAndFlatten (fun x => (snd x)) alts),,Σ |- [ tbranches ] @ lev]
.
Definition RCut' : ∀ Γ Δ Σ₁ Σ₁₂ Σ₂ Σ₃ l,
apply AuCanL.
Defined.
+Definition RLet : ∀ Γ Δ Σ₁ Σ₂ σ₁ σ₂ l,
+ ND Rule ([Γ>Δ> Σ₁ |- [σ₁]@l],,[Γ>Δ> [σ₁@@l],,Σ₂ |- [σ₂]@l ]) [Γ>Δ> Σ₁,,Σ₂ |- [σ₂ ]@l].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RCut ].
+ apply nd_prod.
+ apply nd_id.
+ eapply nd_rule; eapply RArrange; eapply AuCanL.
+ Defined.
+
+Definition RWhere : ∀ Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ l,
+ ND Rule ([Γ>Δ> Σ₁,,([σ₁@@l],,Σ₃) |- [σ₂]@l ],,[Γ>Δ> Σ₂ |- [σ₁]@l]) [Γ>Δ> Σ₁,,(Σ₂,,Σ₃) |- [σ₂ ]@l].
+ intros.
+ eapply nd_comp; [ apply nd_exch | idtac ].
+ eapply nd_rule; eapply RCut.
+ Defined.
+
(* A rule is considered "flat" if it is neither RBrak nor REsc *)
(* TODO: change this to (if RBrak/REsc -> False) *)
Inductive Rule_Flat : forall {h}{c}, Rule h c -> Prop :=
| Flat_RAppCo : ∀ Γ Δ Σ σ₁ σ₂ σ γ q l, Rule_Flat (RAppCo Γ Δ Σ σ₁ σ₂ σ γ q l)
| Flat_RAbsCo : ∀ Γ Σ κ σ σ₁ σ₂ q1 q2 , Rule_Flat (RAbsCo Γ Σ κ σ σ₁ σ₂ q1 q2 )
| Flat_RApp : ∀ Γ Δ Σ tx te p l, Rule_Flat (RApp Γ Δ Σ tx te p l)
-| Flat_RLet : ∀ Γ Δ Σ σ₁ σ₂ p l, Rule_Flat (RLet Γ Δ Σ σ₁ σ₂ p l)
| Flat_RVoid : ∀ q a l, Rule_Flat (RVoid q a l)
| Flat_RCase : ∀ Σ Γ T κlen κ θ l x , Rule_Flat (RCase Σ Γ T κlen κ θ l x)
| Flat_RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂ lev, Rule_Flat (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
destruct X0; destruct s; inversion e.
destruct X0; destruct s; inversion e.
destruct X0; destruct s; inversion e.
- destruct X0; destruct s; inversion e.
- destruct X0; destruct s; inversion e.
Qed.
Lemma systemfc_all_rules_one_conclusion : forall h c1 c2 (r:Rule h (c1,,c2)), False.