(* information needed to define a case branch in a HaskProof *)
Record ProofCaseBranch {n}{tc:TyCon n}{Γ}{lev}{branchtype : HaskType Γ}{avars} :=
-{ cbi_cbi : @StrongAltConInContext n tc Γ avars
-; cbri_freevars : Tree ??(LeveledHaskType Γ)
-; cbri_judg := cbi_Γ cbi_cbi > cbi_Δ cbi_cbi
- > (mapOptionTree weakLT' cbri_freevars),,(unleaves (vec2list (cbi_types cbi_cbi)))
+{ pcb_scb : @StrongCaseBranch n tc Γ avars
+; pcb_freevars : Tree ??(LeveledHaskType Γ)
+; pcb_judg := scb_Γ pcb_scb > scb_Δ pcb_scb
+ > (mapOptionTree weakLT' pcb_freevars),,(unleaves (vec2list (scb_types pcb_scb)))
|- [weakLT' (branchtype @@ lev)]
}.
+Coercion pcb_scb : ProofCaseBranch >-> StrongCaseBranch.
Implicit Arguments ProofCaseBranch [ ].
(* Figure 3, production $\vdash_E$, Uniform rules *)
| RCase : forall Γ Δ lev n tc Σ avars tbranches
(alts:Tree ??(@ProofCaseBranch n tc Γ lev tbranches avars)),
Rule
- ((mapOptionTree cbri_judg alts),,
+ ((mapOptionTree pcb_judg alts),,
[Γ > Δ > Σ |- [ caseType tc avars @@ lev ] ])
- [Γ > Δ > (mapOptionTreeAndFlatten cbri_freevars alts),,Σ |- [ tbranches @@ lev ] ]
+ [Γ > Δ > (mapOptionTreeAndFlatten pcb_freevars alts),,Σ |- [ tbranches @@ lev ] ]
.
Coercion RURule : URule >-> Rule.
Definition UND_to_ND Γ Δ h c : ND (@URule Γ Δ) h c -> ND Rule (mapOptionTree UJudg2judg h) (mapOptionTree UJudg2judg c)
:= @nd_map' _ (@URule Γ Δ ) _ Rule (@UJudg2judg Γ Δ ) (fun h c r => nd_rule (RURule _ _ h c r)) h c.
+Lemma no_urules_with_empty_conclusion : forall Γ Δ c h, @URule Γ Δ c h -> h=[] -> False.
+ intro.
+ intro.
+ induction 1; intros; inversion H.
+ simpl in *; destruct c; try destruct o; simpl in *; try destruct u; inversion H; simpl in *; apply IHX; auto; inversion H1.
+ simpl in *; destruct c; try destruct o; simpl in *; try destruct u; inversion H; simpl in *; apply IHX; auto; inversion H1.
+ Qed.
+
+Lemma no_rules_with_empty_conclusion : forall c h, @Rule c h -> h=[] -> False.
+ intros.
+ destruct X; try destruct c; try destruct o; simpl in *; try inversion H.
+ apply no_urules_with_empty_conclusion in u.
+ apply u.
+ auto.
+ Qed.
+
+Lemma no_urules_with_multiple_conclusions : forall Γ Δ c h,
+ @URule Γ Δ c h -> { h1:Tree ??(UJudg Γ Δ) & { h2:Tree ??(UJudg Γ Δ) & h=(h1,,h2) }} -> False.
+ intro.
+ intro.
+ induction 1; intros.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+
+ apply IHX.
+ destruct X0. destruct s. destruct c; try destruct o; try destruct u; simpl in *.
+ inversion e.
+ inversion e.
+ exists c1. exists c2. auto.
+
+ apply IHX.
+ destruct X0. destruct s. destruct c; try destruct o; try destruct u; simpl in *.
+ inversion e.
+ inversion e.
+ exists c1. exists c2. auto.
+
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
+ Qed.
+
+Lemma no_rules_with_multiple_conclusions : forall c h,
+ Rule c h -> { h1:Tree ??Judg & { h2:Tree ??Judg & h=(h1,,h2) }} -> False.
+ intros.
+ destruct X; try destruct c; try destruct o; simpl in *; try inversion H;
+ try apply no_urules_with_empty_conclusion in u; try apply u.
+ destruct X0; destruct s; inversion e.
+ auto.
+ apply (no_urules_with_multiple_conclusions _ _ h (c1,,c2)) in u. inversion u. exists c1. exists c2. auto.
+ 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.
+ 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.
+ 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.
+ 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.
+ intros.
+ eapply no_rules_with_multiple_conclusions.
+ apply r.
+ exists c1.
+ exists c2.
+ auto.
+ Qed.
+