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preservation_unfixed.v
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preservation_unfixed.v
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Require Import syntax.
Require Import partial.
Require Import heap.
Require Import classTable.
Require Import sframe.
Require Import reductions.
Require Import typing.
Require Import namesAndTypes.
Require Import preservation.
Require Import wf_env.
Import ConcreteEverything.
Section Preservation_fix.
Variable P: Program.
Definition subtypeP := subtype P.
Definition fldP := fld P.
Definition ftypeP := ftype P.
Definition t_frame1' := t_frame1 P.
Definition WF_Frame' := WF_Frame P.
Definition Reduction_SF' := Reduction_SF P .
Definition fieldsP := fields P.
Definition TypeChecksP := TypeChecksTerm P.
Definition TypeChecksExprP := TypeChecksExpr P.
Definition Heap_okP := Heap_ok P.
Definition Heap_dom_okP := Heap_dom_ok P .
Require Import Coq.Lists.List.
Notation "( p +++ a ↦ b )" := (p_env.updatePartFunc
p a b) (at level 0).
Notation "( H +*+ o ↦ obj )" := (p_heap.updatePartFunc
H o obj)
(at level 0).
Notation "( Γ ⊍ x ↦ σ )" := (p_Γ.updatePartFunc
Γ x σ)
(at level 0).
Notation " a ⪳ b " := (subtype P a b) (at level 0).
Notation " a ⪯ b " := (subclass P a b) (at level 0).
Notation "[ L , t ] ^ a" := (ann_frame (sframe L t) a) (at level 0).
Notation "y ∘ f ⟵ z" := (FieldAssignment y f z) (at level 0).
Notation "a ⟿ b" := (Reduction_SF' a b) (at level 0).
Notation " ⊢ H" := (Heap_okP H) (at level 0).
Notation " ⊩ H" := (Heap_dom_okP H) (at level 0).
Notation "( Γ , ef ⊩' t ▷ σ )" := (TypeChecksTerm P Γ ef t σ) (at level 0).
Notation "( Γ , ef ⊩'' e ▷ σ )" := (TypeChecksExpr P Γ ef e σ) (at level 0).
Notation "a ∉ lst" := (~ (In a lst)) (at level 0).
Theorem preservation_case_box :
forall H L σ ann t C x o flds,
WF_Frame' H (ann_frame (sframe L t_let x <- Box C t_in (t)) ann) σ ->
Heap_okP H ->
~ In o (p_heap.domain H) ->
fieldsP C flds ->
WF_Frame'
(p_heap.updatePartFunc H o (obj C (p_FM.newPartFunc flds FM_null)))
(ann_frame (sframe (p_env.updatePartFunc L x (envBox o)) t) ann) σ.
Admitted.
Theorem preservation_case_select :
forall H L σ ann x y f t C FM fmVal o,
WF_Frame' H (ann_frame (sframe L t_let x <- FieldSelection y f t_in (t)) ann) σ ->
Heap_okP H ->
p_env.func L y = Some (envRef o) ->
p_heap.func H o = Some (obj C FM) ->
p_FM.func FM f = Some fmVal ->
WF_Frame'
H
(ann_frame (sframe (p_env.updatePartFunc L x (fm2env fmVal)) t) ann) σ.
intros.
(* 2 *)
rename H1 into t_typ_σb.
rename H2 into _2_cd.
rename H3 into _2_e.
rename H4 into t_is_term.
rename H0 into _2_j.
rename X into _2_k.
(* 3 *)
inversion _2_k.
clear H3 sigma0 H5 ann0 H4 H1 H2.
rename X into ΓL_sub.
rename X0 into wf_vars.
(* 4 *)
inversion ΓL_sub.
clear tau H3 H7 t1 H5 e H4 x0 eff0 H2.
rewrite <- H1 in * ; clear H1 Gamma.
inversion X.
clear H5 f0 H1 x0 eff0 H3 gamma0 H2.
rename C0 into D.
rename H7 into typ_newC.
rename witn into typ_next_frame.
set (E := ftype P D f typ_next_frame).
rewrite <- H4 in *.
fold E in X, X0.
clear H4.
(* rename H4 into _4_c. *)
rename X0 into _4_d.
rename X into _4_e.
set (L' := p_env.updatePartFunc L x (fm2env fmVal)).
clear t0 L0 H0.
(* 5 *)
set (Gamma' := p_gamma.updatePartFunc gamma x (typt_class E)).
fold Gamma' in _4_d.
(* 6 *)
inversion wf_vars.
rename H0 into wf_vars_i.
rename X into wf_vars_ii.
set (_6 := subset_preserved gamma L x (typt_class E) (fm2env fmVal) wf_vars_i).
(* OTHER ORDER COMPARED TO paper proof*)
clear H6 sigma0.
apply (t_frame1 P _ Gamma' eff _ _ _ _ t_is_term _4_d).
split.
apply _6.
intros z tau.
(* END OTHER *)
case_eq (v_eq_dec x z).
(* 7 *)
(* 7 a *)
intros.
rewrite <- e in *; clear e z.
clear H0.
assert (p_env.func L' x = Some (fm2env fmVal)) as gamma_L_subset_a.
unfold L'.
rewrite ( proj1 (p_env.updatedFuncProp L x (fm2env fmVal) x) (eq_refl x)).
reflexivity.
assert (p_gamma.func Gamma' x = Some (typt_class E)) as gamma_x_E.
unfold Gamma'.
rewrite ( proj1 (p_gamma.updatedFuncProp gamma x (typt_class E) x) (eq_refl x)).
reflexivity.
clear H1.
(* 7 b *)
induction fmVal; unfold fm2env in gamma_L_subset_a.
(* 7 b i *)
apply inl. apply inl.
exact gamma_L_subset_a.
(* 7 c *)
rename r into o'.
set (test := _2_j o C FM _2_cd f o').
apply inl. apply inr.
exists (E, o').
unfold Gamma'.
assert (subclass P C D) as C_sub_D.
induction ( wf_vars_ii y (typt_class D) typ_newC).
induction a.
rewrite t_typ_σb in a; discriminate a.
destruct b.
destruct x0.
destruct y0.
destruct a.
destruct H1.
rewrite t_typ_σb in H0.
inversion H0.
rewrite H4 in *.
rewrite typ_newC in H1.
inversion H1.
rewrite <- H5 in *.
clear c H5.
assert (heap_typeof H r x0 = C).
unfold heap_typeof.
(* TODO: rewriting *)
admit.
(* rewrite <- H4. *)
(* rewrite _2_cd. *)
(* reflexivity. *)
rewrite H3 in H2.
inversion H2.
exact H7.
destruct b. destruct x0.
destruct y0.
destruct a.
rewrite t_typ_σb in H0.
inversion H0.
assert (fld P C f) as f_field_C.
apply (field_subclass P C D _ typ_next_frame).
exact C_sub_D.
destruct (test f_field_C _2_e).
clear test.
exists x0.
split.
exact gamma_L_subset_a.
split.
exact gamma_x_E.
(* *)
(*
* know: Gamma' x = E
* typeof(H, o') <: ftype(C, f)
* --------------
* ftype(C, f) = E
* ---------------------
* Goal: heapof(H, o') <: E
*
*)
apply classSub.
assert (E = heap.ftypeP P C f f_field_C).
unfold E.
unfold heap.ftypeP.
rewrite (unique_ftype P C f f_field_C (field_subclass P C D f f_field_C C_sub_D)).
rewrite <- (ftype_subclass P C D f f_field_C C_sub_D).
apply (unique_ftype P D f _ _).
rewrite <- H0 in s.
exact s.
(* 8 *)
intros.
clear H0.
(* 8 a b *)
assert (p_gamma.func gamma z = Some tau).
rewrite <- H1.
symmetry.
set (lem := proj2 (p_gamma.updatedFuncProp gamma x (typt_class E) z)).
firstorder.
rename H0 into _8_b.
assert (In z (p_gamma.domain gamma)) as _8_a.
apply (p_gamma.in_part_func_domain _ z tau _8_b).
(* 8 c *)
assert (In z (p_env.domain L)) as _8_c.
apply (wf_vars_i _ _8_a).
set (_8_e_i := wf_vars_ii z tau _8_b).
unfold WF_Var.
assert (p_env.func L z = p_env.func L' z) as _8_d.
set (lem := proj2 (p_env.updatedFuncProp L x (fm2env fmVal) z)).
symmetry. firstorder.
rewrite <- _8_d.
rewrite <- _8_b in H1.
rewrite H1.
exact _8_e_i.
Admitted.
Theorem preservation_case_assign H L x y f z t C FM o σ envVal ann:
forall witn: is_not_box envVal,
WF_Frame' H (ann_frame (sframe L t_let x <- (FieldAssignment y f z) t_in (t)) ann) σ ->
Heap_okP H ->
p_env.func L y = Some (envRef o) ->
p_heap.func H o = Some (obj C FM) ->
p_env.func L z = Some envVal ->
(* isTerm t -> *)
(WF_Frame'
(p_heap.updatePartFunc H o
(obj C (p_FM.updatePartFunc FM f (env2fm envVal witn))))
(ann_frame (sframe (p_env.updatePartFunc L x envVal ) t) ann) sigma) *
(Heap_okP (p_heap.updatePartFunc H o
(obj C (p_FM.updatePartFunc FM f (env2fm envVal witn))))
).
(* 1 *)
intros.
rename H0 into _1_b.
rename X into _1_c.
(* 2 *)
set (L' := (p_env.updatePartFunc L x envVal)).
set (FM' := (p_FM.updatePartFunc FM f (env2fm envVal witn))).
set (H' := (p_heap.updatePartFunc H o (obj C FM'))).
split.
(* 2b, L z = o_z or L_z = null*)
assert ({o_z | envVal = envRef o_z} + (envVal = envNull)) as wf_H_Γ_L.
case_eq envVal.
intros.
apply inr.
reflexivity.
intros.
apply inl.
exists r.
reflexivity.
intros.
rewrite H0 in *.
inversion _1_c.
inversion X.
inversion X1.
(* L z = b(r), show Gamma z must be Box[D] *)
inversion X0.
set (lem := X4 z _ H23).
inversion lem.
destruct X5.
rewrite e0 in H3.
simplify_eq H3.
destruct s.
destruct x2.
destruct y1.
destruct a.
rewrite H26 in H3.
simplify_eq H3.
destruct X5.
destruct x2.
destruct y1.
destruct a.
destruct H27.
rewrite H23 in H27 .
simplify_eq H27.
rename o into o_y.
rename H1 into _2_c.
rename H2 into _2_d.
(* 2 e says that f in fields(C) by reduction rules (this is wrong!)*)
(* 3 *)
inversion _1_c.
clear H0 H2 t0 H5 ann0 H6 sigma0 H4 L0 H1.
rename X into ΓL_sub.
rename X0 into wf_vars.
(* 4 *)
inversion ΓL_sub.
inversion X.
clear y0 H13 f0 H9 x1 H8 eff1 H11 gamma0 H10.
unfold typing.subtypeP in *.
fold TypeChecksP in *.
fold subtypeP in *.
unfold typing.fldP in *.
fold fldP in *.
clear H2 tau.
clear x0 H4.
clear t0 H6.
clear H1 eff0.
clear e H5.
clear gamma H0.
rewrite <- H12 in *.
clear sigma0 H12.
inversion X1.
clear eff0 H2 x0 H0 H5 f0 gamma H1.
set (typ_newC := p_gamma.in_part_func_domain Gamma y (typt_class C1) H4).
set (typ_next_frame := p_gamma.in_part_func_domain Gamma z (typt_class C0) H14).
rename X1 into _4_c.
(* unfold subtypeP in H15. *)
rename H15 into _4_d.
set (Gamma' := p_gamma.updatePartFunc Gamma x (typt_class C0)).
fold Gamma' in X0.
rename X0 into _4_e.
clear witn2.
rename C0 into D''.
rename C into C'.
rename C1 into C.
rename H6 into _4_f.
(* 5 *)
inversion wf_vars.
rename H0 into _5_a.
rename X0 into _5_b.
(* 6 *)
apply (t_frame1 _ _ Gamma' eff).
inversion H7.
exact H1.
exact _4_e.
split.
exact (subset_preserved _ _ _ _ _ _5_a).
intros s.
case_eq (v_eq_dec s x).
intros.
clear H0; rewrite -> e in *; clear e s.
clear sigma0 H1.
assert (p_gamma.func Gamma' x = p_gamma.func Gamma z ) as _6_a.
set (lem := proj1 (p_gamma.updatedFuncProp Gamma x (typt_class D'') _) (eq_refl _)).
fold Gamma' in lem.
transitivity (Some (typt_class D'')).
exact lem.
symmetry.
exact H14.
assert (p_env.func L' x = p_env.func L z) as _6_b.
transitivity (Some envVal).
exact (proj1 (p_env.updatedFuncProp L x envVal _) (eq_refl _)).
symmetry.
exact H3.
case_eq wf_H_Γ_L.
intros.
clear H0 wf_H_Γ_L.
destruct s.
rewrite -> e in *.
rename x0 into o_z.
assert (In o_z (p_heap.domain H)).
destruct (_5_b z (typt_class D'') H14).
destruct s.
rewrite e0 in H3.
inversion H3.
destruct s. destruct x0.
destruct y0.
destruct a.
rewrite H0 in H3.
inversion H3.
rewrite <- H5 in *.
exact x0.
destruct s.
destruct x0.
destruct y0.
destruct a.
rewrite H0 in H3; inversion H3.
set (stays := p_heap.staysInDomain H o_y (obj C' FM') o_z H0).
assert (heap_typeof H' o_z stays = heap_typeof H o_z H0) as _6_iv.
case_eq (rn_eq_dec o_z o_y).
intros.
clear H1; rewrite e0 in *.
transitivity C'.
unfold heap_typeof.
assert (p_heap.func H' o_z = Some (obj C' FM')).
rewrite e0.
unfold H'.
exact (proj1 (p_heap.updatedFuncProp H o_y (obj C' FM') _) (eq_refl _)).
(* TODO: rewrite *)
admit.
(* rewrite H1. *)
(* reflexivity. *)
symmetry.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite e0. *)
(* rewrite _2_d. *)
(* reflexivity. *)
intros.
clear H1.
set (lem := proj2 (p_heap.updatedFuncProp H o_y (obj C' FM') _) n).
fold H' in lem.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite lem. *)
(* reflexivity. *)
unfold WF_Var.
apply inl.
apply inr.
rewrite _6_b.
rewrite _6_a.
exists (D'', o_z).
exists stays.
split.
exact H3.
split.
exact H14.
rewrite _6_iv.
unfold wf_env.subtypeP.
(* apply classSub. *)
elim (_5_b z (typt_class D'') H14).
intro.
destruct a.
rewrite e0 in H3; inversion H3.
destruct s.
destruct x0.
destruct y0.
destruct a.
rewrite H1 in H3; inversion H3.
rewrite H6 in *.
destruct H2.
rewrite H14 in H2; inversion H2.
rewrite <- H9 in *.
rewrite <- H6 in *.
(* dependent rewrite H6 in x0. *)
assert ((heap_typeof H r x0) = (heap_typeof H o_z H0)).
case_eq (p_heap.func H r).
intros.
destruct b.
transitivity c0.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite H8. *)
(* reflexivity. *)
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite <- H6. *)
(* rewrite H8. *)
(* reflexivity. *)
intros.
elim (proj2 (p_heap.fDomainCompat H r) H8 x0).
rewrite <- H8.
exact H5.
rename _6_iv into _6_d_iv.
(* 6 e *)
intros.
destruct b.
destruct x0.
destruct y0.
destruct a.
destruct H2.
rewrite H14 in H2; discriminate H2.
(* L z = null *)
intros.
clear H0.
rewrite e in *.
apply inl.
apply inl.
rewrite _6_b.
assumption.
(* 7 a *)
intros gamma_L_subset dummy tau gamma_L_subset'; clear dummy.
assert (p_gamma.func Gamma' s = p_gamma.func Gamma s) as gamma_L_subset_a.
apply (proj2 (p_gamma.updatedFuncProp _ _ _ _) gamma_L_subset).
assert (p_env.func L' s = p_env.func L s) as gamma_L_subset_b.
apply (proj2 (p_env.updatedFuncProp _ _ _ _) gamma_L_subset).
rewrite gamma_L_subset' in gamma_L_subset_a.
symmetry in gamma_L_subset_a.
set (gamma_L_subset_c := _5_b s tau gamma_L_subset_a).
case_eq (p_env.func L s).
intros.
case_eq b.
intros.
apply inl. apply inl.
rewrite H1 in H0.
rewrite H0 in gamma_L_subset_b.
assumption.
(* 7 e, b = envRef o_s *)
clear _1_c.
clear ann.
intros.
rename r into o_s.
rewrite H1 in *.
destruct gamma_L_subset_c. (* case analysis on WF-var Gamma L s *)
destruct s0.
rewrite H0 in e; discriminate e.
destruct s0.
destruct x0; destruct y0; destruct a; destruct H5.
rewrite H2 in H0; inversion H0.
rewrite <- H9 in *; clear H9.
clear o_s; rename r into o_s.
clear H0 b H1.
rename H5 into gamma_L_subset_e_i.
rename c into G.
rename H6 into gamma_L_subset_e_ii.
set (stays := p_heap.staysInDomain H o_y (obj C' FM') o_s x0).
fold H' in stays.
(* 7 e iii *)
assert (heap_typeof H o_s x0 = heap_typeof H' o_s stays) as gamma_L_subset_e_iii.
case_eq (rn_eq_dec o_s o_y).
intros.
clear H0. rewrite <- e in *.
transitivity C'.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite _2_d; reflexivity. *)
unfold heap_typeof.
assert (p_heap.func H' o_y = Some (obj C' FM')).
unfold H'.
exact (proj1 (p_heap.updatedFuncProp H o_y (obj C' FM') _) (eq_refl _)).
(* TODO: rewrite *)
admit.
(* rewrite e. *)
(* rewrite H0. *)
(* reflexivity. *)
intros.
clear H0.
set (lem := proj2 (p_heap.updatedFuncProp H o_y (obj C' FM') _) n).
fold H' in lem.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite lem. *)
(* reflexivity. *)
(* end 7 e iii *)
set (lem := _5_b s tau gamma_L_subset_a).
unfold WF_Var.
rewrite gamma_L_subset_b.
rewrite gamma_L_subset'.
(* Case analysis L s = null / o / box with H |- Gamma L s
* leads to Gamma s = G, typeof(H, o_s) <: G
*)
destruct lem.
destruct s0.
apply inl. apply inl. assumption.
apply inl.
apply inr.
destruct s0.
destruct x1.
exists (c, r).
destruct y0.
destruct a.
rewrite H0 in H2; inversion H2.
rewrite H6 in *.
assert (In o_s (p_heap.domain H')).
assumption.
exists H5.
split.
assumption.
split.
destruct H1.
rewrite H1 in gamma_L_subset_a.
symmetry.
assumption.
assert ((heap_typeof H r x1) = (heap_typeof H' o_s H5)).
transitivity (heap_typeof H o_s x0).
case_eq (p_heap.func H r).
intros.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite H8. *)
(* rewrite H6 in H8. *)
(* rewrite H8. *)
(* reflexivity. *)
intros.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite H8. *)
(* rewrite H6 in H8. *)
(* rewrite H8. *)
(* assumption. *)
case_eq (rn_eq_dec o_s o_y).
intros.
clear H8.
transitivity C'.
unfold heap_typeof.
rewrite e in *.
(* TODO: rewrite *)
admit.
(* rewrite _2_d. reflexivity. *)
unfold heap_typeof.
assert (p_heap.func H' o_y = Some (obj C' FM')).
unfold H'.
exact (proj1 (p_heap.updatedFuncProp H o_y (obj C' FM') _) (eq_refl _)).
(* TODO: rewrite *)
admit.
(* rewrite e. *)
(* rewrite H8. *)
(* reflexivity. *)
intros.
clear H8.
set (lem := proj2 (p_heap.updatedFuncProp H o_y (obj C' FM') _) n).
fold H' in lem.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite lem. *)
(* reflexivity. *)
rewrite <- H8.
destruct H1.
exact H9.
destruct s0.
destruct x1.
destruct y0.
destruct a.
destruct H1.
rewrite H0 in H2; inversion H2.
destruct s0.
destruct x0; destruct y0. destruct a. destruct H5.
rewrite H2 in H0; discriminate H0.
(* END 7 e!!! *)
intros.
rename r into o_s.
rewrite H1 in *.
(* Case analysis on H |- Gamma ; L ; s to get box prop: *)
set (lem := _5_b s tau gamma_L_subset_a).
destruct lem.
destruct s0.
rewrite e in H0; discriminate H0.
destruct s0; destruct x0; destruct y0; destruct a.
rewrite H2 in H0; discriminate H0.
destruct s0; destruct x0; destruct y0; destruct a; destruct H5.
rewrite H2 in H0; simplify_eq H0.
intro equRef; rewrite equRef in *.
clear H0.
rewrite gamma_L_subset_a in H5; simplify_eq H5.
intro equTy; rewrite equTy in *.
rename c into F.
rename gamma_L_subset' into gamma_L_subset_f_i_A.
assert (In o_s (p_heap.domain H)).
rewrite <- equRef.
assumption.
assert (In o_s (p_heap.domain H')).
unfold H'.
apply (p_heap.staysInDomain H _ _ o_s).
assumption.
assert (heap_typeof H o_s H0 = heap_typeof H' o_s H8) as gamma_L_subset_f_ii.
case_eq (rn_eq_dec o_s o_y).
intros.
rewrite e in *.
clear H9.
transitivity C'.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite e. *)
(* rewrite _2_d. *)
(* reflexivity. *)
unfold heap_typeof.
assert (p_heap.func H' o_y = Some (obj C' FM')).
apply (p_heap.updatedFuncProp _ _ _ _).
reflexivity.
(* TODO: rewrite *)
admit.
(* rewrite e. *)
(* rewrite H9. *)
(* reflexivity. *)
intros.
clear H9.
case_eq (p_heap.func H o_s).
intros.
destruct b0.
transitivity c.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite H9. *)
(* reflexivity. *)
assert (p_heap.func H' o_s = Some (obj c f0)).
unfold H'.
transitivity (p_heap.func H o_s).
apply (proj2 (p_heap.updatedFuncProp H o_y (obj C' FM') o_s) ).
assumption.
assumption.
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite H10. *)
(* reflexivity. *)
intros.
elim (proj2 (p_heap.fDomainCompat H o_s) H9 H0).
unfold WF_Var.
rewrite gamma_L_subset_b.
rewrite gamma_L_subset_f_i_A.
rewrite H2.
apply inr.
exists (F, o_s).
exists H8.
split.
reflexivity.
split.
reflexivity.
rewrite <- gamma_L_subset_f_ii.
assert ((heap_typeof H o_s H0) = (heap_typeof H r x0)
).
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite equRef. *)
(* induction (p_heap.func H o_s); *)
(* reflexivity. *)
rewrite H9.
assumption.
intro.
set (s_in_gamma := p_gamma.in_part_func_domain Gamma s tau gamma_L_subset_a).
set (lem := _5_a _ s_in_gamma).
elim (proj2 (p_env.fDomainCompat _ _) H0 lem).
(* HEAP OK *)
intro.
rename o into o_y.
rename o0 into o.
case_eq (rn_eq_dec o_y o).
intros; clear H0.
rewrite <- e in *.
set (lem := proj1 (p_heap.updatedFuncProp H o_y (obj C FM') _) (eq_refl _)).
fold H' in lem.
rewrite lem in H4.
inversion H4.
rewrite <- H5 in *.
rewrite <- H6 in *.
clear H5 H6 C0 FM0.
clear lem H4 e o.
intro; intros.
case_eq (fn_eq_dec f0 f).
intros.
clear H4; rewrite e in *.
set (lem := proj1 (p_FM.updatedFuncProp FM f (env2fm envVal witn) f) (eq_refl _ )).
fold FM' in lem.
rewrite H0 in lem.
inversion lem.
induction envVal;
inversion H5.
rewrite <- H6 in *.
rename o into o_z.
rename H3 into _9_b_ii_B.
rename H0 into _9_b_ii_A.
inversion _1_c.
destruct X0.
inversion X.
inversion X0.
set (_9_b_iii_lem := w z (typt_class C0) H23).
unfold WF_Var in _9_b_iii_lem.
assert ({C_o : class * Ref_type |
let (C, o) := C_o in
{witn : In o (p_heap.domain H) |
p_env.func L z = Some (envRef o) /\
p_gamma.func Gamma z = Some (typt_class C) /\
wf_env.subtypeP P (typt_class (heap_typeof H o witn))
(typt_class C)}}).
induction _9_b_iii_lem.
induction a.
rewrite a in _9_b_ii_B; simplify_eq _9_b_ii_B.
exact b.
destruct b; destruct x2; destruct y1; destruct a.
rewrite H25 in _9_b_ii_B; simplify_eq _9_b_ii_B.
clear _9_b_iii_lem.
destruct X3; destruct x2; destruct y1; destruct a.
rewrite H25 in _9_b_ii_B; simplify_eq _9_b_ii_B.
intro ref_eq; rewrite ref_eq in *.
destruct H26.
rename H26 into _9_b_v_C.
rename H27 into _9_b_v_D.
rename H25 into _9_b_v_A.
assert (In o_z (p_heap.domain H)) as _9_b_v_B.
rewrite <- ref_eq; assumption.
rewrite _9_b_v_C in H23; simplify_eq H23; intro simpl_eq; rewrite simpl_eq in *.
inversion X2.
set (lemlem := w y (typt_class C1) H28).
assert ({C_o : class * Ref_type |
let (C, o) := C_o in
{witn : In o (p_heap.domain H) |
p_env.func L y = Some (envRef o) /\
p_gamma.func Gamma y = Some (typt_class C) /\
wf_env.subtypeP P (typt_class (heap_typeof H o witn))
(typt_class C)}}).
induction lemlem.
induction a.
rewrite a in H1; simplify_eq H1.
assumption.
destruct b; destruct x4; destruct y1; destruct a; destruct H32.
rewrite H32 in H28; simplify_eq H28.
destruct X3.
destruct x4.
destruct y1.
destruct a.
destruct H32.
rewrite H31 in H1; simplify_eq H1; intro equu; rewrite equu in *.
rewrite H28 in H32; simplify_eq H32; intro equuu; rewrite <- equuu in *.
assert (heap_typeof H r1 x4 = C).
unfold heap_typeof.
(* TODO: rewrite *)
admit.
(* rewrite equu. *)
(* rewrite H2. *)
(* reflexivity. *)
rewrite H34 in H33.
assert ((heap.ftypeP P C f f_witn) = D).
rewrite <- H30.
symmetry.
inversion H33.
apply (ftype_subclass _ _ _ _ _ H37).
rewrite H35.
exists (p_heap.staysInDomain H o_y (obj C FM') o_z _9_b_v_B).
assert ((heap_typeof H' o_z
(p_heap.staysInDomain H o_y (obj C FM') o_z _9_b_v_B))
= (heap_typeof H o_z _9_b_v_B)
).
unfold heap_typeof.
case_eq (rn_eq_dec o_z o_y).
intros.
(* TODO: rewrite *)
admit.
(* rewrite e1. *)
(* assert (p_heap.func H' o_y = Some (obj C FM')). *)
(* apply (p_heap.updatedFuncProp). *)
(* reflexivity. *)
(* rewrite H37. *)
(* rewrite H2. *)
(* reflexivity. *)
intros.
assert (p_heap.func H' o_z = p_heap.func H o_z).
apply (p_heap.updatedFuncProp).
assumption.
(* TODO: rewrite *)
admit.
(* rewrite H37. *)
(* induction (p_heap.func H o_z). *)
(* reflexivity. *)
Admitted.
(* reflexivity. *)
(* rewrite H36. *)
(* assert ((heap_typeof H o_z _9_b_v_B) = (heap_typeof H r0 x2)). *)
(* unfold heap_typeof. *)
(* rewrite ref_eq. *)
(* induction (p_heap.func H o_z). *)
(* reflexivity. *)
(* reflexivity. *)
(* rewrite H37. *)
(* inversion _9_b_v_D. *)
(* apply (subclass_trans P (heap_typeof H r0 x2) C0 D). *)
(* assumption. *)
(* inversion H24. *)
(* assumption. *)
(* inversion witn. *)
(* intros. *)
(* clear H4. *)
(* assert (p_FM.func FM' f0 = p_FM.func FM f0). *)
(* apply (p_FM.updatedFuncProp). *)
(* assumption. *)
(* rewrite H0 in H4. symmetry in H4. *)
(* set (lem := _1_b o_y C FM H2 f0 o f_witn H4). *)
(* destruct lem. *)
(* exists (p_heap.staysInDomain H o_y (obj C FM') o x0). *)
(* assert ((heap_typeof H' o (p_heap.staysInDomain H o_y (obj C FM') o x0)) *)
(* = (heap_typeof H o x0) *)
(* ). *)
(* case_eq (rn_eq_dec o o_y). *)
(* intros. *)
(* clear H5. *)