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preservation.v
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preservation.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 wf_env.
Import ConcreteEverything.
Section Preservation.
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 .
Theorem preservation_light :
forall gamma H H' L L' t t' sigma eff,
Reduction_SF' ( # H, L, t !)
( # H', L', t' !)
->
TypeChecksP gamma eff t sigma ->
{Gamma' : Gamma_type & TypeChecksP Gamma' eff t' sigma}.
intros.
case_eq t; intros; rewrite H0 in * ; clear H0 t; inversion X;
inversion X0;
exists (p_Γ.updatePartFunc gamma v sigma0); auto;
rewrite H9 in *; auto.
Qed.
Local Open Scope type_scope.
Theorem preservation_case_var :
forall H L x y t sigma ann envVar,
WF_Frame' H (ann_frame (sframe L (t_let x <- Var y t_in (t))) ann) sigma ->
p_env.func L y = Some envVar ->
WF_Frame' H (ann_frame (sframe (p_env.updatePartFunc L x envVar) t) ann) sigma.
intros.
rename H0 into L_y_value.
inversion X.
clear X.
clear H4 t0 H5 H1 H2 L0 ann0 sigma0 H0 H3.
(* rename H6 into let_t_is_term. *)
rename X0 into Gamma_type_let.
rename X1 into wf_env_frame.
inversion Gamma_type_let.
clear Gamma_type_let.
clear gamma H0 x0 H3 eff0 H1 t0 H5 H2 tau H4 e.
rename X into Gamma_type_var_y; rename X0 into Gamma'_type_t.
set (Gamma' := (p_Γ.updatePartFunc Gamma x sigma0)).
fold Gamma' in Gamma'_type_t.
(* split. *)
set (L' := (p_env.updatePartFunc L x envVar)).
apply (t_frame1' H Gamma' eff t
L' ann sigma).
exact Gamma'_type_t.
clear Gamma'_type_t.
split.
destruct wf_env_frame.
apply (subset_preserved _ _ _ _ _ g).
intros.
rename H0 into Gamma'_x.
elim (v_eq_dec x x0).
intro; rewrite <- a in *; clear a.
inversion Gamma_type_var_y.
clear H4 H2 H0 H1 sigma2 gamma eff0 Gamma_type_var_y eff.
rename H3 into Gamma_y_value.
set (wf_var_y := snd wf_env_frame y sigma0 Gamma_y_value).
(* 3.
*)
induction envVar.
{
(*
* 3. (a), L(y) = null
*)
apply inl; apply inl.
apply (proj1 (p_env.updatedFuncProp L x envNull x) (eq_refl x)).
}
(* 3. (b), L(y) = o*)
rename r into o.
rename L_y_value into _3_b_i_A.
induction wf_var_y.
{
induction a. (* case L(y) = null in WF-var *)
{
rewrite a in _3_b_i_A; discriminate.
}
{ (* case L(y) = o', H(o') = <C, FM> in WF-var*)
induction b as [[c r ] [x2 [[Ly_value Gamma_y_value' ]
heap_Γ_y_subtype ]]].
rewrite (_3_b_i_A) in *; inversion Ly_value as [H1].
destruct H1.
rename c into C'.
clear Ly_value.
apply inl. apply inr.
exists (C', o).
exists x2.
split.
{
split.
{
apply p_env.updatedFuncProp; apply eq_refl.
}
{
set (lem := proj1 (p_Γ.updatedFuncProp Gamma x sigma0 x)
(eq_refl x)).
fold Gamma' in lem.
rewrite lem in Gamma'_x; inversion Gamma'_x;
rename H1 into sigma01_eq;
rewrite sigma01_eq in *.
rename sigma1 into sigma'; clear sigma01_eq Gamma'_x.
rewrite Gamma_y_value' in Gamma_y_value; inversion Gamma_y_value;
rename H1 into C'_sigma; rewrite <- C'_sigma in *.
clear C'_sigma sigma'. assumption.
}
}
assumption.
}
}
destruct b;
destruct x2;
destruct y0; destruct p as [[L_y_value Gamma_y_value'] heap_sub_c].
rewrite (_3_b_i_A) in *; inversion L_y_value.
(* 3 c, L(y) = b(o)*)
rename L_y_value into _3_c.
rename r into o.
destruct wf_env_frame as [_3_c_i_A _3_c_i_B].
rename Gamma_y_value into _1_b.
rename sigma0 into sigma'.
simpl in wf_var_y.
rename wf_var_y into _3_c_ii.
induction (_3_c_ii).
induction a. (* 3. c. iii. *)
rewrite a in _3_c; discriminate. (* 3. c. iv.*)
induction b; destruct x2;
destruct p as [r_in_H [[L_y_value Gamma_y_value] gamma_heap_y_r_c] ].
rename L_y_value into _3_c_v.
rewrite _3_c_v in _3_c; discriminate. (* 3. c. v. *)
destruct b; destruct x2. destruct y0. destruct p as
[[L_y_value Gamma_y_value] C'_sub_c].
set (C' := (heap_typeof H r x2)); fold C' in C'_sub_c.
rewrite _3_c in L_y_value; inversion L_y_value.
rename x0 into z.
rename sigma1 into tau.
rename H1 into refs_o_r_equal.
(* intro. *)
(* 3 box *)
destruct refs_o_r_equal.
(* rewrite H5 in *. clear H5 o. rename r into o. *)
clear L_y_value.
unfold WF_Var.
rewrite Gamma'_x.
apply inr.
exists (c, o).
exists x2.
split.
simpl. split.
exact (proj1 (p_env.updatedFuncProp L x (envBox o) x) (eq_refl _)).
simpl in Gamma'_x.
rewrite _1_b in Gamma_y_value.
inversion Gamma_y_value.
rename H1 into sigma'_typt_box_c.
symmetry in sigma'_typt_box_c.
destruct sigma'_typt_box_c.
set (lem := (proj1 (p_Γ.updatedFuncProp Gamma x (typt_box c) x) (eq_refl _))).
fold Gamma' in lem.
simpl in lem.
rewrite lem in Gamma'_x.
inversion Gamma'_x.
rename H1 into typt_box_c_tau.
f_equal.
exact C'_sub_c.
clear Gamma_type_var_y eff ann.
(* 4 *)
rename Gamma'_x into Gamma'z.
intro x_neq_x0.
rename x0 into z.
rename sigma1 into tau.
rename sigma0 into sigma'.
assert (p_Γ.func Gamma z = Some tau) as _4_a.
rewrite <- Gamma'z.
symmetry.
apply p_Γ.updatedFuncProp. firstorder.
destruct wf_env_frame as [ _4_c_i _4_c_ii].
set (_4_d := _4_c_ii z tau _4_a).
unfold WF_Var.
rewrite Gamma'z.
Require Import Coq.Lists.List.
assert (In z (p_Γ.domain Gamma)) as _4_ca.
set (lem := p_Γ.fDomainCompat Gamma z).
set (in_or_not := in_dec v_eq_dec z (p_Γ.domain Gamma)).
firstorder; rename H0 into Gamma_z_None;
rewrite Gamma_z_None in _4_a; discriminate.
set (lem''' := _4_c_i z _4_ca).
case_eq (p_env.func L z).
intros envVar' _4_ca_ii.
assert (p_env.func L' z = Some envVar') as _4_cb.
transitivity (p_env.func L z).
apply p_env.updatedFuncProp. firstorder.
exact _4_ca_ii.
unfold WF_Var in _4_d.
rewrite _4_cb in *.
rewrite _4_ca_ii in _4_d.
rewrite _4_a in _4_d.
exact _4_d.
intro L_z_value.
set (lem'''':= proj2 (p_env.fDomainCompat L z ) L_z_value).
firstorder.
Qed.
Theorem preservation_case_null :
forall H L t sigma ann x ,
WF_Frame' H (ann_frame (sframe L t_let x <- Null t_in (t)) ann) sigma ->
WF_Frame' H (ann_frame (sframe (p_env.updatePartFunc L x envNull) t ) ann) sigma.
intros.
rename X into asm1.
inversion asm1.
clear sigma0 ann0 H3 H5 t0 H4 H1 L0 H0 H2.
rename X0 into _3_b.
rename X into _3_a.
inversion _3_a.
clear H2 tau t0 H5 e H4 x0 H3 eff0 H1 H0 gamma.
rename sigma0 into sigma'.
rename X into _4_a.
rename X0 into _4_b.
destruct _3_b as [_6_a _6_b].
apply (t_frame1' H (p_Γ.updatePartFunc Gamma x sigma' )
eff t
(p_env.updatePartFunc L x envNull) ann sigma).
exact _4_b.
unfold WF_Env.
split.
exact (subset_preserved Gamma L x sigma' envNull _6_a).
intros z tau _8.
elim (v_eq_dec x z).
intro x_is_z; rewrite <- x_is_z in *; clear x_is_z.
(* assert (p_Γ.func (p_Γ.updatePartFunc Gamma x sigma') x = Some sigma') as _10_a. *)
(* exact (proj1 (p_Γ.updatedFuncProp Gamma x sigma' x) (eq_refl x)). *)
(* rewrite -> _10_a in _8; inversion _8. rewrite <- H1 in *; clear H1 tau _8. *)
(* unfold WF_Var. *)
apply inl.
apply inl.
exact (proj1 (p_env.updatedFuncProp L x envNull x) (eq_refl x)).
intro x_neq_z.
assert (p_Γ.func Gamma z = Some tau) as new_4_a.
rewrite <- _8.
symmetry.
apply (proj2 (p_Γ.updatedFuncProp Gamma x sigma' z)).
firstorder.
(* set (new__4_d := _6_b z tau _4_a). *)
unfold WF_Var.
(* rewrite H0. *)
(* Require Import Coq.Lists.List. *)
assert (In z (p_Γ.domain Gamma)) as new_4_ca.
set (lem := p_Γ.fDomainCompat Gamma z).
set (in_or_not := in_dec v_eq_dec z (p_Γ.domain Gamma)).
firstorder. rewrite H0 in new_4_a; discriminate.
set (lem''' := _6_a z new_4_ca).
case_eq (p_env.func L z).
intros envVar new__4_ca_ii.
assert (p_env.func (p_env.updatePartFunc L x envNull ) z = Some envVar) as new_4_cb.
transitivity (p_env.func L z).
apply p_env.updatedFuncProp. firstorder.
exact new__4_ca_ii.
set (new_4d := _6_b z tau new_4_a).
unfold WF_Var in new_4d.
rewrite new_4_cb in *.
rewrite new__4_ca_ii in *.
rewrite new_4_a in *.
rewrite _8.
exact new_4d.
intro.
set (lem'''':= proj2 (p_env.fDomainCompat L z ) H0).
firstorder.
Qed.
Notation "[ L , t ] ^ a" := (ann_frame (sframe L t) a) (at level 0).
Notation "( Γ ⊍ x ↦ σ )" := (p_Γ.updatePartFunc
Γ x σ)
(at level 0).
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).
Theorem preservation_case_new :
forall H L sigma ann t C x o flds,
WF_Frame' H (ann_frame (sframe L t_let x <- New C t_in (t)) ann) sigma ->
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 (envRef o)) t) ann) sigma.
intros H L σ ann t C x o flds wf_F wf_H o_not_in_H C_fields.
set (F := [L, t_let x <- New C t_in (t)] ^ ann).
fold F in wf_F.
(* 2 *)
inversion_clear wf_F as [dummy1 Γ eff dummy2 dummy3 dummy4
dummy5 t_typ_σ wf_H_Γ_L ].
unfold wf_env.TypeChecksP in t_typ_σ.
(* _2_a := t_typ_sigma
* _2_b := wf_H_Γ_L
*)
(* 3 *)
inversion_clear wf_H_Γ_L as [ΓL_sub wf_vars].
(* ΓL_sub := ΓL_sub
* wf_vars := wf_vars
*)
(* 4 *)
inversion_clear t_typ_σ.
rename sigma into C_type; rename X into typ_newC;
rename X0 into typ_next_frame.
(* 5 *)
inversion typ_newC.
{ (* ocap C *)
rewrite <- H2 in *; rewrite <- H4 in *.
clear H1 H0 C0 H2 H4 H2 eff gamma.
rename H3 into C_is_ocap.
(* section between 5, 6 let's call it half_6 *)
(* This is not in the same order on paper *)
(* split. *)
apply ( t_frame1' _
(p_Γ.updatePartFunc Γ x (typt_class C))
eff_ocap _ _ ann σ typ_next_frame).
assert (gamma_env_subset (Γ ⊍ x ↦ typt_class C)
(L +++ x ↦ envRef o)) as gamma_L_subset.
exact (subset_preserved Γ L x (typt_class C)
envNull ΓL_sub). (* actually, 7 *)
split.
{ (* Γ ⊆ L *) exact gamma_L_subset. }
{ (* wf-var z *)
intros z τ.
case_eq (v_eq_dec x z).
{ (* x = z *)
clear wf_vars typ_newC.
intro x_is_z; rewrite <- x_is_z in *; clear x_is_z z.
intro dummy; clear dummy.
(* Section half 6 on paper here somewhere *)
intro half_6.
set (half_6_a_i := proj1
(p_env.updatedFuncProp L x (envRef o) x)
(eq_refl _)).
assert (In o
(p_heap.domain
(H +*+ o ↦ obj C (p_FM.newPartFunc flds FM_null))
))
as o_witn.
apply p_heap.updatedFuncIn.
set (H'o := proj1
(p_heap.updatedFuncProp
H o (obj C (p_FM.newPartFunc flds FM_null)) o)
(eq_refl _)).
assert ((heap_typeof
(p_heap.updatePartFunc H o
(obj C (p_FM.newPartFunc flds FM_null)))
o o_witn) = C) as half_6_a_ii_first.
{ apply (heap_typeof_same _ _ _ _ H'o). }
assert (subtypeP ( typt_class
(heap_typeof
(H +*+ o ↦ obj C (p_FM.newPartFunc flds FM_null))
o o_witn)) (typt_class C)) as half_6_a_ii.
{
rewrite half_6_a_ii_first.
unfold subtypeP.
apply classSub.
apply subclass_refl. (* admitted (actually not defined) *)
}
set (lem := proj1 (p_Γ.updatedFuncProp
Γ x (typt_class C) x) (eq_refl _)).
rewrite half_6 in lem.
inversion lem.
rewrite H1 in *; clear lem H1 τ.
rename half_6 into half_6_a_iii.
apply inl. apply inr. exists (C, o).
exists o_witn.
split.
{
split.
{ exact half_6_a_i. }
{ exact half_6_a_iii. }
}
{ exact half_6_a_ii. }
}
{ (* x != z *)
(* 6 [THIS starts like #4 from E-Null]*)
intros _6_c dummy _6_a.
clear dummy.
assert (p_Γ.func Γ z = Some τ) as e_vi.
rewrite <- _6_a.
symmetry.
apply (proj2 (p_Γ.updatedFuncProp Γ x (typt_class C) z)).
firstorder.
(* set (new__4_d := _6_b z tau typ_newC). *)
(* unfold WF_Var. *)
(* rewrite H0. *)
assert (In z (p_Γ.domain Γ)) as z_in_dom_Gamma.
set (lem := p_Γ.fDomainCompat Γ z).
set (in_or_not := in_dec v_eq_dec z (p_Γ.domain Γ)).
firstorder. rewrite H0 in e_vi; discriminate.
set (z_in_dom_L := ΓL_sub z z_in_dom_Gamma).
case_eq (p_env.func L z).
{
intros envVar L_z_val.
assert (p_env.func
(p_env.updatePartFunc L x (envRef o) )
z = Some envVar) as _6_bb.
{
transitivity (p_env.func L z).
apply p_env.updatedFuncProp. firstorder.
exact L_z_val.
}
destruct envVar.
{
(* 6 d *)
apply inl. apply inl.
exact _6_bb.
}
(* 6 e *)
{ (* L z = o *)
rename r into o'.
assert ({ C : _ &
{witn : _ &
prod (prod (p_env.func L z = Some (envRef o'))
(p_Γ.func Γ z = Some (typt_class C)))
(subtypeP
(typt_class (heap_typeof H o' witn))
(typt_class C))
}}).
{
elim (wf_vars z τ e_vi).
intro.
destruct a.
{ (* L z = null *)
rewrite -> e in L_z_val; discriminate.
}
{ (* L z = o *)
destruct s.
destruct x0.
destruct y.
destruct p.
destruct p.
rewrite L_z_val in e.
inversion e.
rewrite H1 in *; clear H1 o'; rename r into o'.
clear e.
rewrite e_vi in e0.
inversion e0. rewrite -> H1 in *.
clear τ H1.
exists c. exists x0.
split.
{
split.
{ exact L_z_val. }
{ exact e_vi. }
}
{ exact s. }
}
intro.
destruct b. destruct x0. destruct y. destruct p. destruct p.
exists c.
rewrite L_z_val in e.
inversion e.
}
destruct X. destruct s. destruct p. destruct p.
(* clear H0. *)
rename x0 into C'.
rename x1 into o'_witness.
apply inl. apply inr.
exists (C', o').
exists (p_heap.staysInDomain
H o
(obj C (p_FM.newPartFunc flds FM_null)) o' o'_witness).
split.
{
split.
{ exact _6_bb. }
{
rewrite <- e0.
apply (p_Γ.updatedFuncProp). firstorder.
}
}
{ (* L z = o *)
set (H' := (p_heap.updatePartFunc
H o (obj C (p_FM.newPartFunc flds FM_null)))).
apply classSub.
inversion s.
assert (p_heap.func H o' = p_heap.func H' o') as _6_e_ii.
{ apply (p_heap.freshProp _ _ _ o_not_in_H _ o'_witness). }
assert ((heap_typeof
H' o'
(p_heap.staysInDomain
H o (obj C (p_FM.newPartFunc flds FM_null)) o'
o'_witness))
= (heap_typeof H o' o'_witness))
as _6_e_iii.
{
case_eq (p_heap.func H o').
{
intros. destruct b.
{
set (stays_witn := (p_heap.staysInDomain
H o
(obj C
(p_FM.newPartFunc flds FM_null)) o'
o'_witness)).
fold stays_witn.
rewrite <- H0 in *.
rewrite _6_e_ii in H3.
assert (c = C0).
{
symmetry in H0.
set (lem := heap_typeof_impl _ _ _ _ H0).
destruct lem.
rewrite H3 in _6_e_ii.
rewrite e1 in _6_e_ii.
inversion _6_e_ii.
reflexivity.
}
rewrite <- H4.
apply (heap_typeof_same _ _ _ _ H3).
}
}
(* None *)
{
intro is_none;
induction (proj2 (p_heap.fDomainCompat _ _)
is_none o'_witness).
}
}
rewrite _6_e_iii.
exact H2.
}
}
{ (* L z = b(o) *)
rename r into o'.
assert ({ C : _ &
{witn : _ &
prod (prod
(p_env.func L z = Some (envBox o'))
(p_Γ.func Γ z = Some
(typt_box C)))
(subtypeP
(typt_class (heap_typeof H o' witn))
(typt_class C))
}}).
{
elim (wf_vars z τ e_vi).
intro.
destruct a.
{ (* L z = null *)
rewrite -> e in L_z_val; discriminate.
}
{ (* L z = o *)
destruct s.
destruct x0.
destruct y.
destruct p.
destruct p.
rewrite L_z_val in e.
inversion e.
}
{
intro.
destruct b. destruct x0. destruct y. destruct p. destruct p.
rewrite L_z_val in e.
inversion e.
rewrite H1 in *; clear H1 o'; rename r into o'.
clear e.
rewrite e_vi in e0.
inversion e0. rewrite -> H1 in *.
clear τ H1.
exists c. exists x0.
split.
{
split.
{ exact L_z_val. }
{ exact e_vi. }
}
{ exact s. }
}
}
destruct X. destruct s. destruct p. destruct p.
(* clear H0. *)
rename x0 into C'.
rename x1 into o'_witness.
apply inr.
exists (C', o').
exists (p_heap.staysInDomain
H o
(obj C (p_FM.newPartFunc flds FM_null)) o' o'_witness).
split.
{
split.
{ exact _6_bb. }
{
rewrite <- e0.
apply (p_Γ.updatedFuncProp). firstorder.
}
}
{ (* L z = o *)
set (H' := (p_heap.updatePartFunc
H o (obj C (p_FM.newPartFunc flds FM_null)))).
apply classSub.
inversion s.
assert (p_heap.func H o' = p_heap.func H' o') as _6_e_ii.
{ apply (p_heap.freshProp _ _ _ o_not_in_H _ o'_witness). }
assert ((heap_typeof
H' o'
(p_heap.staysInDomain
H o (obj C (p_FM.newPartFunc flds FM_null)) o'
o'_witness))
= (heap_typeof H o' o'_witness))
as _6_e_iii.
{
case_eq (p_heap.func H o').
{
intros. destruct b.
{
set (stays_witn := (p_heap.staysInDomain
H o
(obj C
(p_FM.newPartFunc flds FM_null)) o'
o'_witness)).
fold stays_witn.
rewrite <- H0 in *.
rewrite _6_e_ii in H3.
assert (c = C0).
{
symmetry in H0.
set (lem := heap_typeof_impl _ _ _ _ H0).
destruct lem.
rewrite H3 in _6_e_ii.
rewrite e1 in _6_e_ii.
inversion _6_e_ii.
reflexivity.
}
rewrite <- H4.
apply (heap_typeof_same _ _ _ _ H3).
}
}
(* None *)
{
intro is_none;
induction (proj2 (p_heap.fDomainCompat _ _)
is_none o'_witness).
}
}
rewrite _6_e_iii.
exact H2.
}
}
}
{
intro L_z_is_none;
induction (proj2 (p_env.fDomainCompat _ _) L_z_is_none).
auto.
}
}
}
}
{ (* ocap C *)
rewrite <- H1 in *. rewrite <- H2 in *.
clear H1 H0 C0 H3 H2 eff gamma.
(* section between 5, 6 let's call it half_6 *)
(* This is not in the same order on paper *)
(* split. *)
apply ( t_frame1' _
(p_Γ.updatePartFunc Γ x (typt_class C))
eff_epsilon _ _ ann σ typ_next_frame).
assert (gamma_env_subset (Γ ⊍ x ↦ typt_class C)
(L +++ x ↦ envRef o)) as gamma_L_subset.
exact (subset_preserved Γ L x (typt_class C)
envNull ΓL_sub). (* actually, 7 *)
split.
{ (* Γ ⊆ L *) exact gamma_L_subset. }
{ (* wf-var z *)
intros z τ.
case_eq (v_eq_dec x z).
{ (* x = z *)
clear wf_vars typ_newC.
intro x_is_z; rewrite <- x_is_z in *; clear x_is_z z.
intro dummy; clear dummy.
(* Section half 6 on paper here somewhere *)
intro half_6.
set (half_6_a_i := proj1
(p_env.updatedFuncProp L x (envRef o) x)
(eq_refl _)).
assert (In o
(p_heap.domain
(H +*+ o ↦ obj C (p_FM.newPartFunc flds FM_null))
))
as o_witn.
apply p_heap.updatedFuncIn.
set (H'o := proj1
(p_heap.updatedFuncProp
H o (obj C (p_FM.newPartFunc flds FM_null)) o)
(eq_refl _)).
assert ((heap_typeof
(p_heap.updatePartFunc H o
(obj C (p_FM.newPartFunc flds FM_null)))
o o_witn) = C) as half_6_a_ii_first.
{ apply (heap_typeof_same _ _ _ _ H'o). }
assert (subtypeP ( typt_class
(heap_typeof
(H +*+ o ↦ obj C (p_FM.newPartFunc flds FM_null))
o o_witn)) (typt_class C)) as half_6_a_ii.
{
rewrite half_6_a_ii_first.
unfold subtypeP.
apply classSub.
apply subclass_refl. (* admitted (actually not defined) *)
}
set (lem := proj1 (p_Γ.updatedFuncProp
Γ x (typt_class C) x) (eq_refl _)).
rewrite half_6 in lem.
inversion lem.
rewrite H1 in *; clear lem H1 τ.
rename half_6 into half_6_a_iii.
apply inl. apply inr. exists (C, o).
exists o_witn.
split.
{
split.
{ exact half_6_a_i. }
{ exact half_6_a_iii. }
}
{ exact half_6_a_ii. }
}
{ (* x != z *)
(* 6 [THIS starts like #4 from E-Null]*)
intros _6_c dummy _6_a.
clear dummy.
assert (p_Γ.func Γ z = Some τ) as e_vi.
rewrite <- _6_a.
symmetry.
apply (proj2 (p_Γ.updatedFuncProp Γ x (typt_class C) z)).
firstorder.
(* set (new__4_d := _6_b z tau typ_newC). *)
(* unfold WF_Var. *)
(* rewrite H0. *)
assert (In z (p_Γ.domain Γ)) as z_in_dom_Gamma.
set (lem := p_Γ.fDomainCompat Γ z).
set (in_or_not := in_dec v_eq_dec z (p_Γ.domain Γ)).
firstorder. rewrite H0 in e_vi; discriminate.
set (z_in_dom_L := ΓL_sub z z_in_dom_Gamma).
case_eq (p_env.func L z).
{
intros envVar L_z_val.
assert (p_env.func
(p_env.updatePartFunc L x (envRef o) )
z = Some envVar) as _6_bb.
{
transitivity (p_env.func L z).
apply p_env.updatedFuncProp. firstorder.
exact L_z_val.
}
destruct envVar.
{
(* 6 d *)
apply inl. apply inl.
exact _6_bb.
}
(* 6 e *)
{ (* L z = o *)
rename r into o'.
assert ({ C : _ &
{witn : _ &
prod (prod (p_env.func L z = Some (envRef o'))
(p_Γ.func Γ z = Some (typt_class C)))
(subtypeP
(typt_class (heap_typeof H o' witn))
(typt_class C))
}}).
{
elim (wf_vars z τ e_vi).
intro.
destruct a.
{ (* L z = null *)
rewrite -> e in L_z_val; discriminate.
}
{ (* L z = o *)
destruct s.
destruct x0.
destruct y.
destruct p.
destruct p.
rewrite L_z_val in e.
inversion e.
rewrite H1 in *; clear H1 o'; rename r into o'.
clear e.
rewrite e_vi in e0.
inversion e0. rewrite -> H1 in *.
clear τ H1.
exists c. exists x0.
split.
{
split.
{ exact L_z_val. }
{ exact e_vi. }
}
{ exact s. }
}
intro.
destruct b. destruct x0. destruct y. destruct p. destruct p.
exists c.
rewrite L_z_val in e.
inversion e.
}
destruct X. destruct s. destruct p. destruct p.
(* clear H0. *)
rename x0 into C'.
rename x1 into o'_witness.
apply inl. apply inr.
exists (C', o').
exists (p_heap.staysInDomain
H o
(obj C (p_FM.newPartFunc flds FM_null)) o' o'_witness).
split.
{
split.
{ exact _6_bb. }
{
rewrite <- e0.
apply (p_Γ.updatedFuncProp). firstorder.
}
}
{ (* L z = o *)
set (H' := (p_heap.updatePartFunc
H o (obj C (p_FM.newPartFunc flds FM_null)))).
apply classSub.
inversion s.
assert (p_heap.func H o' = p_heap.func H' o') as _6_e_ii.
{ apply (p_heap.freshProp _ _ _ o_not_in_H _ o'_witness). }
assert ((heap_typeof
H' o'
(p_heap.staysInDomain
H o (obj C (p_FM.newPartFunc flds FM_null)) o'
o'_witness))
= (heap_typeof H o' o'_witness))
as _6_e_iii.
{
case_eq (p_heap.func H o').
{
intros. destruct b.
{
set (stays_witn := (p_heap.staysInDomain
H o
(obj C
(p_FM.newPartFunc flds FM_null)) o'
o'_witness)).
fold stays_witn.
rewrite <- H0 in *.
rewrite _6_e_ii in H3.
assert (c = C0).
{
symmetry in H0.
set (lem := heap_typeof_impl _ _ _ _ H0).
destruct lem.
rewrite H3 in _6_e_ii.
rewrite e1 in _6_e_ii.
inversion _6_e_ii.
reflexivity.
}
rewrite <- H4.
apply (heap_typeof_same _ _ _ _ H3).
}
}
(* None *)
{
intro is_none;
induction (proj2 (p_heap.fDomainCompat _ _)
is_none o'_witness).
}
}
rewrite _6_e_iii.
exact H2.
}
}
{ (* L z = b(o) *)
rename r into o'.
assert ({ C : _ &
{witn : _ &
prod (prod
(p_env.func L z = Some (envBox o'))
(p_Γ.func Γ z = Some
(typt_box C)))
(subtypeP
(typt_class (heap_typeof H o' witn))
(typt_class C))
}}).
{
elim (wf_vars z τ e_vi).
intro.
destruct a.
{ (* L z = null *)
rewrite -> e in L_z_val; discriminate.
}
{ (* L z = o *)
destruct s.
destruct x0.