Coq.IntMap.Mapiter

# Module Coq.IntMap.Mapiter

``Require` Bool.
`Require` Sumbool.
`Require` ZArith.
`Require` Addr.
`Require` Adist.
`Require` Addec.
`Require` Map.
`Require` Mapaxioms.
`Require` Fset.
`Require` PolyList.

`Section` MapIter.

`Variable` A : Set.

`Section` MapSweepDef.

`Variable` f:ad->A->bool.

`Definition` MapSweep2 := [a0:ad; y:A] if (f a0 y) then (SOME ? (a0, y)) else (NONE ?).

`Fixpoint` MapSweep1 [pf:ad->ad; m:(Map A)] : (option (ad * A)) :=
Cases m of
M0 => (NONE ?)
| (M1 a y) => (MapSweep2 (pf a) y)
| (M2 m m') => Cases (MapSweep1 ([a:ad] (pf (ad_double a))) m) of
(SOME r) => (SOME ? r)
| NONE => (MapSweep1 ([a:ad] (pf (ad_double_plus_un a))) m')
end
end.

`Definition` MapSweep := [m:(Map A)] (MapSweep1 ([a:ad] a) m).

`Lemma` MapSweep_semantics_1_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A)
(MapSweep1 pf m)=(SOME ? (a, y)) -> (f a y)=true.
`Proof`.
Induction m. Intros. Discriminate H.
Simpl. Intros a y pf a0 y0. Elim (sumbool_of_bool (f (pf a) y)). Intro H. Unfold MapSweep2.
Rewrite H. Intro H0. Inversion H0. Rewrite <- H3. Assumption.
Intro H. Unfold MapSweep2. Rewrite H. Intro H0. Discriminate H0.
Simpl. Intros. Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)).
Intro H2. Elim H2. Intros r H3. Rewrite H3 in H1. Inversion H1. Rewrite H5 in H3.
Exact (H [a0:ad](pf (ad_double a0)) a y H3).
Intro H2. Rewrite H2 in H1. Exact (H0 [a0:ad](pf (ad_double_plus_un a0)) a y H1).
`Qed`.

`Lemma` MapSweep_semantics_1 : (m:(Map A)) (a:ad) (y:A)
(MapSweep m)=(SOME ? (a, y)) -> (f a y)=true.
`Proof`.
Intros. Exact (MapSweep_semantics_1_1 m [a:ad]a a y H).
`Qed`.

`Lemma` MapSweep_semantics_2_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A)
(MapSweep1 pf m)=(SOME ? (a, y)) -> {a':ad | a=(pf a')}.
`Proof`.
Induction m. Intros. Discriminate H.
Simpl. Unfold MapSweep2. Intros a y pf a0 y0. Case (f (pf a) y). Intros. Split with a.
Inversion H. Reflexivity.
Intro. Discriminate H.
Intros m0 H m1 H0 pf a y. Simpl.
Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). Intro H1. Elim H1.
Intros r H2. Rewrite H2. Intro H3. Inversion H3. Rewrite H5 in H2.
Elim (H [a0:ad](pf (ad_double a0)) a y H2). Intros a0 H6. Split with (ad_double a0).
Assumption.
Intro H1. Rewrite H1. Intro H2. Elim (H0 [a0:ad](pf (ad_double_plus_un a0)) a y H2).
Intros a0 H3. Split with (ad_double_plus_un a0). Assumption.
`Qed`.

`Lemma` MapSweep_semantics_2_2 : (m:(Map A))
(pf,fp:ad->ad) ((a0:ad) (fp (pf a0))=a0) -> (a:ad) (y:A)
(MapSweep1 pf m)=(SOME ? (a, y)) -> (MapGet A m (fp a))=(SOME ? y).
`Proof`.
Induction m. Intros. Discriminate H0.
Simpl. Intros a y pf fp H a0 y0. Unfold MapSweep2. Elim (sumbool_of_bool (f (pf a) y)).
Intro H0. Rewrite H0. Intro H1. Inversion H1. Rewrite (H a). Rewrite (ad_eq_correct a).
Reflexivity.
Intro H0. Rewrite H0. Intro H1. Discriminate H1.
Intros. Rewrite (MapGet_M2_bit_0_if A m0 m1 (fp a)). Elim (sumbool_of_bool (ad_bit_0 (fp a))).
Intro H3. Rewrite H3. Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)).
Intro H4. Simpl in H2. Apply (H0 [a0:ad](pf (ad_double_plus_un a0)) [a0:ad](ad_div_2 (fp a0))).
Intro. Rewrite H1. Apply ad_double_plus_un_div_2.
Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). Intro H5. Elim H5.
Intros r H6. Rewrite H6 in H2. Inversion H2. Rewrite H8 in H6.
Elim (MapSweep_semantics_2_1 m0 [a0:ad](pf (ad_double a0)) a y H6). Intros a0 H9.
Rewrite H9 in H3. Rewrite (H1 (ad_double a0)) in H3. Rewrite (ad_double_bit_0 a0) in H3.
Discriminate H3.
Intro H5. Rewrite H5 in H2. Assumption.
Intro H4. Simpl in H2. Rewrite H4 in H2.
Apply (H0 [a0:ad](pf (ad_double_plus_un a0)) [a0:ad](ad_div_2 (fp a0))). Intro.
Rewrite H1. Apply ad_double_plus_un_div_2.
Assumption.
Intro H3. Rewrite H3. Simpl in H2.
Elim (option_sum ad*A (MapSweep1 [a0:ad](pf (ad_double a0)) m0)). Intro H4. Elim H4.
Intros r H5. Rewrite H5 in H2. Inversion H2. Rewrite H7 in H5.
Apply (H [a0:ad](pf (ad_double a0)) [a0:ad](ad_div_2 (fp a0))). Intro. Rewrite H1.
Apply ad_double_div_2.
Assumption.
Intro H4. Rewrite H4 in H2.
Elim (MapSweep_semantics_2_1 m1 [a0:ad](pf (ad_double_plus_un a0)) a y H2).
Intros a0 H5. Rewrite H5 in H3. Rewrite (H1 (ad_double_plus_un a0)) in H3.
Rewrite (ad_double_plus_un_bit_0 a0) in H3. Discriminate H3.
`Qed`.

`Lemma` MapSweep_semantics_2 : (m:(Map A)) (a:ad) (y:A)
(MapSweep m)=(SOME ? (a, y)) -> (MapGet A m a)=(SOME ? y).
`Proof`.
Intros.
Exact (MapSweep_semantics_2_2 m [a0:ad]a0 [a0:ad]a0 [a0:ad](refl_equal ad a0) a y H).
`Qed`.

`Lemma` MapSweep_semantics_3_1 : (m:(Map A)) (pf:ad->ad)
(MapSweep1 pf m)=(NONE ?) ->
(a:ad) (y:A) (MapGet A m a)=(SOME ? y) -> (f (pf a) y)=false.
`Proof`.
Induction m. Intros. Discriminate H0.
Simpl. Unfold MapSweep2. Intros a y pf. Elim (sumbool_of_bool (f (pf a) y)). Intro H.
Rewrite H. Intro. Discriminate H0.
Intro H. Rewrite H. Intros H0 a0 y0. Elim (sumbool_of_bool (ad_eq a a0)). Intro H1. Rewrite H1.
Intro H2. Inversion H2. Rewrite <- H4. Rewrite <- (ad_eq_complete ? ? H1). Assumption.
Intro H1. Rewrite H1. Intro. Discriminate H2.
Intros. Simpl in H1. Elim (option_sum ad*A (MapSweep1 [a:ad](pf (ad_double a)) m0)).
Intro H3. Elim H3. Intros r H4. Rewrite H4 in H1. Discriminate H1.
Intro H3. Rewrite H3 in H1. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H4.
Rewrite (MapGet_M2_bit_0_1 A a H4 m0 m1) in H2. Rewrite <- (ad_div_2_double_plus_un a H4).
Exact (H0 [a:ad](pf (ad_double_plus_un a)) H1 (ad_div_2 a) y H2).
Intro H4. Rewrite (MapGet_M2_bit_0_0 A a H4 m0 m1) in H2. Rewrite <- (ad_div_2_double a H4).
Exact (H [a:ad](pf (ad_double a)) H3 (ad_div_2 a) y H2).
`Qed`.

`Lemma` MapSweep_semantics_3 : (m:(Map A))
(MapSweep m)=(NONE ?) -> (a:ad) (y:A) (MapGet A m a)=(SOME ? y) ->
(f a y)=false.
`Proof`.
Intros.
Exact (MapSweep_semantics_3_1 m [a0:ad]a0 H a y H0).
`Qed`.

`Lemma` MapSweep_semantics_4_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A)
(MapGet A m a)=(SOME A y) -> (f (pf a) y)=true ->
{a':ad & {y':A | (MapSweep1 pf m)=(SOME ? (a', y'))}}.
`Proof`.
Induction m. Intros. Discriminate H.
Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H1. Split with (pf a1). Split with y.
Rewrite (ad_eq_complete ? ? H1). Unfold MapSweep1 MapSweep2.
Rewrite (ad_eq_complete ? ? H1) in H. Rewrite (M1_semantics_1 ? a1 a0) in H.
Inversion H. Rewrite H0. Reflexivity.

Intro H1. Rewrite (M1_semantics_2 ? a a1 a0 H1) in H. Discriminate H.

Intros. Elim (sumbool_of_bool (ad_bit_0 a)). Intro H3.
Rewrite (MapGet_M2_bit_0_1 ? ? H3 m0 m1) in H1.
Rewrite <- (ad_div_2_double_plus_un a H3) in H2.
Elim (H0 [a0:ad](pf (ad_double_plus_un a0)) (ad_div_2 a) y H1 H2). Intros a'' H4. Elim H4.
Intros y'' H5. Simpl. Elim (option_sum ? (MapSweep1 [a:ad](pf (ad_double a)) m0)).
Intro H6. Elim H6. Intro r. Elim r. Intros a''' y''' H7. Rewrite H7. Split with a'''.
Split with y'''. Reflexivity.
Intro H6. Rewrite H6. Split with a''. Split with y''. Assumption.
Intro H3. Rewrite (MapGet_M2_bit_0_0 ? ? H3 m0 m1) in H1.
Rewrite <- (ad_div_2_double a H3) in H2.
Elim (H [a0:ad](pf (ad_double a0)) (ad_div_2 a) y H1 H2). Intros a'' H4. Elim H4.
Intros y'' H5. Split with a''. Split with y''. Simpl. Rewrite H5. Reflexivity.
`Qed`.

`Lemma` MapSweep_semantics_4 : (m:(Map A)) (a:ad) (y:A)
(MapGet A m a)=(SOME A y) -> (f a y)=true ->
{a':ad & {y':A | (MapSweep m)=(SOME ? (a', y'))}}.
`Proof`.
Intros. Exact (MapSweep_semantics_4_1 m [a0:ad]a0 a y H H0).
`Qed`.

`End` MapSweepDef.

`Variable` B : Set.

`Fixpoint` MapCollect1 [f:ad->A->(Map B); pf:ad->ad; m:(Map A)] : (Map B) :=
Cases m of
M0 => (M0 B)
| (M1 a y) => (f (pf a) y)
| (M2 m1 m2) => (MapMerge B (MapCollect1 f [a0:ad] (pf (ad_double a0)) m1)
(MapCollect1 f [a0:ad] (pf (ad_double_plus_un a0)) m2))
end.

`Definition` MapCollect := [f:ad->A->(Map B); m:(Map A)] (MapCollect1 f [a:ad]a m).

`Section` MapFoldDef.

`Variable` M : Set.
`Variable` neutral : M.
`Variable` op : M -> M -> M.

`Fixpoint` MapFold1 [f:ad->A->M; pf:ad->ad; m:(Map A)] : M :=
Cases m of
M0 => neutral
| (M1 a y) => (f (pf a) y)
| (M2 m1 m2) => (op (MapFold1 f [a0:ad] (pf (ad_double a0)) m1)
(MapFold1 f [a0:ad] (pf (ad_double_plus_un a0)) m2))
end.

`Definition` MapFold := [f:ad->A->M; m:(Map A)] (MapFold1 f [a:ad]a m).

`Lemma` MapFold_empty : (f:ad->A->M) (MapFold f (M0 A))=neutral.
`Proof`.
Trivial.
`Qed`.

`Lemma` MapFold_M1 : (f:ad->A->M) (a:ad) (y:A) (MapFold f (M1 A a y)) = (f a y).
`Proof`.
Trivial.
`Qed`.

`Variable` State : Set.
`Variable` f:State -> ad -> A -> State * M.

`Fixpoint` MapFold1_state [state:State; pf:ad->ad; m:(Map A)]
: State * M :=
Cases m of
M0 => (state, neutral)
| (M1 a y) => (f state (pf a) y)
| (M2 m1 m2) =>
Cases (MapFold1_state state [a0:ad] (pf (ad_double a0)) m1) of
(state1, x1) =>
Cases (MapFold1_state state1 [a0:ad] (pf (ad_double_plus_un a0)) m2) of
(state2, x2) => (state2, (op x1 x2))
end
end
end.

`Definition` MapFold_state := [state:State] (MapFold1_state state [a:ad]a).

`Lemma` pair_sp : (B,C:Set) (x:B*C) x=(Fst x, Snd x).
`Proof`.
Induction x. Trivial.
`Qed`.

`Lemma` MapFold_state_stateless_1 : (m:(Map A)) (g:ad->A->M) (pf:ad->ad)
((state:State) (a:ad) (y:A) (Snd (f state a y))=(g a y)) ->
(state:State)
(Snd (MapFold1_state state pf m))=(MapFold1 g pf m).
`Proof`.
Induction m. Trivial.
Intros. Simpl. Apply H.
Intros. Simpl. Rewrite (pair_sp ? ?
(MapFold1_state state [a0:ad](pf (ad_double a0)) m0)).
Rewrite (H g [a0:ad](pf (ad_double a0)) H1 state).
Rewrite (pair_sp ? ?
(MapFold1_state
(Fst (MapFold1_state state [a0:ad](pf (ad_double a0)) m0))
[a0:ad](pf (ad_double_plus_un a0)) m1)).
Simpl.
Rewrite (H0 g [a0:ad](pf (ad_double_plus_un a0)) H1
(Fst (MapFold1_state state [a0:ad](pf (ad_double a0)) m0))).
Reflexivity.
`Qed`.

`Lemma` MapFold_state_stateless : (g:ad->A->M)
((state:State) (a:ad) (y:A) (Snd (f state a y))=(g a y)) ->
(state:State) (m:(Map A))
(Snd (MapFold_state state m))=(MapFold g m).
`Proof`.
Intros. Exact (MapFold_state_stateless_1 m g [a0:ad]a0 H state).
`Qed`.

`End` MapFoldDef.

`Lemma` MapCollect_as_Fold : (f:ad->A->(Map B)) (m:(Map A))
(MapCollect f m)=(MapFold (Map B) (M0 B) (MapMerge B) f m).
`Proof`.
Induction m;Trivial.
`Qed`.

`Definition` alist := (list (ad*A)).
`Definition` anil := (nil (ad*A)).
`Definition` acons := (!cons (ad*A)).
`Definition` aapp := (!app (ad*A)).

`Definition` alist_of_Map := (MapFold alist anil aapp [a:ad;y:A] (acons (pair ? ? a y) anil)).

`Fixpoint` alist_semantics [l:alist] : ad -> (option A) :=
Cases l of
nil => [_:ad] (NONE A)
| (cons (a, y) l') => [a0:ad] if (ad_eq a a0) then (SOME A y) else (alist_semantics l' a0)
end.

`Lemma` alist_semantics_app : (l,l':alist) (a:ad)
(alist_semantics (aapp l l') a)=
(Cases (alist_semantics l a) of
NONE => (alist_semantics l' a)
| (SOME y) => (SOME A y)
end).
`Proof`.
Unfold aapp. Induction l. Trivial.
Intros. Elim a. Intros a1 y1. Simpl. Case (ad_eq a1 a0). Reflexivity.
Apply H.
`Qed`.

`Lemma` alist_of_Map_semantics_1_1 : (m:(Map A)) (pf:ad->ad) (a:ad) (y:A)
(alist_semantics (MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil) pf m) a)
=(SOME A y) -> {a':ad | a=(pf a')}.
`Proof`.
Induction m. Simpl. Intros. Discriminate H.
Simpl. Intros a y pf a0 y0. Elim (sumbool_of_bool (ad_eq (pf a) a0)). Intro H. Rewrite H.
Intro H0. Split with a. Rewrite (ad_eq_complete ? ? H). Reflexivity.
Intro H. Rewrite H. Intro H0. Discriminate H0.
Intros. Change (alist_semantics
(aapp
(MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil)
[a0:ad](pf (ad_double a0)) m0)
(MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil)
[a0:ad](pf (ad_double_plus_un a0)) m1)) a)=(SOME A y) in H1.
Rewrite (alist_semantics_app
(MapFold1 alist anil aapp [a0:ad][y0:A](acons (a0,y0) anil)
[a0:ad](pf (ad_double a0)) m0)
(MapFold1 alist anil aapp [a0:ad][y0:A](acons (a0,y0) anil)
[a0:ad](pf (ad_double_plus_un a0)) m1) a) in H1.
Elim (option_sum A
(alist_semantics
(MapFold1 alist anil aapp [a0:ad][y0:A](acons (a0,y0) anil)
[a0:ad](pf (ad_double a0)) m0) a)).
Intro H2. Elim H2. Intros y0 H3. Elim (H [a0:ad](pf (ad_double a0)) a y0 H3). Intros a0 H4.
Split with (ad_double a0). Assumption.
Intro H2. Rewrite H2 in H1. Elim (H0 [a0:ad](pf (ad_double_plus_un a0)) a y H1).
Intros a0 H3. Split with (ad_double_plus_un a0). Assumption.
`Qed`.

`Definition` ad_inj := [pf:ad->ad] (a0,a1:ad) (pf a0)=(pf a1) -> a0=a1.

`Lemma` ad_comp_double_inj :
(pf:ad->ad) (ad_inj pf) -> (ad_inj [a0:ad] (pf (ad_double a0))).
`Proof`.
Unfold ad_inj. Intros. Apply ad_double_inj. Exact (H ? ? H0).
`Qed`.

`Lemma` ad_comp_double_plus_un_inj : (pf:ad->ad) (ad_inj pf) ->
(ad_inj [a0:ad] (pf (ad_double_plus_un a0))).
`Proof`.
Unfold ad_inj. Intros. Apply ad_double_plus_un_inj. Exact (H ? ? H0).
`Qed`.

`Lemma` alist_of_Map_semantics_1 : (m:(Map A)) (pf:ad->ad) (ad_inj pf) ->
(a:ad) (MapGet A m a)=(alist_semantics (MapFold1 alist anil aapp
[a0:ad;y:A] (acons (pair ? ? a0 y) anil) pf m)
(pf a)).
`Proof`.
Induction m. Trivial.
Simpl. Intros. Elim (sumbool_of_bool (ad_eq a a1)). Intro H0. Rewrite H0.
Rewrite (ad_eq_complete ? ? H0). Rewrite (ad_eq_correct (pf a1)). Reflexivity.
Intro H0. Rewrite H0. Elim (sumbool_of_bool (ad_eq (pf a) (pf a1))). Intro H1.
Rewrite (H a a1 (ad_eq_complete ? ? H1)) in H0. Rewrite (ad_eq_correct a1) in H0.
Discriminate H0.
Intro H1. Rewrite H1. Reflexivity.
Intros. Change (MapGet A (M2 A m0 m1) a)
=(alist_semantics
(aapp
(MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil)
[a0:ad](pf (ad_double a0)) m0)
(MapFold1 alist anil aapp [a0:ad][y:A](acons (a0,y) anil)
[a0:ad](pf (ad_double_plus_un a0)) m1)) (pf a)).
Rewrite alist_semantics_app. Rewrite (MapGet_M2_bit_0_if A m0 m1 a).
Elim (ad_double_or_double_plus_un a). Intro H2. Elim H2. Intros a0 H3. Rewrite H3.
Rewrite (ad_double_bit_0 a0).
Rewrite <- (H [a1:ad](pf (ad_double a1)) (ad_comp_double_inj pf H1) a0).
Rewrite ad_double_div_2. Case (MapGet A m0 a0).
Elim (option_sum A
(alist_semantics
(MapFold1 alist anil aapp [a1:ad][y:A](acons (a1,y) anil)
[a1:ad](pf (ad_double_plus_un a1)) m1) (pf (ad_double a0)))).
Intro H4. Elim H4. Intros y H5.
Elim (alist_of_Map_semantics_1_1 m1 [a1:ad](pf (ad_double_plus_un a1))
(pf (ad_double a0)) y H5).
Intros a1 H6. Cut (ad_bit_0 (ad_double a0))=(ad_bit_0 (ad_double_plus_un a1)).
Intro. Rewrite (ad_double_bit_0 a0) in H7. Rewrite (ad_double_plus_un_bit_0 a1) in H7.
Discriminate H7.
Rewrite (H1 (ad_double a0) (ad_double_plus_un a1) H6). Reflexivity.
Intro H4. Rewrite H4. Reflexivity.
Trivial.
Intro H2. Elim H2. Intros a0 H3. Rewrite H3. Rewrite (ad_double_plus_un_bit_0 a0).
Rewrite <- (H0 [a1:ad](pf (ad_double_plus_un a1)) (ad_comp_double_plus_un_inj pf H1) a0).
Rewrite ad_double_plus_un_div_2.
Elim (option_sum A
(alist_semantics
(MapFold1 alist anil aapp [a1:ad][y:A](acons (a1,y) anil)
[a1:ad](pf (ad_double a1)) m0) (pf (ad_double_plus_un a0)))).
Intro H4. Elim H4. Intros y H5.
Elim (alist_of_Map_semantics_1_1 m0 [a1:ad](pf (ad_double a1))
(pf (ad_double_plus_un a0)) y H5).
Intros a1 H6. Cut (ad_bit_0 (ad_double_plus_un a0))=(ad_bit_0 (ad_double a1)).
Intro H7. Rewrite (ad_double_plus_un_bit_0 a0) in H7. Rewrite (ad_double_bit_0 a1) in H7.
Discriminate H7.
Rewrite (H1 (ad_double_plus_un a0) (ad_double a1) H6). Reflexivity.
Intro H4. Rewrite H4. Reflexivity.
`Qed`.

`Lemma` alist_of_Map_semantics : (m:(Map A))
(eqm A (MapGet A m) (alist_semantics (alist_of_Map m))).
`Proof`.
Unfold eqm. Intros. Exact (alist_of_Map_semantics_1 m [a0:ad]a0 [a0,a1:ad][p:a0=a1]p a).
`Qed`.

`Fixpoint` Map_of_alist [l:alist] : (Map A) :=
Cases l of
nil => (M0 A)
| (cons (a, y) l') => (MapPut A (Map_of_alist l') a y)
end.

`Lemma` Map_of_alist_semantics : (l:alist)
(eqm A (alist_semantics l) (MapGet A (Map_of_alist l))).
`Proof`.
Unfold eqm. Induction l. Trivial.
Intros r l0 H a. Elim r. Intros a0 y0. Simpl. Elim (sumbool_of_bool (ad_eq a0 a)).
Intro H0. Rewrite H0. Rewrite (ad_eq_complete ? ? H0).
Rewrite (MapPut_semantics A (Map_of_alist l0) a y0 a). Rewrite (ad_eq_correct a).
Reflexivity.
Intro H0. Rewrite H0. Rewrite (MapPut_semantics A (Map_of_alist l0) a0 y0 a).
Rewrite H0. Apply H.
`Qed`.

`Lemma` Map_of_alist_of_Map : (m:(Map A)) (eqmap A (Map_of_alist (alist_of_Map m)) m).
`Proof`.
Unfold eqmap. Intro. Apply eqm_trans with f':=(alist_semantics (alist_of_Map m)).
Apply eqm_sym. Apply Map_of_alist_semantics.
Apply eqm_sym. Apply alist_of_Map_semantics.
`Qed`.

`Lemma` alist_of_Map_of_alist : (l:alist)
(eqm A (alist_semantics (alist_of_Map (Map_of_alist l))) (alist_semantics l)).
`Proof`.
Intro. Apply eqm_trans with f':=(MapGet A (Map_of_alist l)).
Apply eqm_sym. Apply alist_of_Map_semantics.
Apply eqm_sym. Apply Map_of_alist_semantics.
`Qed`.

`Lemma` fold_right_aapp : (M:Set) (neutral:M) (op:M->M->M)
((a,b,c:M) (op (op a b) c)=(op a (op b c))) ->
((a:M) (op neutral a)=a) ->
(f:ad->A->M) (l,l':alist)
(fold_right [r:ad*A][m:M] let (a,y)=r in (op (f a y) m) neutral
(aapp l l'))=
(op (fold_right [r:ad*A][m:M] let (a,y)=r in (op (f a y) m) neutral l)
(fold_right [r:ad*A][m:M] let (a,y)=r in (op (f a y) m) neutral l'))
.
`Proof`.
Induction l. Simpl. Intro. Rewrite H0. Reflexivity.
Intros r l0 H1 l'. Elim r. Intros a y. Simpl. Rewrite H. Rewrite (H1 l'). Reflexivity.
`Qed`.

`Lemma` MapFold_as_fold_1 : (M:Set) (neutral:M) (op:M->M->M)
((a,b,c:M) (op (op a b) c)=(op a (op b c))) ->
((a:M) (op neutral a)=a) ->
((a:M) (op a neutral)=a) ->
(f:ad->A->M) (m:(Map A)) (pf:ad->ad)
(MapFold1 M neutral op f pf m)=
(fold_right [r:(ad*A)][m:M] let (a,y)=r in (op (f a y) m) neutral
(MapFold1 alist anil aapp [a:ad;y:A] (acons (pair ? ?
a y) anil) pf m)).
`Proof`.
Induction m. Trivial.
Intros. Simpl. Rewrite H1. Reflexivity.
Intros. Simpl. Rewrite (fold_right_aapp M neutral op H H0 f).
Rewrite (H2 [a0:ad](pf (ad_double a0))). Rewrite (H3 [a0:ad](pf (ad_double_plus_un a0))).
Reflexivity.
`Qed`.

`Lemma` MapFold_as_fold : (M:Set) (neutral:M) (op:M->M->M)
((a,b,c:M) (op (op a b) c)=(op a (op b c))) ->
((a:M) (op neutral a)=a) ->
((a:M) (op a neutral)=a) ->
(f:ad->A->M) (m:(Map A))
(MapFold M neutral op f m)=
(fold_right [r:(ad*A)][m:M] let (a,y)=r in (op (f a y) m) neutral
(alist_of_Map m)).
`Proof`.
Intros. Exact (MapFold_as_fold_1 M neutral op H H0 H1 f m [a0:ad]a0).
`Qed`.

`Lemma` alist_MapMerge_semantics : (m,m':(Map A))
(eqm A (alist_semantics (aapp (alist_of_Map m') (alist_of_Map m)))
(alist_semantics (alist_of_Map (MapMerge A m m')))).
`Proof`.
Unfold eqm. Intros. Rewrite alist_semantics_app. Rewrite <- (alist_of_Map_semantics m a).
Rewrite <- (alist_of_Map_semantics m' a).
Rewrite <- (alist_of_Map_semantics (MapMerge A m m') a).
Rewrite (MapMerge_semantics A m m' a). Reflexivity.
`Qed`.

`Lemma` alist_MapMerge_semantics_disjoint : (m,m':(Map A))
(eqmap A (MapDomRestrTo A A m m') (M0 A)) ->
(eqm A (alist_semantics (aapp (alist_of_Map m) (alist_of_Map m')))
(alist_semantics (alist_of_Map (MapMerge A m m')))).
`Proof`.
Unfold eqm. Intros. Rewrite alist_semantics_app. Rewrite <- (alist_of_Map_semantics m a).
Rewrite <- (alist_of_Map_semantics m' a).
Rewrite <- (alist_of_Map_semantics (MapMerge A m m') a). Rewrite (MapMerge_semantics A m m' a).
Elim (option_sum ? (MapGet A m a)). Intro H0. Elim H0. Intros y H1. Rewrite H1.
Elim (option_sum ? (MapGet A m' a)). Intro H2. Elim H2. Intros y' H3.
Cut (MapGet A (MapDomRestrTo A A m m') a)=(NONE A).
Rewrite (MapDomRestrTo_semantics A A m m' a). Rewrite H3. Rewrite H1. Intro. Discriminate H4.
Exact (H a).
Intro H2. Rewrite H2. Reflexivity.
Intro H0. Rewrite H0. Case (MapGet A m' a); Trivial.
`Qed`.

`Lemma` alist_semantics_disjoint_comm : (l,l':alist)
(eqmap A (MapDomRestrTo A A (Map_of_alist l) (Map_of_alist l')) (M0 A)) ->
(eqm A (alist_semantics (aapp l l')) (alist_semantics (aapp l' l))).
`Proof`.
Unfold eqm. Intros. Rewrite (alist_semantics_app l l' a). Rewrite (alist_semantics_app l' l a).
Rewrite <- (alist_of_Map_of_alist l a). Rewrite <- (alist_of_Map_of_alist l' a).
Rewrite <- (alist_semantics_app (alist_of_Map (Map_of_alist l))
(alist_of_Map (Map_of_alist l')) a).
Rewrite <- (alist_semantics_app (alist_of_Map (Map_of_alist l'))
(alist_of_Map (Map_of_alist l)) a).
Rewrite (alist_MapMerge_semantics (Map_of_alist l) (Map_of_alist l') a).
Rewrite (alist_MapMerge_semantics_disjoint (Map_of_alist l) (Map_of_alist l') H a).
Reflexivity.
`Qed`.

`End` MapIter.

`

Index