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diff --git a/theories7/IntMap/Mapcanon.v b/theories7/IntMap/Mapcanon.v deleted file mode 100644 index 7beb1fd4..00000000 --- a/theories7/IntMap/Mapcanon.v +++ /dev/null @@ -1,376 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) -(*i $Id: Mapcanon.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) - -Require Bool. -Require Sumbool. -Require Arith. -Require ZArith. -Require Addr. -Require Adist. -Require Addec. -Require Map. -Require Mapaxioms. -Require Mapiter. -Require Fset. -Require PolyList. -Require Lsort. -Require Mapsubset. -Require Mapcard. - -Section MapCanon. - - Variable A : Set. - - Inductive mapcanon : (Map A) -> Prop := - M0_canon : (mapcanon (M0 A)) - | M1_canon : (a:ad) (y:A) (mapcanon (M1 A a y)) - | M2_canon : (m1,m2:(Map A)) (mapcanon m1) -> (mapcanon m2) -> - (le (2) (MapCard A (M2 A m1 m2))) -> (mapcanon (M2 A m1 m2)). - - Lemma mapcanon_M2 : - (m1,m2:(Map A)) (mapcanon (M2 A m1 m2)) -> (le (2) (MapCard A (M2 A m1 m2))). - Proof. - Intros. Inversion H. Assumption. - Qed. - - Lemma mapcanon_M2_1 : (m1,m2:(Map A)) (mapcanon (M2 A m1 m2)) -> (mapcanon m1). - Proof. - Intros. Inversion H. Assumption. - Qed. - - Lemma mapcanon_M2_2 : (m1,m2:(Map A)) (mapcanon (M2 A m1 m2)) -> (mapcanon m2). - Proof. - Intros. Inversion H. Assumption. - Qed. - - Lemma M2_eqmap_1 : (m0,m1,m2,m3:(Map A)) - (eqmap A (M2 A m0 m1) (M2 A m2 m3)) -> (eqmap A m0 m2). - Proof. - Unfold eqmap eqm. Intros. Rewrite <- (ad_double_div_2 a). - Rewrite <- (MapGet_M2_bit_0_0 A ? (ad_double_bit_0 a) m0 m1). - Rewrite <- (MapGet_M2_bit_0_0 A ? (ad_double_bit_0 a) m2 m3). - Exact (H (ad_double a)). - Qed. - - Lemma M2_eqmap_2 : (m0,m1,m2,m3:(Map A)) - (eqmap A (M2 A m0 m1) (M2 A m2 m3)) -> (eqmap A m1 m3). - Proof. - Unfold eqmap eqm. Intros. Rewrite <- (ad_double_plus_un_div_2 a). - Rewrite <- (MapGet_M2_bit_0_1 A ? (ad_double_plus_un_bit_0 a) m0 m1). - Rewrite <- (MapGet_M2_bit_0_1 A ? (ad_double_plus_un_bit_0 a) m2 m3). - Exact (H (ad_double_plus_un a)). - Qed. - - Lemma mapcanon_unique : (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> - (eqmap A m m') -> m=m'. - Proof. - Induction m. Induction m'. Trivial. - Intros a y H H0 H1. Cut (NONE A)=(MapGet A (M1 A a y) a). Simpl. Rewrite (ad_eq_correct a). - Intro. Discriminate H2. - Exact (H1 a). - Intros. Cut (le (2) (MapCard A (M0 A))). Intro. Elim (le_Sn_O ? H4). - Rewrite (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H2). - Intros a y. Induction m'. Intros. Cut (MapGet A (M1 A a y) a)=(NONE A). Simpl. - Rewrite (ad_eq_correct a). Intro. Discriminate H2. - Exact (H1 a). - Intros a0 y0 H H0 H1. Cut (MapGet A (M1 A a y) a)=(MapGet A (M1 A a0 y0) a). Simpl. - Rewrite (ad_eq_correct a). Intro. Elim (sumbool_of_bool (ad_eq a0 a)). Intro H3. - Rewrite H3 in H2. Inversion H2. Rewrite (ad_eq_complete ? ? H3). Reflexivity. - Intro H3. Rewrite H3 in H2. Discriminate H2. - Exact (H1 a). - Intros. Cut (le (2) (MapCard A (M1 A a y))). Intro. Elim (le_Sn_O ? (le_S_n ? ? H4)). - Rewrite (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H2). - Induction m'. Intros. Cut (le (2) (MapCard A (M0 A))). Intro. Elim (le_Sn_O ? H4). - Rewrite <- (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H1). - Intros a y H1 H2 H3. Cut (le (2) (MapCard A (M1 A a y))). Intro. - Elim (le_Sn_O ? (le_S_n ? ? H4)). - Rewrite <- (MapCard_ext A ? ? H3). Exact (mapcanon_M2 ? ? H1). - Intros. Rewrite (H m2). Rewrite (H0 m3). Reflexivity. - Exact (mapcanon_M2_2 ? ? H3). - Exact (mapcanon_M2_2 ? ? H4). - Exact (M2_eqmap_2 ? ? ? ? H5). - Exact (mapcanon_M2_1 ? ? H3). - Exact (mapcanon_M2_1 ? ? H4). - Exact (M2_eqmap_1 ? ? ? ? H5). - Qed. - - Lemma MapPut1_canon : - (p:positive) (a,a':ad) (y,y':A) (mapcanon (MapPut1 A a y a' y' p)). - Proof. - Induction p. Simpl. Intros. Case (ad_bit_0 a). Apply M2_canon. Apply M1_canon. - Apply M1_canon. - Apply le_n. - Apply M2_canon. Apply M1_canon. - Apply M1_canon. - Apply le_n. - Simpl. Intros. Case (ad_bit_0 a). Apply M2_canon. Apply M0_canon. - Apply H. - Simpl. Rewrite MapCard_Put1_equals_2. Apply le_n. - Apply M2_canon. Apply H. - Apply M0_canon. - Simpl. Rewrite MapCard_Put1_equals_2. Apply le_n. - Simpl. Simpl. Intros. Case (ad_bit_0 a). Apply M2_canon. Apply M1_canon. - Apply M1_canon. - Simpl. Apply le_n. - Apply M2_canon. Apply M1_canon. - Apply M1_canon. - Simpl. Apply le_n. - Qed. - - Lemma MapPut_canon : - (m:(Map A)) (mapcanon m) -> (a:ad) (y:A) (mapcanon (MapPut A m a y)). - Proof. - Induction m. Intros. Simpl. Apply M1_canon. - Intros a0 y0 H a y. Simpl. Case (ad_xor a0 a). Apply M1_canon. - Intro. Apply MapPut1_canon. - Intros. Simpl. Elim a. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). - Exact (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). Exact (mapcanon_M2 ? ? H1). - Apply le_plus_plus. Exact (MapCard_Put_lb A m0 ad_z y). - Apply le_n. - Intro. Case p. Intro. Apply M2_canon. Exact (mapcanon_M2_1 m0 m1 H1). - Apply H0. Exact (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). - Exact (mapcanon_M2 m0 m1 H1). - Apply le_reg_l. Exact (MapCard_Put_lb A m1 (ad_x p0) y). - Intro. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). - Exact (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). - Exact (mapcanon_M2 m0 m1 H1). - Apply le_reg_r. Exact (MapCard_Put_lb A m0 (ad_x p0) y). - Apply M2_canon. Apply (mapcanon_M2_1 m0 m1 H1). - Apply H0. Apply (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). - Exact (mapcanon_M2 m0 m1 H1). - Apply le_reg_l. Exact (MapCard_Put_lb A m1 ad_z y). - Qed. - - Lemma MapPut_behind_canon : (m:(Map A)) (mapcanon m) -> - (a:ad) (y:A) (mapcanon (MapPut_behind A m a y)). - Proof. - Induction m. Intros. Simpl. Apply M1_canon. - Intros a0 y0 H a y. Simpl. Case (ad_xor a0 a). Apply M1_canon. - Intro. Apply MapPut1_canon. - Intros. Simpl. Elim a. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). - Exact (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). Exact (mapcanon_M2 ? ? H1). - Apply le_plus_plus. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m0 ad_z y). - Apply le_n. - Intro. Case p. Intro. Apply M2_canon. Exact (mapcanon_M2_1 m0 m1 H1). - Apply H0. Exact (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). - Exact (mapcanon_M2 m0 m1 H1). - Apply le_reg_l. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m1 (ad_x p0) y). - Intro. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). - Exact (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). - Exact (mapcanon_M2 m0 m1 H1). - Apply le_reg_r. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m0 (ad_x p0) y). - Apply M2_canon. Apply (mapcanon_M2_1 m0 m1 H1). - Apply H0. Apply (mapcanon_M2_2 m0 m1 H1). - Simpl. Apply le_trans with m:=(plus (MapCard A m0) (MapCard A m1)). - Exact (mapcanon_M2 m0 m1 H1). - Apply le_reg_l. Rewrite MapCard_Put_behind_Put. Exact (MapCard_Put_lb A m1 ad_z y). - Qed. - - Lemma makeM2_canon : - (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> (mapcanon (makeM2 A m m')). - Proof. - Intro. Case m. Intro. Case m'. Intros. Exact M0_canon. - Intros a y H H0. Exact (M1_canon (ad_double_plus_un a) y). - Intros. Simpl. (Apply M2_canon; Try Assumption). Exact (mapcanon_M2 m0 m1 H0). - Intros a y m'. Case m'. Intros. Exact (M1_canon (ad_double a) y). - Intros a0 y0 H H0. Simpl. (Apply M2_canon; Try Assumption). Apply le_n. - Intros. Simpl. (Apply M2_canon; Try Assumption). - Apply le_trans with m:=(MapCard A (M2 A m0 m1)). Exact (mapcanon_M2 ? ? H0). - Exact (le_plus_r (MapCard A (M1 A a y)) (MapCard A (M2 A m0 m1))). - Simpl. Intros. (Apply M2_canon; Try Assumption). - Apply le_trans with m:=(MapCard A (M2 A m0 m1)). Exact (mapcanon_M2 ? ? H). - Exact (le_plus_l (MapCard A (M2 A m0 m1)) (MapCard A m')). - Qed. - - Fixpoint MapCanonicalize [m:(Map A)] : (Map A) := - Cases m of - (M2 m0 m1) => (makeM2 A (MapCanonicalize m0) (MapCanonicalize m1)) - | _ => m - end. - - Lemma mapcanon_exists_1 : (m:(Map A)) (eqmap A m (MapCanonicalize m)). - Proof. - Induction m. Apply eqmap_refl. - Intros. Apply eqmap_refl. - Intros. Simpl. Unfold eqmap eqm. Intro. - Rewrite (makeM2_M2 A (MapCanonicalize m0) (MapCanonicalize m1) a). - Rewrite MapGet_M2_bit_0_if. Rewrite MapGet_M2_bit_0_if. - Rewrite <- (H (ad_div_2 a)). Rewrite <- (H0 (ad_div_2 a)). Reflexivity. - Qed. - - Lemma mapcanon_exists_2 : (m:(Map A)) (mapcanon (MapCanonicalize m)). - Proof. - Induction m. Apply M0_canon. - Intros. Simpl. Apply M1_canon. - Intros. Simpl. (Apply makeM2_canon; Assumption). - Qed. - - Lemma mapcanon_exists : - (m:(Map A)) {m':(Map A) | (eqmap A m m') /\ (mapcanon m')}. - Proof. - Intro. Split with (MapCanonicalize m). Split. Apply mapcanon_exists_1. - Apply mapcanon_exists_2. - Qed. - - Lemma MapRemove_canon : - (m:(Map A)) (mapcanon m) -> (a:ad) (mapcanon (MapRemove A m a)). - Proof. - Induction m. Intros. Exact M0_canon. - Intros a y H a0. Simpl. Case (ad_eq a a0). Exact M0_canon. - Assumption. - Intros. Simpl. Case (ad_bit_0 a). Apply makeM2_canon. Exact (mapcanon_M2_1 ? ? H1). - Apply H0. Exact (mapcanon_M2_2 ? ? H1). - Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H1). - Exact (mapcanon_M2_2 ? ? H1). - Qed. - - Lemma MapMerge_canon : (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> - (mapcanon (MapMerge A m m')). - Proof. - Induction m. Intros. Exact H0. - Simpl. Intros a y m' H H0. Exact (MapPut_behind_canon m' H0 a y). - Induction m'. Intros. Exact H1. - Intros a y H1 H2. Unfold MapMerge. Exact (MapPut_canon ? H1 a y). - Intros. Simpl. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H3). - Exact (mapcanon_M2_1 ? ? H4). - Apply H0. Exact (mapcanon_M2_2 ? ? H3). - Exact (mapcanon_M2_2 ? ? H4). - Change (le (2) (MapCard A (MapMerge A (M2 A m0 m1) (M2 A m2 m3)))). - Apply le_trans with m:=(MapCard A (M2 A m0 m1)). Exact (mapcanon_M2 ? ? H3). - Exact (MapMerge_Card_lb_l A (M2 A m0 m1) (M2 A m2 m3)). - Qed. - - Lemma MapDelta_canon : (m,m':(Map A)) (mapcanon m) -> (mapcanon m') -> - (mapcanon (MapDelta A m m')). - Proof. - Induction m. Intros. Exact H0. - Simpl. Intros a y m' H H0. Case (MapGet A m' a). Exact (MapPut_canon m' H0 a y). - Intro. Exact (MapRemove_canon m' H0 a). - Induction m'. Intros. Exact H1. - Unfold MapDelta. Intros a y H1 H2. Case (MapGet A (M2 A m0 m1) a). - Exact (MapPut_canon ? H1 a y). - Intro. Exact (MapRemove_canon ? H1 a). - Intros. Simpl. Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H3). - Exact (mapcanon_M2_1 ? ? H4). - Apply H0. Exact (mapcanon_M2_2 ? ? H3). - Exact (mapcanon_M2_2 ? ? H4). - Qed. - - Variable B : Set. - - Lemma MapDomRestrTo_canon : (m:(Map A)) (mapcanon m) -> - (m':(Map B)) (mapcanon (MapDomRestrTo A B m m')). - Proof. - Induction m. Intros. Exact M0_canon. - Simpl. Intros a y H m'. Case (MapGet B m' a). Exact M0_canon. - Intro. Apply M1_canon. - Induction m'. Exact M0_canon. - Unfold MapDomRestrTo. Intros a y. Case (MapGet A (M2 A m0 m1) a). Exact M0_canon. - Intro. Apply M1_canon. - Intros. Simpl. Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 m0 m1 H1). - Apply H0. Exact (mapcanon_M2_2 m0 m1 H1). - Qed. - - Lemma MapDomRestrBy_canon : (m:(Map A)) (mapcanon m) -> - (m':(Map B)) (mapcanon (MapDomRestrBy A B m m')). - Proof. - Induction m. Intros. Exact M0_canon. - Simpl. Intros a y H m'. Case (MapGet B m' a). Assumption. - Intro. Exact M0_canon. - Induction m'. Exact H1. - Intros a y. Simpl. Case (ad_bit_0 a). Apply makeM2_canon. Exact (mapcanon_M2_1 ? ? H1). - Apply MapRemove_canon. Exact (mapcanon_M2_2 ? ? H1). - Apply makeM2_canon. Apply MapRemove_canon. Exact (mapcanon_M2_1 ? ? H1). - Exact (mapcanon_M2_2 ? ? H1). - Intros. Simpl. Apply makeM2_canon. Apply H. Exact (mapcanon_M2_1 ? ? H1). - Apply H0. Exact (mapcanon_M2_2 ? ? H1). - Qed. - - Lemma Map_of_alist_canon : (l:(alist A)) (mapcanon (Map_of_alist A l)). - Proof. - Induction l. Exact M0_canon. - Intro r. Elim r. Intros a y l0 H. Simpl. Apply MapPut_canon. Assumption. - Qed. - - Lemma MapSubset_c_1 : (m:(Map A)) (m':(Map B)) (mapcanon m) -> - (MapSubset A B m m') -> (MapDomRestrBy A B m m')=(M0 A). - Proof. - Intros. Apply mapcanon_unique. Apply MapDomRestrBy_canon. Assumption. - Apply M0_canon. - Exact (MapSubset_imp_2 ? ? m m' H0). - Qed. - - Lemma MapSubset_c_2 : (m:(Map A)) (m':(Map B)) - (MapDomRestrBy A B m m')=(M0 A) -> (MapSubset A B m m'). - Proof. - Intros. Apply MapSubset_2_imp. Unfold MapSubset_2. Rewrite H. Apply eqmap_refl. - Qed. - -End MapCanon. - -Section FSetCanon. - - Variable A : Set. - - Lemma MapDom_canon : (m:(Map A)) (mapcanon A m) -> (mapcanon unit (MapDom A m)). - Proof. - Induction m. Intro. Exact (M0_canon unit). - Intros a y H. Exact (M1_canon unit a ?). - Intros. Simpl. Apply M2_canon. Apply H. Exact (mapcanon_M2_1 A ? ? H1). - Apply H0. Exact (mapcanon_M2_2 A ? ? H1). - Change (le (2) (MapCard unit (MapDom A (M2 A m0 m1)))). Rewrite <- MapCard_Dom. - Exact (mapcanon_M2 A ? ? H1). - Qed. - -End FSetCanon. - -Section MapFoldCanon. - - Variable A, B : Set. - - Lemma MapFold_canon_1 : (m0:(Map B)) (mapcanon B m0) -> - (op : (Map B) -> (Map B) -> (Map B)) - ((m1:(Map B)) (mapcanon B m1) -> (m2:(Map B)) (mapcanon B m2) -> - (mapcanon B (op m1 m2))) -> - (f : ad->A->(Map B)) ((a:ad) (y:A) (mapcanon B (f a y))) -> - (m:(Map A)) (pf : ad->ad) (mapcanon B (MapFold1 A (Map B) m0 op f pf m)). - Proof. - Induction m. Intro. Exact H. - Intros a y pf. Simpl. Apply H1. - Intros. Simpl. Apply H0. Apply H2. - Apply H3. - Qed. - - Lemma MapFold_canon : (m0:(Map B)) (mapcanon B m0) -> - (op : (Map B) -> (Map B) -> (Map B)) - ((m1:(Map B)) (mapcanon B m1) -> (m2:(Map B)) (mapcanon B m2) -> - (mapcanon B (op m1 m2))) -> - (f : ad->A->(Map B)) ((a:ad) (y:A) (mapcanon B (f a y))) -> - (m:(Map A)) (mapcanon B (MapFold A (Map B) m0 op f m)). - Proof. - Intros. Exact (MapFold_canon_1 m0 H op H0 f H1 m [a:ad]a). - Qed. - - Lemma MapCollect_canon : - (f : ad->A->(Map B)) ((a:ad) (y:A) (mapcanon B (f a y))) -> - (m:(Map A)) (mapcanon B (MapCollect A B f m)). - Proof. - Intros. Rewrite MapCollect_as_Fold. Apply MapFold_canon. Apply M0_canon. - Intros. Exact (MapMerge_canon B m1 m2 H0 H1). - Assumption. - Qed. - -End MapFoldCanon. |