ATBR.MxGraph
(**************************************************************************)
(* This is part of ATBR, it is distributed under the terms of the *)
(* GNU Lesser General Public License version 3 *)
(* (see file LICENSE for more details) *)
(* *)
(* Copyright 2009-2011: Thomas Braibant, Damien Pous. *)
(**************************************************************************)
(* This is part of ATBR, it is distributed under the terms of the *)
(* GNU Lesser General Public License version 3 *)
(* (see file LICENSE for more details) *)
(* *)
(* Copyright 2009-2011: Thomas Braibant, Damien Pous. *)
(**************************************************************************)
Basic definitions for matrices: definition of their Graph
Require Import Common.
Require Import Classes.
Require Import Graph.
Require List.
Require Force.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Defs.
Context {G: Graph}.
Variable A: T.
Notation X := (X A A).
Inductive MX (n m: nat) :=
box: (nat -> nat -> X) -> MX n m.
Definition get n m (M: MX n m) := let (f) := M in f.
box: (nat -> nat -> X) -> MX n m.
Definition get n m (M: MX n m) := let (f) := M in f.
matrix equality is bounded pointwise equality
Definition mx_equal n m: relation (MX n m) :=
fun M N => forall i j, i<n -> j<m -> get M i j == get N i j.
matrix Graph, equality is bounded pointwise equality
Program Instance mx_Graph: Graph := {
T := nat;
X := MX;
equal := mx_equal }.
Next Obligation.
split; repeat intro; simpl in *.
reflexivity.
symmetry; auto.
transitivity (get y i j); auto.
Qed.
Lemma mx_equal': forall n m (M N: @Classes.X mx_Graph n m)
(H: forall i j, i<n -> j<m -> get M i j == get N i j), M == N.
Proof (fun _ _ _ _ H => H).
Definition mx_equal_at p q n m n' m' (M : MX n m) (N : MX n' m') :=
forall i j, i < p -> j < q -> get M i j == get N i j.
Lemma mx_equal_at_equal n m (M N: @Classes.X mx_Graph n m) : mx_equal_at n m M N <-> M == N.
Proof. intros. unfold mx_equal_at. intuition. Qed.
End Defs.
Notation MX_ A n m := (@X (mx_Graph A) (n%nat:nat) (m%nat:nat)) (only parsing).
Notation mx_equal_ A n m := (@equal (mx_Graph A) (n%nat) (m%nat)) (only parsing).
Notation "! M" := (get M) (at level 0) : A_scope.
Notation "M == [ n , m ] N" := (@equal (mx_Graph _) n m M N) (at level 80) : A_scope.
Lemma plus_minus : forall m n, S (m+n)-n = S m.
Proof. intros. omega. Qed.
Global Opaque minus.
Lemma lt_n_1 n: ~ (S n<1)%nat.
Proof. omega. Qed.
T := nat;
X := MX;
equal := mx_equal }.
Next Obligation.
split; repeat intro; simpl in *.
reflexivity.
symmetry; auto.
transitivity (get y i j); auto.
Qed.
Lemma mx_equal': forall n m (M N: @Classes.X mx_Graph n m)
(H: forall i j, i<n -> j<m -> get M i j == get N i j), M == N.
Proof (fun _ _ _ _ H => H).
Definition mx_equal_at p q n m n' m' (M : MX n m) (N : MX n' m') :=
forall i j, i < p -> j < q -> get M i j == get N i j.
Lemma mx_equal_at_equal n m (M N: @Classes.X mx_Graph n m) : mx_equal_at n m M N <-> M == N.
Proof. intros. unfold mx_equal_at. intuition. Qed.
End Defs.
Notation MX_ A n m := (@X (mx_Graph A) (n%nat:nat) (m%nat:nat)) (only parsing).
Notation mx_equal_ A n m := (@equal (mx_Graph A) (n%nat) (m%nat)) (only parsing).
Notation "! M" := (get M) (at level 0) : A_scope.
Notation "M == [ n , m ] N" := (@equal (mx_Graph _) n m M N) (at level 80) : A_scope.
Lemma plus_minus : forall m n, S (m+n)-n = S m.
Proof. intros. omega. Qed.
Global Opaque minus.
Lemma lt_n_1 n: ~ (S n<1)%nat.
Proof. omega. Qed.
case analysis on block matrix acesses
Ltac destruct_blocks :=
unfold mx_equal; intros; simpl;
rewrite ? plus_minus;
repeat match goal with
| |- context[S ?i - ?n] => case_eq (S i - n); intros
end.
unfold mx_equal; intros; simpl;
rewrite ? plus_minus;
repeat match goal with
| |- context[S ?i - ?n] => case_eq (S i - n); intros
end.
tactic to pointwise check matrix equality
Ltac mx_intros i j Hi Hj :=
apply mx_equal'; intros i j Hi Hj;
match type of Hi with
| i < 0%nat => elim (lt_n_O _ Hi)
| i < 1%nat => destruct i; [clear Hi | elim (lt_n_1 Hi)]
| _ => idtac
end;
match type of Hj with
| j < 0%nat => elim (lt_n_O _ Hj)
| j < 1%nat => destruct j; [clear Hj | elim (lt_n_1 Hj)]
| _ => idtac
end.
Transparent equal.
Section Props.
Context {G: Graph}.
Variable A: T.
Notation MX n m := (MX_ A n m).
Notation mx_equal n m := (mx_equal_ A n m) (only parsing).
Lemma mx_equal_compat n m: forall M N: MX n m,
M == N ->
forall i j i' j', i<n -> j<m -> i=i' -> j=j' -> !M i j == !N i' j'.
Proof. intros; subst; auto. Qed.
Definition mx_force n m (M: MX n m): MX n m := box n m (Force.id2 n m !M).
Definition mx_print n m (M: MX n m) := Force.print2 n m !M.
Definition mx_noprint n m (M: MX n m) := let _ := mx_force M in (n,m).
Lemma mx_force_id n m (M: MX n m): mx_force M == M.
Proof. repeat intro; unfold mx_force. simpl. rewrite Force.id2_id by assumption. reflexivity. Qed.
Global Instance mx_force_compat n m: Proper (mx_equal n m ==> mx_equal n m) (@mx_force n m).
Proof. intros M N H. rewrite 2 mx_force_id. assumption. Qed.
Global Instance box_compat n m:
Proper (pointwise_relation nat (pointwise_relation nat (equal A A)) ==> mx_equal n m) (box n m).
Proof. intros. intros f g H. mx_intros i j Hi Hj. apply H. Qed.
apply mx_equal'; intros i j Hi Hj;
match type of Hi with
| i < 0%nat => elim (lt_n_O _ Hi)
| i < 1%nat => destruct i; [clear Hi | elim (lt_n_1 Hi)]
| _ => idtac
end;
match type of Hj with
| j < 0%nat => elim (lt_n_O _ Hj)
| j < 1%nat => destruct j; [clear Hj | elim (lt_n_1 Hj)]
| _ => idtac
end.
Transparent equal.
Section Props.
Context {G: Graph}.
Variable A: T.
Notation MX n m := (MX_ A n m).
Notation mx_equal n m := (mx_equal_ A n m) (only parsing).
Lemma mx_equal_compat n m: forall M N: MX n m,
M == N ->
forall i j i' j', i<n -> j<m -> i=i' -> j=j' -> !M i j == !N i' j'.
Proof. intros; subst; auto. Qed.
Definition mx_force n m (M: MX n m): MX n m := box n m (Force.id2 n m !M).
Definition mx_print n m (M: MX n m) := Force.print2 n m !M.
Definition mx_noprint n m (M: MX n m) := let _ := mx_force M in (n,m).
Lemma mx_force_id n m (M: MX n m): mx_force M == M.
Proof. repeat intro; unfold mx_force. simpl. rewrite Force.id2_id by assumption. reflexivity. Qed.
Global Instance mx_force_compat n m: Proper (mx_equal n m ==> mx_equal n m) (@mx_force n m).
Proof. intros M N H. rewrite 2 mx_force_id. assumption. Qed.
Global Instance box_compat n m:
Proper (pointwise_relation nat (pointwise_relation nat (equal A A)) ==> mx_equal n m) (box n m).
Proof. intros. intros f g H. mx_intros i j Hi Hj. apply H. Qed.
sub-matrix
special case of block matrices
Section Subs.
Variables x y n m: nat.
Section Def.
Variable M: MX (x+n) (y+m).
Definition mx_sub00 := mx_sub 0 0 x y M.
Definition mx_sub01 := mx_sub 0 y x m M.
Definition mx_sub10 := mx_sub x 0 n y M.
Definition mx_sub11 := mx_sub x y n m M.
End Def.
Global Instance mx_sub00_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal x y) mx_sub00.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
Global Instance mx_sub01_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal x m) mx_sub01.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
Global Instance mx_sub10_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal n y) mx_sub10.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
Global Instance mx_sub11_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal n m) mx_sub11.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
End Subs.
Section Blocks.
Variables x y n m: nat.
Variables x y n m: nat.
Section Def.
Variable M: MX (x+n) (y+m).
Definition mx_sub00 := mx_sub 0 0 x y M.
Definition mx_sub01 := mx_sub 0 y x m M.
Definition mx_sub10 := mx_sub x 0 n y M.
Definition mx_sub11 := mx_sub x y n m M.
End Def.
Global Instance mx_sub00_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal x y) mx_sub00.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
Global Instance mx_sub01_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal x m) mx_sub01.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
Global Instance mx_sub10_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal n y) mx_sub10.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
Global Instance mx_sub11_compat: Proper (mx_equal (x+n) (y+m) ==> mx_equal n m) mx_sub11.
Proof. repeat intro. simpl. apply H; auto with omega. Qed.
End Subs.
Section Blocks.
Variables x y n m: nat.
block matrix
Definition mx_blocks
(M: MX x y)
(N: MX x m)
(P: MX n y)
(Q: MX n m): MX (x+n) (y+m)
:= box _ _
(fun i j =>
match S i-x, S j-y with
| O, O => !M i j
| S i, O => !P i j
| O, S j => !N i j
| S i, S j => !Q i j
end).
Global Instance mx_blocks_compat:
Proper (
mx_equal x y ==>
mx_equal x m ==>
mx_equal n y ==>
mx_equal n m ==>
mx_equal (x+n) (y+m))
mx_blocks.
Proof.
repeat intro. destruct_blocks; auto with omega.
Qed.
Lemma mx_decompose_blocks :
forall M: MX (x+n) (y+m),
M ==
mx_blocks
(mx_sub00 M)
(mx_sub01 M)
(mx_sub10 M)
(mx_sub11 M).
Proof.
simpl; intros. destruct_blocks; auto with omega.
Qed.
Section Proj.
Variables (a: MX x y) (b: MX x m) (c: MX n y) (d: MX n m).
Lemma mx_block_00: mx_sub00 (mx_blocks a b c d) == a.
Proof.
simpl. intros. destruct_blocks; reflexivity || omega_false.
Qed.
Lemma mx_block_01: mx_sub01 (mx_blocks a b c d) == b.
Proof.
simpl. intros. destruct_blocks; omega_false || auto with compat omega.
Qed.
Lemma mx_block_10: mx_sub10 (mx_blocks a b c d) == c.
Proof.
simpl. intros. destruct_blocks; omega_false || auto with compat omega.
Qed.
Lemma mx_block_11: mx_sub11 (mx_blocks a b c d) == d.
Proof.
simpl. intros. destruct_blocks; omega_false || auto with compat omega.
Qed.
End Proj.
Lemma mx_blocks_equal: forall (a a': MX x y) b b' c c' (d d': MX n m),
mx_blocks a b c d == mx_blocks a' b' c' d' ->
a==a' /\ b==b' /\ c==c' /\ d==d'.
Proof.
intros.
rewrite <- (mx_block_11 a b c d) at 1.
rewrite <- (mx_block_10 a b c d) at 1.
rewrite <- (mx_block_01 a b c d) at 1.
rewrite <- (mx_block_00 a b c d) at 1.
rewrite H.
rewrite mx_block_00, mx_block_01, mx_block_10, mx_block_11.
repeat split; reflexivity.
Qed.
End Blocks.
Lemma mx_blocks_degenerate_00 (a: MX 1 1) b c (d: MX 0 0):
(mx_blocks a b c d: MX 1 1) == a.
Proof.
intros [|] [|] Hi Hj; try omega_false. reflexivity.
Qed.
Lemma mx_blocks_degenerate_11 n m (a: MX 0 0) b c (d: MX n m):
(mx_blocks a b c d: MX n m) == d.
Proof.
mx_intros i j Hi Hj. reflexivity.
Qed.
(M: MX x y)
(N: MX x m)
(P: MX n y)
(Q: MX n m): MX (x+n) (y+m)
:= box _ _
(fun i j =>
match S i-x, S j-y with
| O, O => !M i j
| S i, O => !P i j
| O, S j => !N i j
| S i, S j => !Q i j
end).
Global Instance mx_blocks_compat:
Proper (
mx_equal x y ==>
mx_equal x m ==>
mx_equal n y ==>
mx_equal n m ==>
mx_equal (x+n) (y+m))
mx_blocks.
Proof.
repeat intro. destruct_blocks; auto with omega.
Qed.
Lemma mx_decompose_blocks :
forall M: MX (x+n) (y+m),
M ==
mx_blocks
(mx_sub00 M)
(mx_sub01 M)
(mx_sub10 M)
(mx_sub11 M).
Proof.
simpl; intros. destruct_blocks; auto with omega.
Qed.
Section Proj.
Variables (a: MX x y) (b: MX x m) (c: MX n y) (d: MX n m).
Lemma mx_block_00: mx_sub00 (mx_blocks a b c d) == a.
Proof.
simpl. intros. destruct_blocks; reflexivity || omega_false.
Qed.
Lemma mx_block_01: mx_sub01 (mx_blocks a b c d) == b.
Proof.
simpl. intros. destruct_blocks; omega_false || auto with compat omega.
Qed.
Lemma mx_block_10: mx_sub10 (mx_blocks a b c d) == c.
Proof.
simpl. intros. destruct_blocks; omega_false || auto with compat omega.
Qed.
Lemma mx_block_11: mx_sub11 (mx_blocks a b c d) == d.
Proof.
simpl. intros. destruct_blocks; omega_false || auto with compat omega.
Qed.
End Proj.
Lemma mx_blocks_equal: forall (a a': MX x y) b b' c c' (d d': MX n m),
mx_blocks a b c d == mx_blocks a' b' c' d' ->
a==a' /\ b==b' /\ c==c' /\ d==d'.
Proof.
intros.
rewrite <- (mx_block_11 a b c d) at 1.
rewrite <- (mx_block_10 a b c d) at 1.
rewrite <- (mx_block_01 a b c d) at 1.
rewrite <- (mx_block_00 a b c d) at 1.
rewrite H.
rewrite mx_block_00, mx_block_01, mx_block_10, mx_block_11.
repeat split; reflexivity.
Qed.
End Blocks.
Lemma mx_blocks_degenerate_00 (a: MX 1 1) b c (d: MX 0 0):
(mx_blocks a b c d: MX 1 1) == a.
Proof.
intros [|] [|] Hi Hj; try omega_false. reflexivity.
Qed.
Lemma mx_blocks_degenerate_11 n m (a: MX 0 0) b c (d: MX n m):
(mx_blocks a b c d: MX n m) == d.
Proof.
mx_intros i j Hi Hj. reflexivity.
Qed.
conversions from and to scalars
Definition mx_of_scal (x: X A A): MX 1 1 := box 1 1 (fun _ _ => x).
Definition mx_to_scal (M: MX 1 1): X A A := !M O O.
Global Instance mx_of_scal_compat:
Proper (equal A A ==> mx_equal 1 1) mx_of_scal.
Proof. repeat intro. simpl. trivial. Qed.
Global Instance mx_to_scal_compat:
Proper (mx_equal 1 1 ==> equal A A) mx_to_scal.
Proof. repeat intro. simpl. apply H; auto. Qed.
Lemma mx_to_scal_from_scal (M: MX 1 1):
M == mx_of_scal (mx_to_scal M).
Proof. mx_intros i j Hi Hj. reflexivity. Qed.
Lemma Meq_to_eq: forall a b, mx_of_scal a == mx_of_scal b -> a == b.
Proof.
intros a b H. apply (H O O); auto.
Qed.
Lemma eq_to_Meq: forall a b, mx_to_scal a == mx_to_scal b -> a == b.
Proof.
intros a b H. mx_intros i j Hi Hj. apply H.
Qed.
Definition mx_to_scal (M: MX 1 1): X A A := !M O O.
Global Instance mx_of_scal_compat:
Proper (equal A A ==> mx_equal 1 1) mx_of_scal.
Proof. repeat intro. simpl. trivial. Qed.
Global Instance mx_to_scal_compat:
Proper (mx_equal 1 1 ==> equal A A) mx_to_scal.
Proof. repeat intro. simpl. apply H; auto. Qed.
Lemma mx_to_scal_from_scal (M: MX 1 1):
M == mx_of_scal (mx_to_scal M).
Proof. mx_intros i j Hi Hj. reflexivity. Qed.
Lemma Meq_to_eq: forall a b, mx_of_scal a == mx_of_scal b -> a == b.
Proof.
intros a b H. apply (H O O); auto.
Qed.
Lemma eq_to_Meq: forall a b, mx_to_scal a == mx_to_scal b -> a == b.
Proof.
intros a b H. mx_intros i j Hi Hj. apply H.
Qed.
transposition
Definition mx_transpose n m (M : MX n m): MX m n := box m n (fun i j => !M j i).
Global Instance mx_transpose_compat n m:
Proper (mx_equal n m ==> mx_equal m n) (@mx_transpose n m).
Proof. repeat intro. simpl. apply H; trivial. Qed.
Lemma mx_transpose_blocks x y n m (a: MX x y) b c (d: MX n m):
mx_transpose (mx_blocks a b c d)
== mx_blocks (mx_transpose a) (mx_transpose c) (mx_transpose b) (mx_transpose d).
Proof.
repeat intro. simpl. destruct_blocks; reflexivity.
Qed.
End Props.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub00_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub01_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub10_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub11_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_transpose_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_force_compat: compat algebra.
Hint Extern 4 (mx_equal_ _ _ _ _ _) => apply mx_blocks_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_of_scal_compat: compat algebra.
Hint Extern 1 (equal _ _ _ _) => apply mx_to_scal_compat: compat algebra.
Hint Extern 5 (equal _ _ _ _) => apply mx_equal_compat : compat algebra.
(* Hint Resolve @equal_compat: compat algebra. *)
(* Hint Resolve *)
(* @mx_sub_compat @mx_blocks_compat *)
(* @scal_to_Mat_compat @Mat_to_scal_compat *)
(* : compat algebra. *)
Global Instance mx_transpose_compat n m:
Proper (mx_equal n m ==> mx_equal m n) (@mx_transpose n m).
Proof. repeat intro. simpl. apply H; trivial. Qed.
Lemma mx_transpose_blocks x y n m (a: MX x y) b c (d: MX n m):
mx_transpose (mx_blocks a b c d)
== mx_blocks (mx_transpose a) (mx_transpose c) (mx_transpose b) (mx_transpose d).
Proof.
repeat intro. simpl. destruct_blocks; reflexivity.
Qed.
End Props.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub00_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub01_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub10_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_sub11_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_transpose_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_force_compat: compat algebra.
Hint Extern 4 (mx_equal_ _ _ _ _ _) => apply mx_blocks_compat: compat algebra.
Hint Extern 1 (mx_equal_ _ _ _ _ _) => apply mx_of_scal_compat: compat algebra.
Hint Extern 1 (equal _ _ _ _) => apply mx_to_scal_compat: compat algebra.
Hint Extern 5 (equal _ _ _ _) => apply mx_equal_compat : compat algebra.
(* Hint Resolve @equal_compat: compat algebra. *)
(* Hint Resolve *)
(* @mx_sub_compat @mx_blocks_compat *)
(* @scal_to_Mat_compat @Mat_to_scal_compat *)
(* : compat algebra. *)