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Diffstat (limited to 'theories/Sets/Permut.v')
| -rwxr-xr-x | theories/Sets/Permut.v | 86 |
1 files changed, 43 insertions, 43 deletions
diff --git a/theories/Sets/Permut.v b/theories/Sets/Permut.v index 03a8b7428..c3a1da01c 100755 --- a/theories/Sets/Permut.v +++ b/theories/Sets/Permut.v @@ -15,77 +15,77 @@ Section Axiomatisation. -Variable U: Set. +Variable U : Set. -Variable op: U -> U -> U. +Variable op : U -> U -> U. Variable cong : U -> U -> Prop. -Hypothesis op_comm : (x,y:U)(cong (op x y) (op y x)). -Hypothesis op_ass : (x,y,z:U)(cong (op (op x y) z) (op x (op y z))). +Hypothesis op_comm : forall x y:U, cong (op x y) (op y x). +Hypothesis op_ass : forall x y z:U, cong (op (op x y) z) (op x (op y z)). -Hypothesis cong_left : (x,y,z:U)(cong x y)->(cong (op x z) (op y z)). -Hypothesis cong_right : (x,y,z:U)(cong x y)->(cong (op z x) (op z y)). -Hypothesis cong_trans : (x,y,z:U)(cong x y)->(cong y z)->(cong x z). -Hypothesis cong_sym : (x,y:U)(cong x y)->(cong y x). +Hypothesis cong_left : forall x y z:U, cong x y -> cong (op x z) (op y z). +Hypothesis cong_right : forall x y z:U, cong x y -> cong (op z x) (op z y). +Hypothesis cong_trans : forall x y z:U, cong x y -> cong y z -> cong x z. +Hypothesis cong_sym : forall x y:U, cong x y -> cong y x. (** Remark. we do not need: [Hypothesis cong_refl : (x:U)(cong x x)]. *) Lemma cong_congr : - (x,y,z,t:U)(cong x y)->(cong z t)->(cong (op x z) (op y t)). + forall x y z t:U, cong x y -> cong z t -> cong (op x z) (op y t). Proof. -Intros; Apply cong_trans with (op y z). -Apply cong_left; Trivial. -Apply cong_right; Trivial. +intros; apply cong_trans with (op y z). +apply cong_left; trivial. +apply cong_right; trivial. Qed. -Lemma comm_right : (x,y,z:U)(cong (op x (op y z)) (op x (op z y))). +Lemma comm_right : forall x y z:U, cong (op x (op y z)) (op x (op z y)). Proof. -Intros; Apply cong_right; Apply op_comm. +intros; apply cong_right; apply op_comm. Qed. -Lemma comm_left : (x,y,z:U)(cong (op (op x y) z) (op (op y x) z)). +Lemma comm_left : forall x y z:U, cong (op (op x y) z) (op (op y x) z). Proof. -Intros; Apply cong_left; Apply op_comm. +intros; apply cong_left; apply op_comm. Qed. -Lemma perm_right : (x,y,z:U)(cong (op (op x y) z) (op (op x z) y)). +Lemma perm_right : forall x y z:U, cong (op (op x y) z) (op (op x z) y). Proof. -Intros. -Apply cong_trans with (op x (op y z)). -Apply op_ass. -Apply cong_trans with (op x (op z y)). -Apply cong_right; Apply op_comm. -Apply cong_sym; Apply op_ass. +intros. +apply cong_trans with (op x (op y z)). +apply op_ass. +apply cong_trans with (op x (op z y)). +apply cong_right; apply op_comm. +apply cong_sym; apply op_ass. Qed. -Lemma perm_left : (x,y,z:U)(cong (op x (op y z)) (op y (op x z))). +Lemma perm_left : forall x y z:U, cong (op x (op y z)) (op y (op x z)). Proof. -Intros. -Apply cong_trans with (op (op x y) z). -Apply cong_sym; Apply op_ass. -Apply cong_trans with (op (op y x) z). -Apply cong_left; Apply op_comm. -Apply op_ass. +intros. +apply cong_trans with (op (op x y) z). +apply cong_sym; apply op_ass. +apply cong_trans with (op (op y x) z). +apply cong_left; apply op_comm. +apply op_ass. Qed. -Lemma op_rotate : (x,y,z,t:U)(cong (op x (op y z)) (op z (op x y))). +Lemma op_rotate : forall x y z t:U, cong (op x (op y z)) (op z (op x y)). Proof. -Intros; Apply cong_trans with (op (op x y) z). -Apply cong_sym; Apply op_ass. -Apply op_comm. +intros; apply cong_trans with (op (op x y) z). +apply cong_sym; apply op_ass. +apply op_comm. Qed. (* Needed for treesort ... *) -Lemma twist : (x,y,z,t:U) - (cong (op x (op (op y z) t)) (op (op y (op x t)) z)). +Lemma twist : + forall x y z t:U, cong (op x (op (op y z) t)) (op (op y (op x t)) z). Proof. -Intros. -Apply cong_trans with (op x (op (op y t) z)). -Apply cong_right; Apply perm_right. -Apply cong_trans with (op (op x (op y t)) z). -Apply cong_sym; Apply op_ass. -Apply cong_left; Apply perm_left. +intros. +apply cong_trans with (op x (op (op y t) z)). +apply cong_right; apply perm_right. +apply cong_trans with (op (op x (op y t)) z). +apply cong_sym; apply op_ass. +apply cong_left; apply perm_left. Qed. -End Axiomatisation. +End Axiomatisation.
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