We consider a Set U , given with a commutative-associative operator op , and a congruence cong ; we show permutation lemmas
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Section
Axiomatisation.
Variable
U: Set.
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
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).
Remark. we do not need: .
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Lemma
cong_congr :
(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.
Qed
.
Lemma
comm_right : (x,y,z:U)(cong (op x (op y z)) (op x (op z y))).
Proof
.
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)).
Proof
.
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)).
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.
Qed
.
Lemma
perm_left : (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.
Qed
.
Lemma
op_rotate : (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.
Qed
.
Lemma
twist : (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.
Qed
.
End
Axiomatisation.