set f = Lukasiewicz_norm ;
A1: for a, b being Element of [.0,1.] holds Lukasiewicz_norm . (a,b) = Lukasiewicz_norm . (b,a)
proof
let a, b be Element of [.0,1.]; :: thesis: Lukasiewicz_norm . (a,b) = Lukasiewicz_norm . (b,a)
Lukasiewicz_norm . (a,b) = max (0,((a + b) - 1)) by LukNorm
.= Lukasiewicz_norm . (b,a) by LukNorm ;
hence Lukasiewicz_norm . (a,b) = Lukasiewicz_norm . (b,a) ; :: thesis: verum
end;
C1: for a, b, c being Element of [.0,1.] holds Lukasiewicz_norm . ((Lukasiewicz_norm . (a,b)),c) = Lukasiewicz_norm . (a,(Lukasiewicz_norm . (b,c)))
proof
let a, b, c be Element of [.0,1.]; :: thesis: Lukasiewicz_norm . ((Lukasiewicz_norm . (a,b)),c) = Lukasiewicz_norm . (a,(Lukasiewicz_norm . (b,c)))
set B = max (0,((a + b) - 1));
set C = max (0,((b + c) - 1));
G1: max (0,((a + b) - 1)) in [.0,1.] by Lemma2;
G2: max (0,((b + c) - 1)) in [.0,1.] by Lemma2;
Lukasiewicz_norm . ((Lukasiewicz_norm . (a,b)),c) = Lukasiewicz_norm . ((max (0,((a + b) - 1))),c) by LukNorm
.= max (0,(((max (0,((a + b) - 1))) + c) - 1)) by
.= max (0,((a + (max (0,((b + c) - 1)))) - 1)) by Lemma3
.= Lukasiewicz_norm . (a,(max (0,((b + c) - 1)))) by
.= Lukasiewicz_norm . (a,(Lukasiewicz_norm . (b,c))) by LukNorm ;
hence Lukasiewicz_norm . ((Lukasiewicz_norm . (a,b)),c) = Lukasiewicz_norm . (a,(Lukasiewicz_norm . (b,c))) ; :: thesis: verum
end;
D1: for a, b, c, d being Element of [.0,1.] st a <= c & b <= d holds
Lukasiewicz_norm . (a,b) <= Lukasiewicz_norm . (c,d)
proof
let a, b, c, d be Element of [.0,1.]; :: thesis: ( a <= c & b <= d implies Lukasiewicz_norm . (a,b) <= Lukasiewicz_norm . (c,d) )
assume ( a <= c & b <= d ) ; :: thesis: Lukasiewicz_norm . (a,b) <= Lukasiewicz_norm . (c,d)
then a + b <= c + d by XREAL_1:7;
then (a + b) - 1 <= (c + d) - 1 by XREAL_1:9;
then max (0,((a + b) - 1)) <= max (0,((c + d) - 1)) by XXREAL_0:26;
then max (0,((a + b) - 1)) <= Lukasiewicz_norm . (c,d) by LukNorm;
hence Lukasiewicz_norm . (a,b) <= Lukasiewicz_norm . (c,d) by LukNorm; :: thesis: verum
end;
for a being Element of [.0,1.] holds Lukasiewicz_norm . (a,1) = a
proof
let a be Element of [.0,1.]; :: thesis: Lukasiewicz_norm . (a,1) = a
T1: 1 in [.0,1.] by XXREAL_1:1;
T2: 0 <= a by XXREAL_1:1;
Lukasiewicz_norm . (a,1) = max (0,((a + 1) - 1)) by
.= a by ;
hence Lukasiewicz_norm . (a,1) = a ; :: thesis: verum
end;
hence ( Lukasiewicz_norm is commutative & Lukasiewicz_norm is associative & Lukasiewicz_norm is monotonic & Lukasiewicz_norm is with-1-identity ) by ; :: thesis: verum