:: Fundamental Properties of Fuzzy Implications
:: by Adam Grabowski
::
:: Received September 29, 2018
:: Copyright (c) 2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, XXREAL_1, REAL_1, XXREAL_0, FUZIMPL1, SUBSET_1, CARD_1,
TARSKI, ARYTM_1, ARYTM_3, POWER, FUNCT_1, XBOOLE_0, RELAT_1, BINOP_1,
FUZIMPL2, PREPOWER, LIMFUNC1, FDIFF_1, ORDINAL2, JGRAPH_2, RCOMP_1;
notations TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, XXREAL_0, NUMBERS, BINOP_1,
VALUED_0, VALUED_1, XREAL_0, RELAT_1, FUNCT_1, RCOMP_1, PARTFUN1,
FUZNORM1, FCONT_1, PREPOWER, POWER, FDIFF_1, FUZIMPL1, LIMFUNC1,
TAYLOR_1;
constructors REAL_1, SEQ_4, TOPMETR, PREPOWER, POWER, FUZNORM1, FUZIMPL1,
TAYLOR_1, LIMFUNC1, FDIFF_1, FCONT_1;
registrations XBOOLE_0, RELSET_1, XXREAL_0, XREAL_0, MEMBERED, ORDINAL1,
FUZNORM1, VALUED_0, RCOMP_1, NUMBERS, FCONT_3, TAYLOR_1;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
definitions FUZIMPL1;
equalities SUBSET_1, XCMPLX_0, LIMFUNC1;
expansions TARSKI, FUZIMPL1;
theorems FUNCT_1, FUNCT_2, RELAT_1, XREAL_0, POWER, XCMPLX_1, XREAL_1,
XXREAL_0, XXREAL_1, FUZNORM1, FUZIMPL1, TAYLOR_1, PREPOWER, ROLLE,
FDIFF_1, INTEGRA7, VALUED_1, XBOOLE_1, FCONT_1, RCOMP_1;
begin :: Preliminaries
notation
synonym I_LK for Lukasiewicz_implication;
synonym I_GD for Goedel_implication;
synonym I_RC for Reichenbach_implication;
synonym I_KD for Kleene-Dienes_implication;
synonym I_GG for Goguen_implication;
synonym I_RS for Rescher_implication;
synonym I_YG for Yager_implication;
synonym I_WB for Weber_implication;
synonym I_FD for Fodor_implication;
end;
reserve x,y for Element of [.0,1.];
Lemma00:
x * y + (1 - x) * 1 in [.y,1.]
proof
0 <= x & x <= 1 & y <= 1 by XXREAL_1:1; then
A1: y <= (1-x)*1+x*y by XREAL_1:173;
0 <= x & x <= 1 & y <= 1 by XXREAL_1:1; then
(1 - x) * 1 + x * y <= 1 by XREAL_1:174;
hence thesis by A1,XXREAL_1:1;
end;
LemmaHalf:
(1 / 2) to_power (1 / 2) <> 1
proof
(1 / 2) to_power (1 / 2) < 1 to_power (1 / 2) by POWER:37;
hence thesis by POWER:26;
end;
theorem
#R 1 = AffineMap (1,0) | right_open_halfline 0
proof
set f = #R 1;
set g = AffineMap (1,0) | right_open_halfline 0;
A1: dom f = right_open_halfline 0 by TAYLOR_1:def 4;
dom AffineMap (1,0) = REAL by FUNCT_2:def 1; then
BA: dom f = dom g by A1,RELAT_1:62;
reconsider p = 1 as Real;
for x being object st x in dom f holds f.x = g.x
proof
let x be object;
assume x in dom f; then
reconsider xx = x as Element of right_open_halfline 0 by TAYLOR_1:def 4;
A2: xx > 0 by XXREAL_1:4;
f.x = xx #R p by TAYLOR_1:def 4 .= 1 * xx + 0 by PREPOWER:72,A2
.= (AffineMap (1,0)).xx by FCONT_1:def 4
.= g.x by FUNCT_1:49;
hence thesis;
end;
hence thesis by BA,FUNCT_1:2;
end;
theorem LemmaAffine:
for a,b being Real holds
AffineMap (a,b) is_differentiable_on REAL &
for x being Real holds diff (AffineMap (a,b), x) = a
proof
let a,b be Real;
reconsider Z = [#]REAL as open Subset of REAL;
A1: dom AffineMap (a,b) = REAL by FUNCT_2:def 1;
A2: for x being Real st x in Z holds
AffineMap (a,b).x = a * x + b by FCONT_1:def 4;
hence
Ka: AffineMap (a,b) is_differentiable_on REAL by A1,FDIFF_1:23;
let x be Real;
AC: x in Z by XREAL_0:def 1;
((AffineMap(a,b))`|Z).x = a by A1,A2,FDIFF_1:23,XREAL_0:def 1;
hence thesis by AC,FDIFF_1:def 7,Ka;
end;
theorem
0 < x < 1 & 0 < y < 1 implies
( #R x + AffineMap (-x, x-1)) | ].0,1.[ is increasing
proof
assume ZZ: 0 < x < 1 & 0 < y < 1;
set f1 = #R x;
set f2 = AffineMap (-x, x-1);
reconsider Y = ].0,1.[ as open Subset of REAL;
set f = f1 + f2;
set A = right_open_halfline 0;
G0: A c= dom f1 by TAYLOR_1:def 4;
dom f1 = ].0,+infty.[ by TAYLOR_1:def 4; then
G1: dom (f1 | A) = A by RELAT_1:62;
K2: f2 is_differentiable_on A by FDIFF_1:26,LemmaAffine;
TR: (f1 | A) | A = f1 | A by RELAT_1:72;
f1 | A is_differentiable_on A
proof
for z being Real st z in A holds
(f1 | A) | A is_differentiable_in z
proof
let z be Real;
assume
W0: z in A; then
z > 0 by XXREAL_1:235; then
consider N being Neighbourhood of z such that ::: lemma!
R1: N c= dom f1 & ex L being LinearFunc, R being RestFunc st
for x being Real st x in N holds f1.x - f1.z = L.(x-z) + R.(x-z)
by FDIFF_1:def 4,TAYLOR_1:21;
consider L being LinearFunc, R being RestFunc such that
R2: for x being Real st x in N holds
f1.x - f1.z = L.(x-z) + R.(x-z) by R1;
set V = A;
consider V1 being Neighbourhood of z such that
Wa: V1 c= V by RCOMP_1:18,W0;
consider N1 being Neighbourhood of z such that
FX: N1 c= N & N1 c= V1 by RCOMP_1:17;
R4: N1 c= dom (f1 | V) by FX,Wa,G1;
for x being Real st x in N1 holds
(f1 | V).x - (f1 | V).z = L.(x-z) + R.(x-z)
proof
let x be Real;
assume
F1: x in N1; then
f1.x = (f1 | V).x & f1.z = (f1 | V).z by FUNCT_1:49,Wa,FX,W0;
hence thesis by F1,R2,FX;
end;
hence thesis by TR,FDIFF_1:def 4,R4;
end;
hence thesis by G1,FDIFF_1:def 6;
end; then
G2: f1 is_differentiable_on A by INTEGRA7:5; then
g2: f1 is_differentiable_on Y by FDIFF_1:26,XXREAL_1:247;
k2: f2 is_differentiable_on Y by FDIFF_1:26,LemmaAffine;
dom f2 = REAL by FUNCT_2:def 1; then
A c= dom f1 /\ dom f2 by G0,XBOOLE_1:19; then
A c= dom (f1 + f2) by VALUED_1:def 1; then
f is_differentiable_on A by K2,FDIFF_1:18,G2; then
ga: f is_differentiable_on Y by FDIFF_1:26,XXREAL_1:247;
az: dom f = dom f1 /\ dom f2 by VALUED_1:def 1
.= right_open_halfline 0 /\ dom f2 by TAYLOR_1:def 4
.= right_open_halfline 0 /\ REAL by FUNCT_2:def 1
.= right_open_halfline 0 by XBOOLE_1:28;
for y being Real st y in Y holds 0 < diff(f,y)
proof
let y be Real;
assume
Sa: y in Y; then
Sb: 0 < y < 1 by XXREAL_1:4;
f1 is_differentiable_in y & f2 is_differentiable_in y
by k2,g2,Sa,FDIFF_1:9; then
H1: diff (f,y) = diff(f1,y) + diff(f2,y) by FDIFF_1:13;
f1 is_differentiable_in y & diff(f1,y) = x
* (y #R (x-1)) by TAYLOR_1:21,Sb; then
diff (f1,y) = x * y to_power (x - 1) by POWER:def 2,Sb; then
H3: diff (f,y) = x * y to_power (x - 1) - x by H1,LemmaAffine
.= x * (y to_power (x - 1) - 1);
x - 1 < 1 - 1 by ZZ,XREAL_1:9; then
y to_power (x - 1) > y to_power 0 by POWER:40,Sb; then
y to_power (x - 1) > 1 by POWER:24; then
y to_power (x - 1) - 1 > 1 - 1 by XREAL_1:9;
hence thesis by XREAL_1:129,H3,ZZ;
end;
hence thesis by ROLLE:9,ga,az,XXREAL_1:247;
end;
theorem
for u being Real st u in ].0,1.] holds
( #R x + AffineMap (-x, x-1)).u = u to_power x - 1 + x - x * u
proof
set f1 = #R x;
set f2 = AffineMap (-x, x-1);
reconsider Y = ].0,1.[ as open Subset of REAL;
set f = f1 + f2;
set A = right_open_halfline 0;
BX: dom f = dom f1 /\ dom f2 by VALUED_1:def 1
.= right_open_halfline 0 /\ dom f2 by TAYLOR_1:def 4
.= right_open_halfline 0 /\ REAL by FUNCT_2:def 1
.= right_open_halfline 0 by XBOOLE_1:28;
let u be Real;
assume
S1: u in ].0,1.]; then
ZE: u > 0 by XXREAL_1:2;
1 < +infty by XXREAL_0:9,XREAL_0:def 1; then
h0: ].0,1.] c= ].0,+infty.[ by XXREAL_1:49; then
f.u = f1.u + f2.u by S1,BX,VALUED_1:def 1
.= (u #R x) + f2.u by TAYLOR_1:def 4,h0,S1
.= (u to_power x) + f2.u by ZE,POWER:def 2
.= (u to_power x) + ((-x)*u + (x-1)) by FCONT_1:def 4
.= (u to_power x) + (-x*u + x-1);
hence thesis;
end;
begin :: On the Ordering of Fuzzy Implications
theorem Lemma01:
(x <= y implies I_LK.(x,y) = 1) &
(x > y implies I_LK.(x,y) = 1 - x + y)
proof
thus x <= y implies I_LK.(x,y) = 1
proof
assume x <= y; then
1 - x >= 1 - y by XREAL_1:10; then
a1: 1 - x + y >= 1 - y + y by XREAL_1:6;
I_LK.(x,y) = min (1, 1 - x + y) by FUZIMPL1:def 14
.= 1 by a1,XXREAL_0:def 9;
hence thesis;
end;
assume x > y; then
1 - x <= 1 - y by XREAL_1:10; then
a1: 1 - x + y <= 1 - y + y by XREAL_1:6;
I_LK.(x,y) = min (1, 1 - x + y) by FUZIMPL1:def 14
.= 1 - x + y by a1,XXREAL_0:def 9;
hence thesis;
end;
theorem
(x = 0 implies I_GG.(x,y) = 1) &
(x > 0 implies I_GG.(x,y) = min (1, y / x))
proof
thus x = 0 implies I_GG.(x,y) = 1
proof
assume x = 0; then
x <= y by XXREAL_1:1;
hence thesis by FUZIMPL1:def 19;
end;
assume
S0: x > 0;
A1: y >= 0 by XXREAL_1:1;
per cases;
suppose
A2: x <= y; then
S1: I_GG.(x,y) = 1 by FUZIMPL1:def 19;
y / x >= 1 by A2,S0,XREAL_1:181;
hence thesis by S1,XXREAL_0:def 9;
end;
suppose
S2: x > y; then
I_GG.(x,y) = y / x by FUZIMPL1:def 19;
hence thesis by S2,A1,XREAL_1:183,XXREAL_0:def 9;
end;
end;
Lemma03:
max (1 - x, y) in [.0,1.]
proof
1 - x in [.0,1.] by FUZNORM1:7;
hence thesis by FUZNORM1:2;
end;
Lemma11:
x <> 0 implies y <= y / x
proof
assume x <> 0; then
x <= 1 & y >= 0 & x > 0 by XXREAL_1:1; then
y / 1 <= y / x by XREAL_1:118;
hence thesis;
end;
Lemacik1:
x > y implies 1 - x + y >= y / x
proof
assume
S0: x > y;
1 - x in [.0,1.] by FUZNORM1:7; then
1 - x >= 0 by XXREAL_1:1; then
(1 - x) * x >= (1 - x) * y by S0,XREAL_1:64; then
a1: (1 - x) * x + x * y >= (1 - x) * y + x * y by XREAL_1:6;
S2: x > 0 by S0,XXREAL_1:1; then
x * (1 - x + y) / x >= y / x by a1,XREAL_1:72;
hence thesis by XCMPLX_1:89,S2;
end;
Lemma111:
I_KD <= I_RC
proof
set f = I_KD;
set g = I_RC;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
x * y in [.0,1.] by FUZNORM1:3; then
x * y >= 0 by XXREAL_1:1; then
A1: 1 - x + 0 <= 1 - x + x * y by XREAL_1:6;
x * y + (1 - x) * 1 in [.y,1.] by Lemma00; then
y <= 1 - x + x * y by XXREAL_1:1; then
max (1 - x, y) <= 1 - x + x * y by A1,XXREAL_0:28; then
f.(x,y) <= 1 - x + x * y by FUZIMPL1:def 18;
hence thesis by FUZIMPL1:def 17;
end;
hence thesis by FUZNORM1:def 16;
end;
Lemma112:
I_RC <= I_LK
proof
set f = I_RC;
set g = I_LK;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
1 - x + x * y in [.0,1.] by FUZIMPL1:3; then
A1: 1 - x + x * y <= 1 by XXREAL_1:1;
A2: x >= 0 & y >= 0 by XXREAL_1:1;
x <= 1 by XXREAL_1:1; then
x * y <= 1 * y by A2,XREAL_1:64; then
1 - x + x * y <= 1 - x + y by XREAL_1:6; then
1 - x + x * y <= min (1,1 - x + y) by A1,XXREAL_0:20; then
f.(x,y) <= min (1,1 - x + y) by FUZIMPL1:def 17;
hence thesis by FUZIMPL1:def 14;
end;
hence thesis by FUZNORM1:def 16;
end;
Lemma113:
I_LK <= I_WB
proof
set f = I_LK;
set g = I_WB;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
A1: x <= 1 & y <= 1 by XXREAL_1:1;
per cases;
suppose
A2: x = 1; then
B1: g.(x,y) = y by FUZIMPL1:def 22;
min (1, 1 - x + y) = y by A1,A2,XXREAL_0:def 9;
hence thesis by B1,FUZIMPL1:def 14;
end;
suppose x <> 1; then
x < 1 by A1,XXREAL_0:1; then
g.(x,y) = 1 by FUZIMPL1:def 22;
hence thesis by XXREAL_1:1;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
theorem :: Example 1.1.6 (1.1) p. 3
I_KD <= I_RC <= I_LK <= I_WB by Lemma111,Lemma112,Lemma113;
Lemma121:
I_RS <= I_GD
proof
set f = I_RS;
set g = I_GD;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose
G0: x <= y; then
f.(x,y) = 1 by FUZIMPL1:def 20;
hence thesis by G0,FUZIMPL1:def 16;
end;
suppose x > y; then
f.(x,y) = 0 by FUZIMPL1:def 20;
hence thesis by XXREAL_1:1;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
Lemma122:
I_GD <= I_GG
proof
set f = I_GD;
set g = I_GG;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose
B0: x <= y; then
f.(x,y) = 1 by FUZIMPL1:def 16;
hence thesis by B0,FUZIMPL1:def 19;
end;
suppose
C1: x > y; then
C2: f.(x,y) = y by FUZIMPL1:def 16;
C3: g.(x,y) = y / x by FUZIMPL1:def 19,C1;
y >= 0 by XXREAL_1:1;
hence thesis by C2,C3,Lemma11,C1;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
Lemma123:
I_GG <= I_LK
proof
set f = I_GG;
set g = I_LK;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose
C1: x <= y; then
g.(x,y) = 1 by Lemma01;
hence thesis by C1,FUZIMPL1:def 19;
end;
suppose
C1: x > y; then
C2: f.(x,y) = y / x by FUZIMPL1:def 19;
g.(x,y) = 1 - x + y by C1,Lemma01;
hence thesis by C2,C1,Lemacik1;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
theorem :: Example 1.1.6 (1.2) p. 3
I_RS <= I_GD <= I_GG <= I_LK <= I_WB
by Lemma121,Lemma122,Lemma123,Lemma113;
Lemma132:
I_RC <= I_LK
proof
set f = I_RC;
set g = I_LK;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose x <= y; then
g.(x,y) = 1 by Lemma01;
hence thesis by XXREAL_1:1;
end;
suppose x > y; then
C2: g.(x,y) = 1 - x + y by Lemma01;
C3: f.(x,y) = 1 - x + x * y by FUZIMPL1:def 17;
y >= 0 & x <= 1 by XXREAL_1:1; then
x * y <= 1 * y by XREAL_1:64;
hence thesis by C2,C3,XREAL_1:6;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
theorem :: Example 1.1.6 (1.3) p. 3
:::I_YG <=
I_RC <= I_LK <= I_WB by Lemma132,Lemma113;
Lemma141:
I_KD <= I_FD
proof
set f = I_KD;
set g = I_FD;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose x <= y; then
g.(x,y) = 1 by FUZIMPL1:def 23;
hence thesis by XXREAL_1:1;
end;
suppose x > y; then
g.(x,y) = max (1 - x, y) by FUZIMPL1:def 23;
hence thesis by FUZIMPL1:def 18;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
Lemma142:
I_FD <= I_LK
proof
set f = I_FD;
set g = I_LK;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose
B1: x <= y; then
f.(x,y) = 1 by FUZIMPL1:def 23;
hence thesis by B1,Lemma01;
end;
suppose x > y; then
C3: f.(x,y) = max (1 - x, y) by FUZIMPL1:def 23;
C2: g.(x,y) = min (1, 1 - x + y) by FUZIMPL1:def 14;
max (1 - x, y) in [.0,1.] by Lemma03; then
C1: max (1 - x, y) <= 1 by XXREAL_1:1;
y >= 0 & x >= 0 by XXREAL_1:1; then
D1: 1 - x + 0 <= 1 - x + y by XREAL_1:6;
1 - x in [.0,1.] by FUZNORM1:7; then
1 - x >= 0 by XXREAL_1:1; then
0 + y <= 1 - x + y by XREAL_1:6; then
max (1 - x, y) <= 1 - x + y by D1,XXREAL_0:28;
hence thesis by C1,C2,C3,XXREAL_0:20;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
theorem :: Example 1.1.6 (1.4) p. 3
I_KD <= I_FD <= I_LK <= I_WB by Lemma141,Lemma142,Lemma113;
Lemma151:
I_RS <= I_GD
proof
set f = I_RS;
set g = I_GD;
for x,y being Element of [.0,1.] holds f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose x <= y; then
g.(x,y) = 1 by FUZIMPL1:def 16;
hence thesis by XXREAL_1:1;
end;
suppose x > y; then
f.(x,y) = 0 by FUZIMPL1:def 20;
hence thesis by XXREAL_1:1;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
Lemma152:
I_GD <= I_FD
proof
set f = I_GD;
set g = I_FD;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose
A0: x <= y; then
f.(x,y) = 1 by FUZIMPL1:def 16;
hence thesis by A0,FUZIMPL1:def 23;
end;
suppose
A2: x > y; then
A3: g.(x,y) = max(1 - x, y) by FUZIMPL1:def 23;
f.(x,y) = y by A2,FUZIMPL1:def 16;
hence thesis by A3,XXREAL_0:25;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
Lemma153:
I_FD <= I_LK
proof
set f = I_FD;
set g = I_LK;
for x,y being Element of [.0,1.] holds
f.(x,y) <= g.(x,y)
proof
let x,y be Element of [.0,1.];
per cases;
suppose
A0: x <= y; then
g.(x,y) = 1 by Lemma01;
hence thesis by A0,FUZIMPL1:def 23;
end;
suppose
A2: x > y; then
A3: f.(x,y) = max(1 - x, y) by FUZIMPL1:def 23;
A4: g.(x,y) = 1 - x + y by A2,Lemma01;
0 <= y by XXREAL_1:1; then
A5: 1 - x + 0 <= 1 - x + y by XREAL_1:6;
1 - x in [.0,1.] by FUZNORM1:7; then
0 <= 1 - x by XXREAL_1:1; then
0 + y <= 1 - x + y by XREAL_1:6;
hence thesis by A3,XXREAL_0:28,A4,A5;
end;
end;
hence thesis by FUZNORM1:def 16;
end;
theorem :: Example 1.1.6 (1.5) p. 3
I_RS <= I_GD <= I_FD <= I_LK <= I_WB by Lemma151,Lemma152,Lemma153,Lemma113;
begin :: Additional Properties of Fuzzy Implications
definition
let I be BinOp of [.0,1.];
attr I is satisfying_(NP) means
for y being Element of [.0,1.] holds I.(1,y) = y;
attr I is satisfying_(EP) means
for x,y,z being Element of [.0,1.] holds I.(x,I.(y,z)) = I.(y,I.(x,z));
attr I is satisfying_(IP) means
for x being Element of [.0,1.] holds I.(x,x) = 1;
attr I is satisfying_(OP) means
for x,y being Element of [.0,1.] holds
I.(x,y) = 1 iff x <= y;
end;
reserve I for BinOp of [.0,1.];
notation
let I be BinOp of [.0,1.];
synonym I is satisfying_(NC) for I is 01-dominant;
synonym I is satisfying_(I1) for I is decreasing_on_1st;
synonym I is satisfying_(I2) for I is increasing_on_2nd;
synonym I is satisfying_(I3) for I is 00-dominant;
synonym I is satisfying_(I4) for I is 11-dominant;
synonym I is satisfying_(I5) for I is 10-weak;
end;
begin
theorem LemmaXA:
I is satisfying_(LB) implies I is satisfying_(I3) satisfying_(NC)
proof
assume
a1: I is satisfying_(LB);
0 in [.0,1.] by XXREAL_1:1;
hence I is satisfying_(I3) by a1;
1 in [.0,1.] by XXREAL_1:1;
hence thesis by a1;
end;
registration
cluster satisfying_(LB) -> satisfying_(I3) satisfying_(NC)
for BinOp of [.0,1.];
coherence by LemmaXA;
end;
theorem LemmaVA:
I is satisfying_(RB) implies I is satisfying_(I4) satisfying_(NC)
proof
assume
a1: I is satisfying_(RB);
1 in [.0,1.] by XXREAL_1:1;
hence I is satisfying_(I4) by a1;
0 in [.0,1.] by XXREAL_1:1;
hence thesis by a1;
end;
registration
cluster satisfying_(RB) -> satisfying_(I4) satisfying_(NC)
for BinOp of [.0,1.];
coherence by LemmaVA;
end;
theorem LemmaWA:
I is satisfying_(NP) implies I is satisfying_(I4) satisfying_(I5)
proof
assume
a1: I is satisfying_(NP);
1 in [.0,1.] by XXREAL_1:1;
hence I is satisfying_(I4) by a1;
0 in [.0,1.] by XXREAL_1:1;
hence thesis by a1;
end;
registration
cluster satisfying_(NP) -> satisfying_(I4) satisfying_(I5)
for BinOp of [.0,1.];
coherence by LemmaWA;
end;
theorem LemmaZA:
I is satisfying_(IP) implies I is satisfying_(I3) satisfying_(I4)
proof
assume
a1: I is satisfying_(IP);
A2: 0 in [.0,1.] by XXREAL_1:1;
1 in [.0,1.] by XXREAL_1:1;
hence thesis by a1,A2;
end;
registration
cluster satisfying_(IP) -> satisfying_(I3) satisfying_(I4)
for BinOp of [.0,1.];
coherence by LemmaZA;
end;
theorem LemmaAA:
I is satisfying_(OP) implies I is satisfying_(I3) satisfying_(I4)
satisfying_(NC) satisfying_(LB) satisfying_(RB) satisfying_(IP)
proof
assume
a1: I is satisfying_(OP);
A2: 0 in [.0,1.] by XXREAL_1:1;
A3: 1 in [.0,1.] by XXREAL_1:1;
T3: I is satisfying_(LB)
proof
let y be Element of [.0,1.];
0 <= y by XXREAL_1:1;
hence thesis by a1,A2;
end;
I is satisfying_(RB)
proof
let x be Element of [.0,1.];
x <= 1 by XXREAL_1:1;
hence thesis by a1,A3;
end;
hence thesis by T3,a1;
end;
registration
cluster satisfying_(OP) -> satisfying_(I3) satisfying_(I4)
satisfying_(NC) satisfying_(LB) satisfying_(RB) satisfying_(IP)
for BinOp of [.0,1.];
coherence by LemmaAA;
end;
theorem LemmaAB:
I is satisfying_(EP) satisfying_(OP) implies
I is satisfying_(I1) satisfying_(I3) satisfying_(I4) satisfying_(I5)
satisfying_(LB) satisfying_(RB) satisfying_(NC) satisfying_(NP)
satisfying_(IP)
proof
assume
AA: I is satisfying_(EP) satisfying_(OP);
tt: for x1,x2,y being Element of [.0,1.] st
x1 <= x2 holds I.(x1,y) >= I.(x2,y)
proof
let x1,x2,y be Element of [.0,1.];
assume
Z1: x1 <= x2;
I.(x2,I.(I.(x2,y),y)) = I.(I.(x2,y), I.(x2,y)) by AA
.= 1 by AA; then
x2 <= I.(I.(x2,y),y) by AA; then
1 = I.(x1,I.(I.(x2,y),y)) by AA,Z1,XXREAL_0:2
.= I.(I.(x2,y),I.(x1,y)) by AA;
hence thesis by AA;
end;
for y being Element of [.0,1.] holds I.(1,y) = y
proof
let y be Element of [.0,1.];
S1: 1 in [.0,1.] by XXREAL_1:1;
reconsider i = 1 as Element of [.0,1.] by XXREAL_1:1;
S2: I.(i,y) in [.0,1.];
I.(y,I.(1,y)) = I.(1,I.(y,y)) by S1,AA
.= I.(1,1) by AA
.= 1 by S1,AA; then
Z1: y <= I.(1,y) by AA,S2;
I.(i,I.(I.(i,y),y)) = I.(I.(i,y),I.(i,y)) by AA
.= 1 by AA; then
Z2: 1 <= I.(I.(i,y),y) by AA;
1 >= I.(I.(i,y),y) by XXREAL_1:1; then
I.(I.(i,y),y) = 1 by Z2,XXREAL_0:1; then
I.(1,y) <= y by AA;
hence thesis by Z1,XXREAL_0:1;
end; then
I is satisfying_(NP);
hence thesis by AA,tt;
end;
registration
cluster satisfying_(EP) satisfying_(OP) -> satisfying_(I1)
satisfying_(I5) satisfying_(NP) for BinOp of [.0,1.];
coherence by LemmaAB;
end;
begin
registration
cluster I_LK -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
coherence
proof
set I = I_LK;
A1: I_LK is satisfying_(OP)
proof
let x,y be Element of [.0,1.];
I.(x,y) = 1 implies x <= y
proof
assume I.(x,y) = 1; then
min (1, 1 - x + y) = 1 by FUZIMPL1:def 14; then
1 - x + y - 1 >= 1 - 1 by XREAL_1:9,XXREAL_0:def 9; then
y - x + x >= 0 + x by XREAL_1:6;
hence thesis;
end;
hence thesis by Lemma01;
end;
for x,y,z being Element of [.0,1.] holds I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
U3: 1 in [.0,1.] by XXREAL_1:1;
U2: x <= 1 & y <= 1 by XXREAL_1:1;
per cases;
suppose
U1: y <= z & x <= z; then
O1: I.(x,I.(y,z)) = I.(x,1) by A1
.= 1 by U2,U3,A1;
I.(y,I.(x,z)) = I.(y,1) by A1,U1
.= 1 by U2,A1,U3;
hence thesis by O1;
end;
suppose
y > z & x > z; then
PO: I.(x,z) = 1 - x + z & I.(y,z) = 1 - y + z by Lemma01;
per cases;
suppose
P1: x > 1 - y + z; then
x + y > 1 + z - y + y by XREAL_1:6; then
o9: x + y - x > 1 + z - x by XREAL_1:9;
O1: I.(x,I.(y,z)) = 1 - x + (1 - y + z) by Lemma01,P1,PO;
I.(y,I.(x,z)) = 1 - y + (1 - x + z) by Lemma01,o9,PO;
hence thesis by O1;
end;
suppose
S1: x <= 1 - y + z; then
x + y <= 1 - y + z + y by XREAL_1:6; then
O9: x + y - x <= 1 + z - x by XREAL_1:9;
I.(x,I.(y,z)) = 1 by Lemma01,PO,S1;
hence thesis by PO,O9,Lemma01;
end;
end;
suppose
VV: y <= z & x > z; then
O3: I.(x,z) = 1 - x + z by Lemma01;
1 - x in [.0,1.] by FUZNORM1:7; then
1 - x >= 0 by XXREAL_1:1; then
o2: 1 - x + z >= 0 + z by XREAL_1:6;
I.(x,I.(y,z)) = I.(x,1) by VV,Lemma01
.= 1 by Lemma01,U2,U3;
hence thesis by o2,Lemma01,O3,VV,XXREAL_0:2;
end;
suppose
VV: y > z & x <= z; then
O8: I.(y,z) = 1 - y + z by Lemma01;
1 - y in [.0,1.] by FUZNORM1:7; then
0 <= 1 - y by XXREAL_1:1; then
o2: 0 + z <= 1 - y + z by XREAL_1:6;
I.(y,I.(x,z)) = I.(y,1) by VV,Lemma01
.= 1 by Lemma01,U2,U3;
hence thesis by o2,Lemma01,O8,VV,XXREAL_0:2;
end;
end; then
I_LK is satisfying_(EP);
hence thesis by A1;
end;
cluster I_GD -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
coherence
proof
set I = I_GD;
A1: I is satisfying_(OP)
proof
let x,y be Element of [.0,1.];
I.(x,y) = 1 implies x <= y
proof
assume
T2: I.(x,y) = 1;
assume
T3: x > y; then
I.(x,y) = y by FUZIMPL1:def 16;
hence thesis by T3,XXREAL_1:1,T2;
end;
hence thesis by FUZIMPL1:def 16;
end;
for x,y,z being Element of [.0,1.] holds I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x, y, z be Element of [.0,1.];
AA: 1 in [.0,1.] by XXREAL_1:1;
per cases;
suppose
YY: y <= z & x <= z; then
Y1: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 16
.= 1 by FUZIMPL1:def 16,AA;
I.(y,I.(x,z)) = I.(y,1) by YY,FUZIMPL1:def 16
.= 1 by AA,FUZIMPL1:def 16;
hence thesis by Y1;
end;
suppose
YY: y > z & x <= z; then
Y1: I.(x,I.(y,z)) = I.(x,z) by FUZIMPL1:def 16
.= 1 by FUZIMPL1:def 16,YY;
I.(y,I.(x,z)) = I.(y,1) by YY,FUZIMPL1:def 16
.= 1 by AA,FUZIMPL1:def 16;
hence thesis by Y1;
end;
suppose
YY: y > z & x > z; then
Y1: I.(x,I.(y,z)) = I.(x,z) by FUZIMPL1:def 16
.= z by FUZIMPL1:def 16,YY;
I.(y,I.(x,z)) = I.(y,z) by YY,FUZIMPL1:def 16
.= z by YY,FUZIMPL1:def 16;
hence thesis by Y1;
end;
suppose
YY: y <= z & x > z; then
Y1: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 16
.= 1 by FUZIMPL1:def 16,AA;
I.(y,I.(x,z)) = I.(y,z) by YY,FUZIMPL1:def 16
.= 1 by YY,FUZIMPL1:def 16;
hence thesis by Y1;
end;
end; then
I is satisfying_(EP);
hence thesis by A1;
end;
cluster I_RC -> satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
coherence
proof
set I = I_RC;
I0: 1 / 2 in [.0,1.] by XXREAL_1:1; then
p1: I.(1/2, 1/2) = 1 - 1 / 2 + (1 / 2) * (1 / 2) by FUZIMPL1:def 17
.= 3 / 4;
at: for y being Element of [.0,1.] holds I.(1,y) = y
proof
let y be Element of [.0,1.];
1 in [.0,1.] by XXREAL_1:1; then
I.(1,y) = 1 - 1 + 1 * y by FUZIMPL1:def 17
.= y;
hence thesis;
end;
for x,y,z being Element of [.0,1.] holds I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
i0: I.(y,z) = 1 - y + y * z by FUZIMPL1:def 17;
i2: I.(x,z) = 1 - x + x * z by FUZIMPL1:def 17;
I1: I.(x,I.(y,z)) = 1 - x + x * (1 - y + y * z) by i0,FUZIMPL1:def 17;
I.(y,I.(x,z)) = 1 - y + y * (1 - x + x * z) by i2,FUZIMPL1:def 17;
hence thesis by I1;
end;
hence thesis by at,p1,I0;
end;
cluster I_KD -> satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
coherence
proof
set I = I_KD;
K1: 1 in [.0,1.] & 1/2 in [.0,1.] by XXREAL_1:1;
ka: for y being Element of [.0,1.] holds I.(1,y) = y
proof
let y be Element of [.0,1.];
T1: 0 <= y by XXREAL_1:1;
I.(1,y) = max (1 - 1,y) by K1,FUZIMPL1:def 18
.= y by T1,XXREAL_0:def 10;
hence thesis;
end;
n1: I.(1/2,1/2) = max (1 - 1 / 2, 1 / 2) by FUZIMPL1:def 18,K1
.= max (1 / 2, 1 / 2);
for x,y,z being Element of [.0,1.] holds
I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
y2: I.(y,z) = max (1-y,z) by FUZIMPL1:def 18;
y3: I.(x,z) = max (1-x,z) by FUZIMPL1:def 18;
Y1: I.(x,I.(y,z)) = max (1-x, max (1-y,z)) by y2,FUZIMPL1:def 18
.= max (max(1-x, 1-y),z) by XXREAL_0:34;
I.(y,I.(x,z)) = max (1-y, max (1-x,z)) by y3,FUZIMPL1:def 18
.= max (max (1-y, 1-x),z) by XXREAL_0:34;
hence thesis by Y1;
end;
hence thesis by ka,n1,K1;
end;
cluster I_GG -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
coherence
proof
set I = I_GG;
for x,y being Element of [.0,1.] holds
I.(x,y) = 1 iff x <= y
proof
let x,y be Element of [.0,1.];
thus I.(x,y) = 1 implies x <= y
proof
assume
T2: I.(x,y) = 1;
assume
T3: x > y; then
T4: x > 0 by XXREAL_1:1;
I.(x,y) = y / x by T3,FUZIMPL1:def 19; then
y = 1 * x by XCMPLX_1:87,T4,T2;
hence thesis by T3;
end;
thus thesis by FUZIMPL1:def 19;
end; then
A1: I is satisfying_(OP);
for x,y,z being Element of [.0,1.] holds I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x, y, z be Element of [.0,1.];
AB: 0 <= x <= 1 by XXREAL_1:1;
AC: 0 <= y <= 1 by XXREAL_1:1;
AA: 1 in [.0,1.] by XXREAL_1:1;
per cases;
suppose
YY: y <= z & x <= z; then
Y1: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 19
.= 1 by FUZIMPL1:def 19,AA,AB;
I.(y,I.(x,z)) = I.(y,1) by YY,FUZIMPL1:def 19
.= 1 by AC,FUZIMPL1:def 19,AA;
hence thesis by Y1;
end;
suppose
YY: y > z & x <= z;
x * y <= x by XREAL_1:153,AC,AB; then
x * y <= z & y > 0 by YY,XXREAL_0:2; then
F1: x <= z / y by XREAL_1:77;
F2: z / y in [.0,1.] by YY,FUZIMPL1:6;
Y1: I.(x,I.(y,z)) = I.(x, z / y) by FUZIMPL1:def 19,YY
.= 1 by FUZIMPL1:def 19,F1,F2;
I.(y,I.(x,z)) = I.(y,1) by YY,FUZIMPL1:def 19
.= 1 by AC,AA,FUZIMPL1:def 19;
hence thesis by Y1;
end;
suppose
YY: y > z & x > z; then
JJ: x > 0 & y > 0 by XXREAL_1:1;
per cases;
suppose
IK: x <= z / y;
J1: y > 0 by XXREAL_1:1,YY; then
x * y <= z / y * y by IK,XREAL_1:64; then
x * y <= z by J1,XCMPLX_1:87; then
(x * y) / x <= z / x by JJ,XREAL_1:72; then
FF: y <= z / x by JJ,XCMPLX_1:89;
F2: z / x in [.0,1.] by YY,FUZIMPL1:6;
F4: z / y in [.0,1.] by YY,FUZIMPL1:6;
Y1: I.(x,I.(y,z)) = I.(x,z/y) by FUZIMPL1:def 19,YY
.= 1 by FUZIMPL1:def 19,F4,IK;
I.(y,I.(x,z)) = I.(y,z / x) by YY,FUZIMPL1:def 19
.= 1 by FUZIMPL1:def 19,F2,FF;
hence thesis by Y1;
end;
suppose
F5: x > z / y; then
x * y > z / y * y by JJ,XREAL_1:68; then
x * y > z by JJ,XCMPLX_1:87; then
x * y / x > z / x by XREAL_1:74,JJ; then
F1: y > z / x by XCMPLX_1:89,JJ;
F2: z / x in [.0,1.] by YY,FUZIMPL1:6;
F4: z / y in [.0,1.] by YY,FUZIMPL1:6;
Y1: I.(x,I.(y,z)) = I.(x,z/y) by FUZIMPL1:def 19,YY
.= (z / y) / x by FUZIMPL1:def 19,F5,F4;
I.(y,I.(x,z)) = I.(y,z / x) by YY,FUZIMPL1:def 19
.= (z / x) / y by FUZIMPL1:def 19,F2,F1;
hence thesis by Y1;
end;
end;
suppose
YY: y <= z & x > z; then
F2: z / x in [.0,1.] by FUZIMPL1:6;
x * y <= y by XREAL_1:153,AC,AB; then
x * y <= z & x > 0 by YY,XXREAL_0:2; then
F1: y <= z/x by XREAL_1:77;
Y1: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 19,YY
.= 1 by FUZIMPL1:def 19,AA,AB;
I.(y,I.(x,z)) = I.(y,z/x) by YY,FUZIMPL1:def 19
.= 1 by FUZIMPL1:def 19,F2,F1;
hence thesis by Y1;
end;
end; then
I is satisfying_(EP);
hence thesis by A1;
end;
cluster I_RS -> non satisfying_(NP) non satisfying_(EP) satisfying_(IP)
satisfying_(OP);
coherence
proof
set I = I_RS;
A1: I is satisfying_(OP) by FUZIMPL1:def 20;
K1: 1 in [.0,1.] & 1/2 in [.0,1.] & 0 in [.0,1.] by XXREAL_1:1; then
I.(1,1/2) <> 1/2 by FUZIMPL1:def 20; then
I is non satisfying_(NP) by K1;
hence thesis by A1;
end;
cluster I_YG -> satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
coherence
proof
set I = I_YG;
A1: 1 / 2 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1;
BA: for x,y,z being Element of [.0,1.] holds
I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
per cases by XXREAL_1:1;
suppose
Y0: x = 0 & y = 0 & z = 0; then
Y1: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 21
.= 1 to_power x by A1,FUZIMPL1:def 21
.= 1 by POWER:26;
I.(y,I.(x,z)) = I.(y,1) by FUZIMPL1:def 21,Y0
.= 1 to_power y by A1,FUZIMPL1:def 21
.= 1 by POWER:26;
hence thesis by Y1;
end;
suppose
R1: z > 0; then
r4: I.(y,z) = z to_power y by FUZIMPL1:def 21;
r6: I.(x,z) = z to_power x by R1,FUZIMPL1:def 21;
R2: z to_power y > 0 by R1,POWER:34;
R3: z to_power x > 0 by R1,POWER:34;
R5: I.(x,I.(y,z)) = (z to_power y) to_power x by R2,r4,FUZIMPL1:def 21
.= z to_power (y * x) by R1,POWER:33;
I.(y,I.(x,z)) = (z to_power x) to_power y by R3,r6,FUZIMPL1:def 21
.= z to_power (x * y) by R1,POWER:33;
hence thesis by R5;
end;
suppose
R1: y > 0 & x > 0 & z = 0; then
r4: I.(y,z) = z to_power y by FUZIMPL1:def 21;
r6: I.(x,z) = z to_power x by R1,FUZIMPL1:def 21;
R2: z to_power y = 0 by R1,POWER:def 2;
R3: z to_power x = 0 by R1,POWER:def 2;
R5: I.(x,I.(y,z)) = (z to_power y) to_power x by R1,r4,FUZIMPL1:def 21
.= 0 by R1,POWER:def 2,R2;
I.(y,I.(x,z)) = (z to_power x) to_power y by R1,r6,FUZIMPL1:def 21
.= 0 by R1,POWER:def 2,R3;
hence thesis by R5;
end;
suppose
R1: y = 0 & x > 0 & z = 0;
R3: z to_power x = 0 by R1,POWER:def 2;
R5: I.(x,I.(y,z)) = I.(x,1) by R1,FUZIMPL1:def 21
.= 1 to_power x by FUZIMPL1:def 21,A1
.= 1 by POWER:26;
I.(y,I.(x,z)) = I.(y,z to_power x) by R1,FUZIMPL1:def 21
.= 1 by R1,R3,FUZIMPL1:def 21;
hence thesis by R5;
end;
suppose
R1: y > 0 & x = 0 & z = 0; then
R2: z to_power y = 0 by POWER:def 2;
R5: I.(x,I.(y,z)) = I.(x,z to_power y) by R1,FUZIMPL1:def 21
.= 1 by R1,R2,FUZIMPL1:def 21;
I.(y,I.(x,z)) = I.(y,1) by R1,FUZIMPL1:def 21
.= 1 to_power y by FUZIMPL1:def 21,A1
.= 1 by POWER:26;
hence thesis by R5;
end;
suppose
R1: y > 0 & x > 0 & z = 0; then
r6: I.(x,z) = z to_power x by FUZIMPL1:def 21;
R2: z to_power y = 0 by R1,POWER:def 2;
R3: z to_power x = 0 by R1,POWER:def 2;
R5: I.(x,I.(y,z)) = I.(x,z to_power y) by R1,FUZIMPL1:def 21
.= (z to_power y) to_power x by R1,R2,FUZIMPL1:def 21
.= 0 by R1,POWER:def 2,R2;
I.(y,I.(x,z)) = (z to_power x) to_power y by R1,r6,FUZIMPL1:def 21
.= 0 by R1,POWER:def 2,R3;
hence thesis by R5;
end;
end;
B2: for y being Element of [.0,1.] holds I.(1,y) = y
proof
let y be Element of [.0,1.];
I.(1,y) = y to_power 1 by FUZIMPL1:def 21,A1;
hence thesis by POWER:25;
end;
I.(1/2,1/2) = (1 / 2) to_power (1 / 2) by A1,FUZIMPL1:def 21;
hence thesis by BA,B2,A1,LemmaHalf;
end;
cluster I_WB -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
non satisfying_(OP);
coherence
proof
set I = I_WB;
a0: for x being Element of [.0,1.] holds I.(x,x) = 1
proof
let x be Element of [.0,1.];
x <= 1 by XXREAL_1:1; then
per cases by XXREAL_0:1;
suppose x < 1;
hence thesis by FUZIMPL1:def 22;
end;
suppose x = 1;
hence thesis by FUZIMPL1:def 22;
end;
end;
BB: 1 in [.0,1.] by XXREAL_1:1;
CA: for x,y,z being Element of [.0,1.] holds
I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
x <= 1 & y <= 1 & z <= 1 by XXREAL_1:1; then
per cases by XXREAL_0:1;
suppose
Y0: x = 1 & y = 1; then
Y1: I.(x,I.(y,z)) = I.(x,z) by FUZIMPL1:def 22
.= z by FUZIMPL1:def 22,Y0;
I.(y,I.(x,z)) = I.(y,z) by Y0,FUZIMPL1:def 22
.= z by Y0,FUZIMPL1:def 22;
hence thesis by Y1;
end;
suppose
Y0: x < 1 & y = 1; then
I.(y,I.(x,z)) = I.(y,1) by FUZIMPL1:def 22
.= 1 by Y0,FUZIMPL1:def 22;
hence thesis by Y0,FUZIMPL1:def 22;
end;
suppose
Y0: x = 1 & y < 1; then
I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 22
.= 1 by FUZIMPL1:def 22,Y0;
hence thesis by Y0,FUZIMPL1:def 22;
end;
suppose
Y0: x < 1 & y < 1; then
I.(x,I.(y,z)) = 1 by FUZIMPL1:def 22;
hence thesis by Y0,FUZIMPL1:def 22;
end;
end;
ex x,y being Element of [.0,1.] st not (I.(x,y) = 1 iff x <= y)
proof
set x = 1 / 2, y = 0;
reconsider x,y as Element of [.0,1.] by XXREAL_1:1;
take x,y;
thus thesis by FUZIMPL1:def 22;
end;
hence thesis by CA,a0,BB,FUZIMPL1:def 22;
end;
cluster I_FD -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
coherence
proof
set I = I_FD;
A1: I is satisfying_(OP)
proof
let x,y be Element of [.0,1.];
thus I.(x,y) = 1 implies x <= y
proof
assume
T2: I.(x,y) = 1;
assume
T3: x > y; then
I.(x,y) = max(1 - x, y) by FUZIMPL1:def 23; then
1 - x = 1 or y = 1 by XXREAL_0:16,T2;
hence thesis by T3,XXREAL_1:1;
end;
thus thesis by FUZIMPL1:def 23;
end;
for x,y,z being Element of [.0,1.] holds
I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
Ff: 1 in [.0,1.] by XXREAL_1:1;
F0: x <= 1 & y <= 1 & z >= 0 by XXREAL_1:1;
per cases;
suppose
F1: y <= z & x <= z; then
F2: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 23
.= 1 by A1,F0,Ff;
I.(y,I.(x,z)) = I.(y,1) by F1,FUZIMPL1:def 23
.= 1 by A1,F0,Ff;
hence thesis by F2;
end;
suppose
F1: y <= z & x > z;
ff: z <= max (1-x,z) by XXREAL_0:25;
f4: I.(x,z) = max (1-x,z) by FUZIMPL1:def 23,F1;
I.(x,I.(y,z)) = I.(x,1) by F1,FUZIMPL1:def 23
.= 1 by A1,F0,Ff;
hence thesis by A1,ff,F1,XXREAL_0:2,f4;
end;
suppose
F1: y > z & x > z;
x2: I.(y,z) = max (1-y,z) by F1,FUZIMPL1:def 23;
x4: I.(x,z) = max (1-x,z) by F1,FUZIMPL1:def 23;
per cases;
suppose
X1: x <= max (1-y,z); then
x <= 1 - y or x <= z by XXREAL_0:def 10; then
x + y <= 1 - y + y by XREAL_1:6,F1; then
y1: x + y - x <= 1 - x by XREAL_1:9;
1 - x <= max (1-x,z) by XXREAL_0:25; then
X3: y <= max (1-x,z) by y1,XXREAL_0:2;
I.(x,I.(y,z)) = 1 by X1,x2,FUZIMPL1:def 23;
hence thesis by X3,x4,FUZIMPL1:def 23;
end;
suppose
we: 1 - x < y; then
X3: y > max (1-x,z) by XXREAL_0:29,F1;
1 - x - y < y - y by we,XREAL_1:9; then
1 - x - y + x < 0 + x by XREAL_1:6; then
X1: x > max (1-y,z) by XXREAL_0:def 10,F1;
F2: I.(x,I.(y,z)) = max (1-x,max(1-y,z))
by x2,FUZIMPL1:def 23,X1;
I.(y,I.(x,z)) = max (1-y,max(1-x,z))
by FUZIMPL1:def 23,X3,x4;
hence thesis by F2,XXREAL_0:34;
end;
suppose
SE: 1 - x >= y; then
1 - x + x >= y + x by XREAL_1:6; then
ww: 1 - y >= y + x - y by XREAL_1:9; then
SA: max (1-y,z) = 1 - y by XXREAL_0:def 10,XXREAL_0:2,F1;
P0: 1 - x in [.0,1.] & 1 - y in [.0,1.] by FUZNORM1:7;
F2: I.(x,I.(y,z)) = I.(x,max (1-y,z)) by F1,FUZIMPL1:def 23
.= 1 by P0,FUZIMPL1:def 23,ww,SA;
I.(y,I.(x,z)) = I.(y,max (1-x,z)) by F1,FUZIMPL1:def 23
.= I.(y,1-x) by SE,F1,XXREAL_0:2,XXREAL_0:def 10
.= 1 by A1,P0,SE;
hence thesis by F2;
end;
end;
suppose
F1: y > z & x <= z;
f3: z <= max (1-y,z) by XXREAL_0:25;
f4: I.(y,z) = max (1-y,z) by FUZIMPL1:def 23,F1;
I.(y,I.(x,z)) = I.(y,1) by F1,FUZIMPL1:def 23
.= 1 by A1,F0,Ff;
hence thesis by f4,A1,F1,f3,XXREAL_0:2;
end;
end; then
I is satisfying_(EP);
hence thesis by A1;
end;
end;
registration
cluster I_{0} -> non satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
coherence
proof
set I = I_{0};
a2: ex x being Element of [.0,1.] st I.(x,x) <> 1
proof
reconsider x = 1/2 as Element of [.0,1.] by XXREAL_1:1;
I.(x,x) = 0 by FUZIMPL1:def 24;
hence thesis;
end;
SS: 1 in [.0,1.] by XXREAL_1:1;
SA: 0 in [.0,1.] by XXREAL_1:1;
a3: not for y being Element of [.0,1.] holds I.(1,y) = y
proof
reconsider y = 1/2 as Element of [.0,1.] by XXREAL_1:1;
I.(1,y) = 0 by FUZIMPL1:def 24,SS;
hence thesis;
end;
for x,y,z being Element of [.0,1.] holds
I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
x >= 0 & y >= 0 & z <= 1 by XXREAL_1:1; then
per cases by XXREAL_0:1;
suppose
B1: z = 1; then
B2: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 24 .= 1 by FUZIMPL1:def 24,SS;
I.(y,I.(x,z)) = I.(y,1) by FUZIMPL1:def 24,B1
.= 1 by FUZIMPL1:def 24,SS;
hence thesis by B2;
end;
suppose
B1: x = 0; then
I.(y,I.(x,z)) = I.(y,1) by FUZIMPL1:def 24
.= 1 by FUZIMPL1:def 24,SS;
hence thesis by B1,FUZIMPL1:def 24;
end;
suppose
B1: y = 0; then
I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 24
.= 1 by FUZIMPL1:def 24,SS;
hence thesis by B1,FUZIMPL1:def 24;
end;
suppose
F1: x > 0 & z < 1 & y > 0; then
F2: I.(y,I.(x,z)) = I.(y,0) by FUZIMPL1:def 24
.= 0 by SA,F1,FUZIMPL1:def 24;
I.(x,I.(y,z)) = I.(x,0) by F1,FUZIMPL1:def 24
.= 0 by SA,FUZIMPL1:def 24,F1;
hence thesis by F2;
end;
end;
hence thesis by a2,a3;
end;
cluster I_{1} -> non satisfying_(NP) satisfying_(EP) satisfying_(IP)
non satisfying_(OP);
coherence
proof
set I = I_{1};
a2: for x being Element of [.0,1.] holds I.(x,x) = 1
proof
let x be Element of [.0,1.];
0 <= x <= 1 by XXREAL_1:1; then
per cases by XXREAL_0:1;
suppose x = 0;
hence thesis by FUZIMPL1:def 25;
end;
suppose 0 < x < 1;
hence thesis by FUZIMPL1:def 25;
end;
suppose x = 1;
hence thesis by FUZIMPL1:def 25;
end;
end;
SS: 1 in [.0,1.] by XXREAL_1:1;
not for y being Element of [.0,1.] holds I.(1,y) = y
proof
reconsider y = 1/2 as Element of [.0,1.] by XXREAL_1:1;
I.(1,y) = 1 by SS,FUZIMPL1:def 25;
hence thesis;
end; then
A3: I is non satisfying_(NP);
for x,y,z being Element of [.0,1.] holds
I.(x,I.(y,z)) = I.(y,I.(x,z))
proof
let x,y,z be Element of [.0,1.];
y <= 1 & x <= 1 & z >= 0 by XXREAL_1:1; then
per cases by XXREAL_0:1;
suppose
B1: z = 0 & x = 1 & y = 1; then
B2: I.(x,I.(y,z)) = I.(x,0) by FUZIMPL1:def 25
.= 0 by B1,FUZIMPL1:def 25;
I.(y,I.(x,z)) = I.(y,0) by FUZIMPL1:def 25,B1
.= 0 by B1,FUZIMPL1:def 25;
hence thesis by B2;
end;
suppose
B1: z > 0 & x = 1 & y = 1; then
B2: I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 25
.= 1 by FUZIMPL1:def 25,SS;
I.(y,I.(x,z)) = I.(y,1) by FUZIMPL1:def 25,B1
.= 1 by FUZIMPL1:def 25,SS;
hence thesis by B2;
end;
suppose
B1: z = 0 & x = 1 & y < 1; then
I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 25
.= 1 by FUZIMPL1:def 25,SS;
hence thesis by B1,FUZIMPL1:def 25;
end;
suppose
B1: z = 0 & x < 1 & y = 1; then
I.(y,I.(x,z)) = I.(y,1) by FUZIMPL1:def 25
.= 1 by FUZIMPL1:def 25,SS;
hence thesis by B1,FUZIMPL1:def 25;
end;
suppose
B1: z = 0 & x < 1 & y < 1; then
I.(y,I.(x,z)) = 1 by FUZIMPL1:def 25;
hence thesis by B1,FUZIMPL1:def 25;
end;
suppose
B1: z > 0 & x < 1 & y = 1; then
I.(y,I.(x,z)) = I.(y,1) by FUZIMPL1:def 25
.= 1 by FUZIMPL1:def 25,SS;
hence thesis by B1,FUZIMPL1:def 25;
end;
suppose
B1: z > 0 & x = 1 & y < 1; then
I.(x,I.(y,z)) = I.(x,1) by FUZIMPL1:def 25
.= 1 by FUZIMPL1:def 25,SS;
hence thesis by FUZIMPL1:def 25,B1;
end;
suppose
F1: z > 0 & x < 1 & y < 1; then
I.(y,I.(x,z)) = 1 by FUZIMPL1:def 25;
hence thesis by FUZIMPL1:def 25,F1;
end;
end; then
I is satisfying_(EP);
hence thesis by a2,A3;
end;
end;