:: Fundamental Properties of Fuzzy Implications
:: by Adam Grabowski
::
:: Received September 29, 2018
:: Copyright (c) 2018-2021 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;
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.];
theorem :: FUZIMPL2:1
#R 1 = AffineMap (1,0) | right_open_halfline 0;
theorem :: FUZIMPL2:2
for a,b being Real holds
AffineMap (a,b) is_differentiable_on REAL &
for x being Real holds diff (AffineMap (a,b), x) = a;
theorem :: FUZIMPL2:3
0 < x < 1 & 0 < y < 1 implies
( #R x + AffineMap (-x, x-1)) | ].0,1.[ is increasing;
theorem :: FUZIMPL2:4
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;
begin :: On the Ordering of Fuzzy Implications
theorem :: FUZIMPL2:5
(x <= y implies I_LK.(x,y) = 1) &
(x > y implies I_LK.(x,y) = 1 - x + y);
theorem :: FUZIMPL2:6
(x = 0 implies I_GG.(x,y) = 1) &
(x > 0 implies I_GG.(x,y) = min (1, y / x));
theorem :: FUZIMPL2:7 :: Example 1.1.6 (1.1) p. 3
I_KD <= I_RC <= I_LK <= I_WB;
theorem :: FUZIMPL2:8 :: Example 1.1.6 (1.2) p. 3
I_RS <= I_GD <= I_GG <= I_LK <= I_WB;
theorem :: FUZIMPL2:9 :: Example 1.1.6 (1.3) p. 3
:::I_YG <=
I_RC <= I_LK <= I_WB;
theorem :: FUZIMPL2:10 :: Example 1.1.6 (1.4) p. 3
I_KD <= I_FD <= I_LK <= I_WB;
theorem :: FUZIMPL2:11 :: Example 1.1.6 (1.5) p. 3
I_RS <= I_GD <= I_FD <= I_LK <= I_WB;
begin :: Additional Properties of Fuzzy Implications
definition
let I be BinOp of [.0,1.];
attr I is satisfying_(NP) means
:: FUZIMPL2:def 1
for y being Element of [.0,1.] holds I.(1,y) = y;
attr I is satisfying_(EP) means
:: FUZIMPL2:def 2
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
:: FUZIMPL2:def 3
for x being Element of [.0,1.] holds I.(x,x) = 1;
attr I is satisfying_(OP) means
:: FUZIMPL2:def 4
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;
theorem :: FUZIMPL2:12
I is satisfying_(LB) implies I is satisfying_(I3) satisfying_(NC);
registration
cluster satisfying_(LB) -> satisfying_(I3) satisfying_(NC)
for BinOp of [.0,1.];
end;
theorem :: FUZIMPL2:13
I is satisfying_(RB) implies I is satisfying_(I4) satisfying_(NC);
registration
cluster satisfying_(RB) -> satisfying_(I4) satisfying_(NC)
for BinOp of [.0,1.];
end;
theorem :: FUZIMPL2:14
I is satisfying_(NP) implies I is satisfying_(I4) satisfying_(I5);
registration
cluster satisfying_(NP) -> satisfying_(I4) satisfying_(I5)
for BinOp of [.0,1.];
end;
theorem :: FUZIMPL2:15
I is satisfying_(IP) implies I is satisfying_(I3) satisfying_(I4);
registration
cluster satisfying_(IP) -> satisfying_(I3) satisfying_(I4)
for BinOp of [.0,1.];
end;
theorem :: FUZIMPL2:16
I is satisfying_(OP) implies I is satisfying_(I3) satisfying_(I4)
satisfying_(NC) satisfying_(LB) satisfying_(RB) satisfying_(IP);
registration
cluster satisfying_(OP) -> satisfying_(I3) satisfying_(I4)
satisfying_(NC) satisfying_(LB) satisfying_(RB) satisfying_(IP)
for BinOp of [.0,1.];
end;
theorem :: FUZIMPL2:17
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);
registration
cluster satisfying_(EP) satisfying_(OP) -> satisfying_(I1)
satisfying_(I5) satisfying_(NP) for BinOp of [.0,1.];
end;
registration
cluster I_LK -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
cluster I_GD -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
cluster I_RC -> satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
cluster I_KD -> satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
cluster I_GG -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
cluster I_RS -> non satisfying_(NP) non satisfying_(EP) satisfying_(IP)
satisfying_(OP);
cluster I_YG -> satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
cluster I_WB -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
non satisfying_(OP);
cluster I_FD -> satisfying_(NP) satisfying_(EP) satisfying_(IP)
satisfying_(OP);
end;
registration
cluster I_{0} -> non satisfying_(NP) satisfying_(EP) non satisfying_(IP)
non satisfying_(OP);
cluster I_{1} -> non satisfying_(NP) satisfying_(EP) satisfying_(IP)
non satisfying_(OP);
end;