Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### Partial Functions from a Domain to a Domain

**by****Jaroslaw Kotowicz**- Received May 31, 1990
- MML identifier: PARTFUN2

- [ Mizar article, MML identifier index ]

environ vocabulary PARTFUN1, RELAT_1, FINSEQ_4, FUNCT_1, BOOLE, FUNCOP_1, PARTFUN2; notation TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2, FUNCOP_1, PARTFUN1, FINSEQ_4; constructors FUNCOP_1, PARTFUN1, FINSEQ_4, MEMBERED, XBOOLE_0; clusters RELAT_1, FUNCT_1, RELSET_1, FUNCOP_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin reserve x,y,X,Y for set; reserve C,D,E for non empty set; reserve SC for Subset of C; reserve SD for Subset of D; reserve SE for Subset of E; reserve c,c1,c2 for Element of C; reserve d,d1,d2 for Element of D; reserve e for Element of E; reserve f,f1,g for PartFunc of C,D; reserve t for PartFunc of D,C; reserve s for PartFunc of D,E; reserve h for PartFunc of C,E; reserve F for PartFunc of D,D; canceled 2; theorem :: PARTFUN2:3 dom f = dom g & (for c st c in dom f holds f/.c = g/.c) implies f = g; theorem :: PARTFUN2:4 :: ograniczyc do implikacji y in rng f iff ex c st c in dom f & y = f/.c; canceled; theorem :: PARTFUN2:6 h = s*f iff (for c holds c in dom h iff c in dom f & f/.c in dom s) & for c st c in dom h holds h/.c = s/.(f/.c); canceled 2; theorem :: PARTFUN2:9 c in dom f & f/.c in dom s implies (s*f)/.c = s/.(f/.c); theorem :: PARTFUN2:10 rng f c= dom s & c in dom f implies (s*f)/.c = s/.(f/.c); definition let D; let SD; redefine func id SD -> PartFunc of D,D; end; canceled; theorem :: PARTFUN2:12 F = id SD iff dom F = SD & for d st d in SD holds F/.d = d; canceled; theorem :: PARTFUN2:14 d in dom F /\ SD implies F/.d = (F*id SD)/.d; theorem :: PARTFUN2:15 d in dom((id SD)*F) iff d in dom F & F/.d in SD; theorem :: PARTFUN2:16 (for c1,c2 st c1 in dom f & c2 in dom f & f/.c1 = f/.c2 holds c1 = c2) implies f is one-to-one; theorem :: PARTFUN2:17 f is one-to-one & x in dom f & y in dom f & f/.x = f/.y implies x = y; definition cluster {} -> one-to-one; end; definition let X,Y; cluster one-to-one PartFunc of X,Y; end; definition let X,Y; let f be one-to-one PartFunc of X,Y; redefine func f" -> PartFunc of Y,X; end; theorem :: PARTFUN2:18 for f being one-to-one PartFunc of C,D holds for g be PartFunc of D,C holds g = f" iff dom g = rng f & for d,c holds d in rng f & c = g/.d iff c in dom f & d = f/.c; canceled 3; theorem :: PARTFUN2:22 for f being one-to-one PartFunc of C,D st c in dom f holds c = f"/.(f/.c) & c = (f"*f)/.c; theorem :: PARTFUN2:23 for f being one-to-one PartFunc of C,D st d in rng f holds d = f/.(f"/.d) & d = (f*(f"))/.d; theorem :: PARTFUN2:24 f is one-to-one & dom f = rng t & rng f = dom t & (for c,d st c in dom f & d in dom t holds f/.c = d iff t/.d = c) implies t = f"; canceled 7; theorem :: PARTFUN2:32 g = f|X iff dom g = dom f /\ X & for c st c in dom g holds g/.c = f/.c; canceled; theorem :: PARTFUN2:34 c in dom f /\ X implies f|X/.c = f/.c; theorem :: PARTFUN2:35 c in dom f & c in X implies f|X/.c = f/.c; theorem :: PARTFUN2:36 c in dom f & c in X implies f/.c in rng (f|X); definition let C,D; let X,f; redefine func X|f -> PartFunc of C,D; end; theorem :: PARTFUN2:37 g = X|f iff (for c holds c in dom g iff c in dom f & f/.c in X) & (for c st c in dom g holds g/.c = f/.c); theorem :: PARTFUN2:38 c in dom (X|f) iff c in dom f & f/.c in X; theorem :: PARTFUN2:39 c in dom (X|f) implies X|f/.c = f/.c; theorem :: PARTFUN2:40 SD = f.: X iff for d holds d in SD iff ex c st c in dom f & c in X & d = f/.c; theorem :: PARTFUN2:41 d in (f qua Relation of C,D).:X iff ex c st c in dom f & c in X & d = f/.c; theorem :: PARTFUN2:42 c in dom f implies f.:{c} = {f/.c}; theorem :: PARTFUN2:43 c1 in dom f & c2 in dom f implies f.:{c1,c2} = {f/.c1,f/.c2}; theorem :: PARTFUN2:44 SC = f"X iff for c holds c in SC iff c in dom f & f/.c in X; canceled; theorem :: PARTFUN2:46 for f ex g being Function of C,D st for c st c in dom f holds g.c = f/.c; theorem :: PARTFUN2:47 f tolerates g iff for c st c in dom f /\ dom g holds f/.c = g/.c; scheme PartFuncExD{D,C()->non empty set, P[set,set]}: ex f being PartFunc of D(),C() st (for d be Element of D() holds d in dom f iff (ex c be Element of C() st P[d,c])) & (for d be Element of D() st d in dom f holds P[d,f/.d]); scheme LambdaPFD{D,C()->non empty set, F(set)->Element of C(), P[set]}: ex f being PartFunc of D(),C() st (for d be Element of D() holds d in dom f iff P[d]) & (for d be Element of D() st d in dom f holds f/.d = F(d)); scheme UnPartFuncD{C,D()->non empty set, X()->set, F(set)->Element of D()}: for f,g being PartFunc of C(),D() st (dom f=X() & for c be Element of C() st c in dom f holds f/.c = F(c)) & (dom g=X() & for c be Element of C() st c in dom g holds g/.c = F(c)) holds f = g; definition let C,D; let SC,d; redefine func SC --> d -> PartFunc of C,D; end; theorem :: PARTFUN2:48 c in SC implies (SC --> d)/.c = d; theorem :: PARTFUN2:49 (for c st c in dom f holds f/.c = d) implies f = dom f --> d; theorem :: PARTFUN2:50 c in dom f implies f*(SE --> c) = SE --> f/.c; theorem :: PARTFUN2:51 (id SC) is total iff SC = C; theorem :: PARTFUN2:52 (SC --> d) is total implies SC <> {}; theorem :: PARTFUN2:53 (SC --> d) is total iff SC = C; :: :: PARTIAL FUNCTION CONSTANT ON SET :: definition let C,D; let f,X; canceled 2; pred f is_constant_on X means :: PARTFUN2:def 3 ex d st for c st c in X /\ dom f holds f/.c = d; end; canceled; theorem :: PARTFUN2:55 f is_constant_on X iff for c1,c2 st c1 in X /\ dom f & c2 in X /\ dom f holds f/.c1=f/.c2; theorem :: PARTFUN2:56 X meets dom f implies (f is_constant_on X iff ex d st rng (f|X) = {d}); theorem :: PARTFUN2:57 f is_constant_on X & Y c= X implies f is_constant_on Y; theorem :: PARTFUN2:58 X misses dom f implies f is_constant_on X; theorem :: PARTFUN2:59 f|SC = dom (f|SC) --> d implies f is_constant_on SC; theorem :: PARTFUN2:60 f is_constant_on {x}; theorem :: PARTFUN2:61 f is_constant_on X & f is_constant_on Y & X /\ Y meets dom f implies f is_constant_on X \/ Y; theorem :: PARTFUN2:62 f is_constant_on Y implies f|X is_constant_on Y; theorem :: PARTFUN2:63 SC --> d is_constant_on SC; :: :: OF PARTIAL FUNCTION FROM A DOMAIN TO A DOMAIN :: theorem :: PARTFUN2:64 f c= g iff dom f c= dom g & (for c st c in dom f holds f/.c = g/.c); theorem :: PARTFUN2:65 c in dom f & d = f/.c iff [c,d] in f; theorem :: PARTFUN2:66 [c,e] in (s*f) implies [c,f/.c] in f & [f/.c,e] in s; theorem :: PARTFUN2:67 f = {[c,d]} implies f/.c = d; theorem :: PARTFUN2:68 dom f = {c} implies f = {[c,f/.c]}; theorem :: PARTFUN2:69 f1 = f /\ g & c in dom f1 implies f1/.c = f/.c & f1/.c = g/.c; theorem :: PARTFUN2:70 c in dom f & f1 = f \/ g implies f1/.c = f/.c; theorem :: PARTFUN2:71 c in dom g & f1 = f \/ g implies f1/.c = g/.c; theorem :: PARTFUN2:72 c in dom f1 & f1 = f \/ g implies f1/.c = f/.c or f1/.c = g/.c; theorem :: PARTFUN2:73 c in dom f & c in SC iff [c,f/.c] in (f|SC); theorem :: PARTFUN2:74 c in dom f & f/.c in SD iff [c,f/.c] in (SD|f); theorem :: PARTFUN2:75 c in f"SD iff [c,f/.c] in f & f/.c in SD; theorem :: PARTFUN2:76 f is_constant_on X iff ex d st for c st c in X /\ dom f holds f.c = d; theorem :: PARTFUN2:77 f is_constant_on X iff for c1,c2 st c1 in X /\ dom f & c2 in X /\ dom f holds f.c1=f.c2; theorem :: PARTFUN2:78 d in f.:X implies ex c st c in dom f & c in X & d = f.c; theorem :: PARTFUN2:79 f is one-to-one implies (d in rng f & c = (f").d iff c in dom f & d = f.c);

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