:: Technical Preliminaries to Algebraic Specifications
:: by Grzegorz Bancerek
::
:: Received September 7, 1999
:: Copyright (c) 1999 Association of Mizar Users
theorem Th1: :: ALGSPEC1:1
theorem Th2: :: ALGSPEC1:2
theorem Th3: :: ALGSPEC1:3
theorem Th4: :: ALGSPEC1:4
theorem Th5: :: ALGSPEC1:5
theorem Th6: :: ALGSPEC1:6
:: deftheorem defines -indexing ALGSPEC1:def 1 :
theorem Th7: :: ALGSPEC1:7
theorem Th8: :: ALGSPEC1:8
theorem Th9: :: ALGSPEC1:9
theorem Th10: :: ALGSPEC1:10
theorem Th11: :: ALGSPEC1:11
theorem Th12: :: ALGSPEC1:12
theorem Th13: :: ALGSPEC1:13
theorem :: ALGSPEC1:14
theorem :: ALGSPEC1:15
canceled;
theorem :: ALGSPEC1:16
theorem :: ALGSPEC1:17
theorem Th18: :: ALGSPEC1:18
theorem Th19: :: ALGSPEC1:19
theorem Th20: :: ALGSPEC1:20
theorem Th21: :: ALGSPEC1:21
theorem Th22: :: ALGSPEC1:22
theorem :: ALGSPEC1:23
:: deftheorem Def2 defines rng-retract ALGSPEC1:def 2 :
theorem Th24: :: ALGSPEC1:24
theorem Th25: :: ALGSPEC1:25
theorem :: ALGSPEC1:26
theorem :: ALGSPEC1:27
theorem Th28: :: ALGSPEC1:28
theorem :: ALGSPEC1:29
:: deftheorem Def3 defines form_a_replacement_in ALGSPEC1:def 3 :
theorem Th30: :: ALGSPEC1:30
theorem Th31: :: ALGSPEC1:31
theorem Th32: :: ALGSPEC1:32
theorem :: ALGSPEC1:33
theorem Th34: :: ALGSPEC1:34
theorem :: ALGSPEC1:35
definition
let S be non
empty non
void ManySortedSign ;
let f,
g be
Function;
assume A1:
f,
g form_a_replacement_in S
;
func S with-replacement f,
g -> non
empty non
void strict ManySortedSign means :
Def4:
:: ALGSPEC1:def 4
( the
carrier of
S -indexing f,the
carrier' of
S -indexing g form_morphism_between S,
it & the
carrier of
it = rng (the carrier of S -indexing f) & the
carrier' of
it = rng (the carrier' of S -indexing g) );
uniqueness
for b1, b2 being non empty non void strict ManySortedSign st the carrier of S -indexing f,the carrier' of S -indexing g form_morphism_between S,b1 & the carrier of b1 = rng (the carrier of S -indexing f) & the carrier' of b1 = rng (the carrier' of S -indexing g) & the carrier of S -indexing f,the carrier' of S -indexing g form_morphism_between S,b2 & the carrier of b2 = rng (the carrier of S -indexing f) & the carrier' of b2 = rng (the carrier' of S -indexing g) holds
b1 = b2
existence
ex b1 being non empty non void strict ManySortedSign st
( the carrier of S -indexing f,the carrier' of S -indexing g form_morphism_between S,b1 & the carrier of b1 = rng (the carrier of S -indexing f) & the carrier' of b1 = rng (the carrier' of S -indexing g) )
end;
:: deftheorem Def4 defines with-replacement ALGSPEC1:def 4 :
theorem Th36: :: ALGSPEC1:36
theorem Th37: :: ALGSPEC1:37
theorem Th38: :: ALGSPEC1:38
theorem Th39: :: ALGSPEC1:39
theorem Th40: :: ALGSPEC1:40
theorem Th41: :: ALGSPEC1:41
theorem Th42: :: ALGSPEC1:42
theorem :: ALGSPEC1:43
theorem Th44: :: ALGSPEC1:44
theorem Th45: :: ALGSPEC1:45
:: deftheorem Def5 defines Extension ALGSPEC1:def 5 :
theorem :: ALGSPEC1:46
canceled;
theorem Th47: :: ALGSPEC1:47
theorem Th48: :: ALGSPEC1:48
theorem Th49: :: ALGSPEC1:49
theorem Th50: :: ALGSPEC1:50
theorem Th51: :: ALGSPEC1:51
for
S1,
S2,
S being non
empty ManySortedSign for
f1,
g1,
f2,
g2 being
Function st
f1 tolerates f2 &
f1,
g1 form_morphism_between S1,
S &
f2,
g2 form_morphism_between S2,
S holds
f1 +* f2,
g1 +* g2 form_morphism_between S1 +* S2,
S
theorem :: ALGSPEC1:52
theorem Th53: :: ALGSPEC1:53
theorem Th54: :: ALGSPEC1:54
theorem :: ALGSPEC1:55
theorem :: ALGSPEC1:56
:: deftheorem Def6 defines Algebra ALGSPEC1:def 6 :
:: deftheorem Def7 defines Algebra ALGSPEC1:def 7 :
theorem :: ALGSPEC1:57
theorem :: ALGSPEC1:58
theorem Th59: :: ALGSPEC1:59
theorem Th60: :: ALGSPEC1:60
theorem :: ALGSPEC1:61
theorem Th62: :: ALGSPEC1:62
theorem Th63: :: ALGSPEC1:63
theorem Th64: :: ALGSPEC1:64
theorem Th65: :: ALGSPEC1:65
theorem :: ALGSPEC1:66