:: CIRCTRM1 semantic presentation
theorem Th1: :: CIRCTRM1:1
definition
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
V5 ManySortedSet of the
carrier of c
1;
let c
3 be non
empty Subset of
(c1 -Terms c2);
func c
3 -CircuitStr -> non
empty strict ManySortedSign equals :: CIRCTRM1:def 1
ManySortedSign(#
(Subtrees a3),
([:the OperSymbols of a1,{the carrier of a1}:] -Subtrees a3),
([:the OperSymbols of a1,{the carrier of a1}:] -ImmediateSubtrees a3),
(incl ([:the OperSymbols of a1,{the carrier of a1}:] -Subtrees a3)) #);
correctness
coherence
ManySortedSign(# (Subtrees c3),([:the OperSymbols of c1,{the carrier of c1}:] -Subtrees c3),([:the OperSymbols of c1,{the carrier of c1}:] -ImmediateSubtrees c3),(incl ([:the OperSymbols of c1,{the carrier of c1}:] -Subtrees c3)) #) is non empty strict ManySortedSign ;
;
end;
:: deftheorem Def1 defines -CircuitStr CIRCTRM1:def 1 :
theorem Th2: :: CIRCTRM1:2
theorem Th3: :: CIRCTRM1:3
theorem Th4: :: CIRCTRM1:4
theorem Th5: :: CIRCTRM1:5
theorem Th6: :: CIRCTRM1:6
theorem Th7: :: CIRCTRM1:7
theorem Th8: :: CIRCTRM1:8
theorem Th9: :: CIRCTRM1:9
theorem Th10: :: CIRCTRM1:10
theorem Th11: :: CIRCTRM1:11
theorem Th12: :: CIRCTRM1:12
theorem Th13: :: CIRCTRM1:13
:: deftheorem Def2 defines the_sort_of CIRCTRM1:def 2 :
:: deftheorem Def3 defines the_action_of CIRCTRM1:def 3 :
:: deftheorem Def4 defines -CircuitSorts CIRCTRM1:def 4 :
theorem Th14: :: CIRCTRM1:14
:: deftheorem Def5 defines -CircuitCharact CIRCTRM1:def 5 :
:: deftheorem Def6 defines -Circuit CIRCTRM1:def 6 :
theorem Th15: :: CIRCTRM1:15
theorem Th16: :: CIRCTRM1:16
theorem Th17: :: CIRCTRM1:17
theorem Th18: :: CIRCTRM1:18
theorem Th19: :: CIRCTRM1:19
theorem Th20: :: CIRCTRM1:20
:: deftheorem Def7 defines @ CIRCTRM1:def 7 :
definition
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
V5 ManySortedSet of the
carrier of c
1;
let c
3 be
SetWithCompoundTerm of c
1,c
2;
let c
4 be
non-empty locally-finite MSAlgebra of c
1;
let c
5 be
State of
(c3 -Circuit c4);
set c
6 =
[:the OperSymbols of c1,{the carrier of c1}:];
defpred S
1[
set ,
set ,
set ] means (
root-tree [a2,a1] in Subtrees c
3 implies a
3 = c
5 . (root-tree [a2,a1]) );
E24:
for b
1 being
Vertex of c
1for b
2 being
Element of c
2 . b
1 holds
ex b
3 being
Element of the
Sorts of c
4 . b
1 st
S
1[b
1,b
2,b
3]
mode CompatibleValuation of c
5 -> ManySortedFunction of a
2,the
Sorts of a
4 means :
Def8:
:: CIRCTRM1:def 8
for b
1 being
Vertex of a
1for b
2 being
Element of a
2 . b
1 holds
(
root-tree [b2,b1] in Subtrees a
3 implies
(a6 . b1) . b
2 = a
5 . (root-tree [b2,b1]) );
existence
ex b1 being ManySortedFunction of c2,the Sorts of c4 st
for b2 being Vertex of c1
for b3 being Element of c2 . b2 holds
( root-tree [b3,b2] in Subtrees c3 implies (b1 . b2) . b3 = c5 . (root-tree [b3,b2]) )
end;
:: deftheorem Def8 defines CompatibleValuation CIRCTRM1:def 8 :
theorem Th21: :: CIRCTRM1:21
theorem Th22: :: CIRCTRM1:22
theorem Th23: :: CIRCTRM1:23
:: deftheorem Def9 defines are_equivalent_wrt CIRCTRM1:def 9 :
theorem Th24: :: CIRCTRM1:24
theorem Th25: :: CIRCTRM1:25
theorem Th26: :: CIRCTRM1:26
theorem Th27: :: CIRCTRM1:27
theorem Th28: :: CIRCTRM1:28
for b
1, b
2, b
3 being non
empty ManySortedSign for b
4, b
5, b
6, b
7 being
Function holds
( b
1,b
2 are_equivalent_wrt b
4,b
5 & b
2,b
3 are_equivalent_wrt b
6,b
7 implies b
1,b
3 are_equivalent_wrt b
6 * b
4,b
7 * b
5 )
theorem Th29: :: CIRCTRM1:29
definition
let c
1, c
2 be non
empty ManySortedSign ;
pred c
1,c
2 are_equivalent means :: CIRCTRM1:def 10
ex b
1, b
2 being
one-to-one Function st a
1,a
2 are_equivalent_wrt b
1,b
2;
reflexivity
for b1 being non empty ManySortedSign holds
ex b2, b3 being one-to-one Function st b1,b1 are_equivalent_wrt b2,b3
symmetry
for b1, b2 being non empty ManySortedSign holds
not ( ex b3, b4 being one-to-one Function st b1,b2 are_equivalent_wrt b3,b4 & ( for b3, b4 being one-to-one Function holds
not b2,b1 are_equivalent_wrt b3,b4 ) )
end;
:: deftheorem Def10 defines are_equivalent CIRCTRM1:def 10 :
theorem Th30: :: CIRCTRM1:30
:: deftheorem Def11 defines preserves_inputs_of CIRCTRM1:def 11 :
theorem Th31: :: CIRCTRM1:31
theorem Th32: :: CIRCTRM1:32
theorem Th33: :: CIRCTRM1:33
theorem Th34: :: CIRCTRM1:34
:: deftheorem Def12 defines form_embedding_of CIRCTRM1:def 12 :
theorem Th35: :: CIRCTRM1:35
theorem Th36: :: CIRCTRM1:36
for b
1, b
2, b
3 being non
empty ManySortedSign for b
4, b
5, b
6, b
7 being
Functionfor b
8 being
non-empty MSAlgebra of b
1for b
9 being
non-empty MSAlgebra of b
2for b
10 being
non-empty MSAlgebra of b
3 holds
( b
4,b
5 form_embedding_of b
8,b
9 & b
6,b
7 form_embedding_of b
9,b
10 implies b
6 * b
4,b
7 * b
5 form_embedding_of b
8,b
10 )
:: deftheorem Def13 defines are_similar_wrt CIRCTRM1:def 13 :
theorem Th37: :: CIRCTRM1:37
theorem Th38: :: CIRCTRM1:38
theorem Th39: :: CIRCTRM1:39
theorem Th40: :: CIRCTRM1:40
theorem Th41: :: CIRCTRM1:41
for b
1, b
2, b
3 being non
empty ManySortedSign for b
4, b
5, b
6, b
7 being
Functionfor b
8 being
non-empty MSAlgebra of b
1for b
9 being
non-empty MSAlgebra of b
2for b
10 being
non-empty MSAlgebra of b
3 holds
( b
8,b
9 are_similar_wrt b
4,b
5 & b
9,b
10 are_similar_wrt b
6,b
7 implies b
8,b
10 are_similar_wrt b
6 * b
4,b
7 * b
5 )
:: deftheorem Def14 defines are_similar CIRCTRM1:def 14 :
theorem Th42: :: CIRCTRM1:42
theorem Th43: :: CIRCTRM1:43
theorem Th44: :: CIRCTRM1:44
theorem Th45: :: CIRCTRM1:45
theorem Th46: :: CIRCTRM1:46
theorem Th47: :: CIRCTRM1:47
theorem Th48: :: CIRCTRM1:48
theorem Th49: :: CIRCTRM1:49
theorem Th50: :: CIRCTRM1:50
theorem Th51: :: CIRCTRM1:51
theorem Th52: :: CIRCTRM1:52
theorem Th53: :: CIRCTRM1:53
theorem Th54: :: CIRCTRM1:54
theorem Th55: :: CIRCTRM1:55
theorem Th56: :: CIRCTRM1:56
theorem Th57: :: CIRCTRM1:57
theorem Th58: :: CIRCTRM1:58
:: deftheorem Def15 defines calculates CIRCTRM1:def 15 :
:: deftheorem Def16 defines specifies CIRCTRM1:def 16 :
definition
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
V5 ManySortedSet of the
carrier of c
1;
let c
3 be
non-empty MSAlgebra of c
1;
let c
4 be non
empty Subset of
(c1 -Terms c2);
let c
5 be non
empty non
void Circuit-like ManySortedSign ;
let c
6 be
non-empty Circuit of c
5;
assume
c
6 calculates c
4,c
3
;
then consider c
7, c
8 being
Function such that E58:
c
7,c
8 form_embedding_of c
4 -Circuit c
3,c
6
and E59:
c
7 preserves_inputs_of c
4 -CircuitStr ,c
5
by Def15;
E60:
c
7 is
one-to-one
by E58, Def12;
mode SortMap of c
4,c
3,c
6 -> one-to-one Function means :
Def17:
:: CIRCTRM1:def 17
( a
7 preserves_inputs_of a
4 -CircuitStr ,a
5 & ex b
1 being
Function st a
7,b
1 form_embedding_of a
4 -Circuit a
3,a
6 );
existence
ex b1 being one-to-one Function st
( b1 preserves_inputs_of c4 -CircuitStr ,c5 & ex b2 being Function st b1,b2 form_embedding_of c4 -Circuit c3,c6 )
by E58, E59, E60;
end;
:: deftheorem Def17 defines SortMap CIRCTRM1:def 17 :
definition
let c
1 be non
empty non
void ManySortedSign ;
let c
2 be
V5 ManySortedSet of the
carrier of c
1;
let c
3 be
non-empty MSAlgebra of c
1;
let c
4 be non
empty Subset of
(c1 -Terms c2);
let c
5 be non
empty non
void Circuit-like ManySortedSign ;
let c
6 be
non-empty Circuit of c
5;
assume E59:
c
6 calculates c
4,c
3
;
let c
7 be
SortMap of c
4,c
3,c
6;
consider c
8 being
Function such that E60:
c
7,c
8 form_embedding_of c
4 -Circuit c
3,c
6
by E59, Def17;
E61:
c
8 is
one-to-one
by E60, Def12;
mode OperMap of c
4,c
3,c
6,c
7 -> one-to-one Function means :: CIRCTRM1:def 18
a
7,a
8 form_embedding_of a
4 -Circuit a
3,a
6;
existence
ex b1 being one-to-one Function st c7,b1 form_embedding_of c4 -Circuit c3,c6
by E60, E61;
end;
:: deftheorem Def18 defines OperMap CIRCTRM1:def 18 :
theorem Th59: :: CIRCTRM1:59
theorem Th60: :: CIRCTRM1:60
theorem Th61: :: CIRCTRM1:61
theorem Th62: :: CIRCTRM1:62
theorem Th63: :: CIRCTRM1:63