:: Preliminaries to Circuits, I
:: by Yatsuka Nakamura, Piotr Rudnicki, Andrzej Trybulec and Pauline N. Kawamoto
::
:: Received November 17, 1994
:: Copyright (c) 1994-2021 Association of Mizar Users

scheme :: PRE_CIRC:sch 1
FraenkelFinIm{ F₁() -> non empty finite set , F₂( set ) -> set , P₁[ set ] } :

{ F₂(x) where x is Element of F₁() : P₁[x] } is finite

proof end;

theorem :: PRE_CIRC:1

canceled;

::---------------------------------------------------------------------------
:: Many Sorted Sets and Functions
::---------------------------------------------------------------------------
::$CT

theorem :: PRE_CIRC:2

for I being set
for MSS being ManySortedSet of I holds (MSS #) . (<*> I) = {{}}

proof end;

scheme :: PRE_CIRC:sch 2
MSSLambda2Part{ F₁() -> set , P₁[ object ], F₂( object ) -> object , F₃( object ) -> object } :

ex f being ManySortedSet of F₁() st
for i being Element of F₁() st i in F₁() holds
( ( P₁[i] implies f . i = F₂(i) ) & ( P₁[i] implies f . i = F₃(i) ) )

proof end;

registration

let I be set ;

cluster Relation-like V2() I -defined Function-like total V44() for set ;

existence
ex b₁ being ManySortedSet of I st
( b₁ is V2() & b₁ is V44() )

proof end;

end;

registration

let I be set ;
let M be ManySortedSet of I;
let A be Subset of I;

cluster M | A -> A -defined total for A -defined Function;

coherence
for b₁ being A -defined Function st b₁ = M | A holds
b₁ is total

proof end;

end;

registration

let I be set ;
let M be ManySortedSet of I;
let A be Subset of I;

cluster M | A -> total ;

coherence
M | A is total ;

end;

registration

let M be non-empty Function;
let A be set ;

cluster M | A -> non-empty ;

coherence
M | A is non-empty

proof end;

end;

theorem Th2: :: PRE_CIRC:3

for I being non empty set
for B being V2() ManySortedSet of I holds not union (rng B) is empty

proof end;

theorem Th3: :: PRE_CIRC:4

for I being set holds uncurry (I --> {}) = {}

proof end;

theorem :: PRE_CIRC:5

for I being non empty set
for A being set
for B being V2() ManySortedSet of I
for F being ManySortedFunction of I --> A,B holds dom (commute F) = A

proof end;

scheme :: PRE_CIRC:sch 3
LambdaRecCorrD{ F₁() -> non empty set , F₂() -> Element of F₁(), F₃( Nat, Element of F₁()) -> Element of F₁() } :

( ex f being sequence of F₁() st
( f . 0 = F₂() & ( for i being Nat holds f . (i + 1) = F₃(i,(f . i)) ) ) & ( for f1, f2 being sequence of F₁() st f1 . 0 = F₂() & ( for i being Nat holds f1 . (i + 1) = F₃(i,(f1 . i)) ) & f2 . 0 = F₂() & ( for i being Nat holds f2 . (i + 1) = F₃(i,(f2 . i)) ) holds
f1 = f2 ) )

proof end;

scheme :: PRE_CIRC:sch 4
LambdaMSFD{ F₁() -> non empty set , F₂() -> Subset of F₁(), F₃() -> ManySortedSet of F₂(), F₄() -> ManySortedSet of F₂(), F₅( object ) -> set } :

ex f being ManySortedFunction of F₃(),F₄() st
for i being Element of F₁() st i in F₂() holds
f . i = F₅(i)

provided

A1: for i being Element of F₁() st i in F₂() holds
F₅(i) is Function of (F₃() . i),(F₄() . i)

proof end;

theorem :: PRE_CIRC:6

for I being set
for f being V2() ManySortedSet of I
for g being Function
for s being Element of product f st dom g c= dom f & ( for x being set st x in dom g holds
g . x in f . x ) holds
s +* g is Element of product f

proof end;

theorem :: PRE_CIRC:7

for A, B being non empty set
for C being V2() ManySortedSet of A
for InpFs being ManySortedFunction of A --> B,C
for b being Element of B ex c being ManySortedSet of A st
( c = (commute InpFs) . b & c in C )

proof end;

theorem :: PRE_CIRC:8

for n being Nat
for a being set holds product (n |-> {a}) = {(n |-> a)}

proof end;

registration

let D be non empty set ;

cluster -> finite for Element of FinTrees D;

coherence
for b₁ being Element of FinTrees D holds b₁ is finite

proof end;

end;

registration

let T be finite DecoratedTree;
let t be Element of dom T;

cluster T | t -> finite ;

coherence
T | t is finite

proof end;

end;

theorem Th8: :: PRE_CIRC:9

for T being finite Tree
for p being Element of T holds T | p, { t where t is Element of T : p is_a_prefix_of t } are_equipotent

proof end;

registration

let T be finite DecoratedTree-like Function;
let t be Element of dom T;
let T1 be finite DecoratedTree;

cluster T with-replacement (t,T1) -> finite ;

coherence
T with-replacement (t,T1) is finite

proof end;

end;

theorem Th9: :: PRE_CIRC:10

for T, T1 being finite Tree
for p being Element of T holds T with-replacement (p,T1) = { t where t is Element of T : not p is_a_prefix_of t } \/ { (p ^ t1) where t1 is Element of T1 : verum }

proof end;

theorem Th10: :: PRE_CIRC:11

for T, T1 being finite Tree
for p being Element of T
for f being FinSequence of NAT st f in T with-replacement (p,T1) & p is_a_prefix_of f holds
ex t1 being Element of T1 st f = p ^ t1

proof end;

theorem Th11: :: PRE_CIRC:12

for p being Tree-yielding FinSequence
for k being Element of NAT st k + 1 in dom p holds
(tree p) | <*k*> = p . (k + 1)

proof end;

theorem :: PRE_CIRC:13

for q being DTree-yielding FinSequence
for k being Nat st k + 1 in dom q holds
<*k*> in tree (doms q)

proof end;

theorem Th13: :: PRE_CIRC:14

for p, q being Tree-yielding FinSequence
for k being Element of NAT st len p = len q & k + 1 in dom p & ( for i being Nat st i in dom p & i <> k + 1 holds
p . i = q . i ) holds
for t being Tree st q . (k + 1) = t holds
tree q = (tree p) with-replacement (<*k*>,t)

proof end;

theorem :: PRE_CIRC:15

for e1, e2 being finite DecoratedTree
for x being set
for k being Element of NAT
for p being DTree-yielding FinSequence st <*k*> in dom e1 & e1 = x -tree p holds
ex q being DTree-yielding FinSequence st
( e1 with-replacement (<*k*>,e2) = x -tree q & len q = len p & q . (k + 1) = e2 & ( for i being Nat st i in dom p & i <> k + 1 holds
q . i = p . i ) )

proof end;

theorem :: PRE_CIRC:16

for T being finite Tree
for p being Element of T st p <> {} holds
card (T | p) < card T

proof end;

theorem Th16: :: PRE_CIRC:17

for T, T1 being finite Tree
for p being Element of T holds (card (T with-replacement (p,T1))) + (card (T | p)) = (card T) + (card T1)

proof end;

theorem :: PRE_CIRC:18

for T, T1 being finite DecoratedTree
for p being Element of dom T holds (card (T with-replacement (p,T1))) + (card (T | p)) = (card T) + (card T1)

proof end;

registration

let x be object ;

cluster root-tree x -> finite ;

coherence
root-tree x is finite

proof end;

end;

theorem :: PRE_CIRC:19

for x being set holds card (root-tree x) = 1

proof end;

:: from AMISTD_2

theorem Th19: :: PRE_CIRC:20

for F being non empty finite set holds (card F) - 1 = (card F) -' 1

proof end;

:: from AMISTD_2, 2006.03.26, A.T.

theorem :: PRE_CIRC:21

for f, g being Function holds
( dom f, dom g are_equipotent iff f,g are_equipotent )

proof end;

theorem :: PRE_CIRC:22

for f, g being finite Function st dom f misses dom g holds
card (f +* g) = (card f) + (card g)

proof end;

:: from SCMPDS_9. 2008.05.06, A.T.

theorem :: PRE_CIRC:23

for n being Nat holds { k where k is Nat : k > n } is infinite

proof end;

theorem Th23: :: PRE_CIRC:24

for D being finite set
for k being Nat st card D = k + 1 holds
ex x being Element of D ex C being Subset of D st
( D = C \/ {x} & card C = k )

proof end;

theorem Th24: :: PRE_CIRC:25

for D being finite set st card D = 1 holds
ex x being Element of D st D = {x}

proof end;

scheme :: PRE_CIRC:sch 5
MinValue{ F₁() -> non empty finite set , F₂( Element of F₁()) -> Real } :

ex x being Element of F₁() st
for y being Element of F₁() holds F₂(x) <= F₂(y)

proof end;