:: WSIERP_1 semantic presentation
theorem Th1: :: WSIERP_1:1
canceled;
theorem Th2: :: WSIERP_1:2
theorem Th3: :: WSIERP_1:3
theorem Th4: :: WSIERP_1:4
canceled;
theorem Th5: :: WSIERP_1:5
for b
1, b
2 being
real number for b
3 being
Nat holds
( b
1 >= 0 & b
2 >= 0 & b
3 > 0 & b
1 |^ b
3 = b
2 |^ b
3 implies b
1 = b
2 )
theorem Th6: :: WSIERP_1:6
Lemma4:
for b1, b2 being real number holds
( ( b1 >= 0 implies b1 + b2 >= b2 ) & ( b1 + b2 >= b2 implies b1 >= 0 ) & not ( b1 > 0 & not b1 + b2 > b2 ) & not ( b1 + b2 > b2 & not b1 > 0 ) & ( b1 >= 0 implies b2 >= b2 - b1 ) & ( b2 >= b2 - b1 implies b1 >= 0 ) & not ( b1 > 0 & not b2 > b2 - b1 ) & not ( b2 > b2 - b1 & not b1 > 0 ) )
Lemma5:
for b1, b2, b3 being real number holds
( b1 >= 0 & b2 >= b3 implies ( b1 + b2 >= b3 & b2 >= b3 - b1 ) )
Lemma6:
for b1, b2, b3 being real number holds
( ( ( b1 >= 0 & b2 > b3 ) or ( b1 > 0 & b2 >= b3 ) ) implies ( b1 + b2 > b3 & b2 > b3 - b1 ) )
theorem Th7: :: WSIERP_1:7
theorem Th8: :: WSIERP_1:8
Lemma7:
for b1, b2 being Nat holds
( b1 divides b2 iff b1 divides b2 )
Lemma8:
for b1 being Nat holds abs b1 = b1
by ABSVALUE:def 1;
Lemma9:
for b1 being Nat holds
not for b2 being Nat holds
( not b1 = 2 * b2 & not b1 = (2 * b2) + 1 )
Lemma10:
for b1, b2, b3 being Nat holds
( b1 > 0 & b2 |^ b1 = b3 |^ b1 implies b2 = b3 )
by Th5;
theorem Th9: :: WSIERP_1:9
theorem Th10: :: WSIERP_1:10
Lemma13:
for b1, b2 being Nat holds b1 gcd b2 = b1 hcf b2
Lemma14:
for b1, b2 being Nat holds
( b1,b2 are_relative_prime iff b1,b2 are_relative_prime )
theorem Th11: :: WSIERP_1:11
theorem Th12: :: WSIERP_1:12
for b
1, b
2, b
3 being
Nat holds
( b
1 hcf b
2 = 1 & b
3 hcf b
2 = 1 implies
(b1 * b3) hcf b
2 = 1 )
theorem Th13: :: WSIERP_1:13
theorem Th14: :: WSIERP_1:14
theorem Th15: :: WSIERP_1:15
theorem Th16: :: WSIERP_1:16
theorem Th17: :: WSIERP_1:17
Lemma22:
for b1, b2, b3, b4 being Nat holds
( b1 hcf b2 = 1 implies ( b1 hcf (b2 |^ b3) = 1 & (b1 |^ b4) hcf (b2 |^ b3) = 1 ) )
theorem Th18: :: WSIERP_1:18
theorem Th19: :: WSIERP_1:19
theorem Th20: :: WSIERP_1:20
theorem Th21: :: WSIERP_1:21
Lemma26:
for b1, b2 being Nat holds
( b1 <> 0 implies ( b1 divides b2 iff b2 / b1 is Nat ) )
theorem Th22: :: WSIERP_1:22
Lemma27:
for b1, b2 being Nat
for b3 being Integer holds
( b1 <> 0 & b2 * b3 = b1 implies b3 is Nat )
Lemma28:
for b1 being Nat
for b2 being Integer holds
( b1 <= b2 implies b2 is Nat )
by INT_1:16;
Lemma29:
for b1, b2, b3 being Nat holds
( b1 + b2 <= b3 implies ( b1 <= b3 & b2 <= b3 ) )
theorem Th23: :: WSIERP_1:23
for b
1, b
2, b
3 being
Nat holds
( b
1 <= b
2 - b
3 implies ( b
1 <= b
2 & b
3 <= b
2 ) )
Lemma30:
for b1, b2 being Integer holds
( b1 < b2 iff b1 <= b2 - 1 )
Lemma31:
for b1, b2 being Integer holds
( b1 < b2 + 1 iff b1 <= b2 )
Lemma32:
for b1 being real number
for b2 being Function holds
( ( b1 in dom b2 & b2 . b1 in rng b2 ) or b2 . b1 = {} )
Lemma33:
for b1 being Nat
for b2 being set holds
( 0 in b2 implies for b3 being FinSequence of b2 holds
b3 . b1 is Element of b2 )
Lemma34:
for b1 being Nat
for b2 being FinSequence holds
not ( b1 in dom b2 & ( for b3, b4 being FinSequence holds
not ( b2 = (b3 ^ <*(b2 . b1)*>) ^ b4 & len b3 = b1 - 1 & len b4 = (len b2) - b1 ) ) )
:: deftheorem Def1 defines Del WSIERP_1:def 1 :
for b
1 being
Natfor b
2 being
FinSequencefor b
3 being
set holds
( ( not b
1 in dom b
2 implies ( b
3 = Del b
2,b
1 iff b
3 = b
2 ) ) & ( b
1 in dom b
2 implies ( b
3 = Del b
2,b
1 iff (
(len b3) + 1
= len b
2 & ( for b
4 being
Nat holds
( ( b
4 < b
1 implies b
3 . b
4 = b
2 . b
4 ) & ( b
4 >= b
1 implies b
3 . b
4 = b
2 . (b4 + 1) ) ) ) ) ) ) );
Lemma36:
for b1 being Nat
for b2, b3, b4 being FinSequence
for b5 being set holds
( b1 in dom b2 & b2 = (b3 ^ <*b5*>) ^ b4 & len b3 = b1 - 1 implies Del b2,b1 = b3 ^ b4 )
Lemma37:
for b1 being Nat
for b2 being FinSequence holds dom (Del b2,b1) c= dom b2
Lemma38:
for b1 being Nat
for b2, b3 being FinSequence holds
( ( b1 <= len b2 implies Del (b2 ^ b3),b1 = (Del b2,b1) ^ b3 ) & ( b1 >= 1 implies Del (b2 ^ b3),((len b2) + b1) = b2 ^ (Del b3,b1) ) )
Lemma39:
for b1 being FinSequence
for b2 being set holds
( Del (<*b2*> ^ b1),1 = b1 & Del (b1 ^ <*b2*>),((len b1) + 1) = b1 )
theorem Th24: :: WSIERP_1:24
canceled;
theorem Th25: :: WSIERP_1:25
canceled;
theorem Th26: :: WSIERP_1:26
for b
1, b
2, b
3 being
set holds
(
Del <*b1*>,1
= {} &
Del <*b1,b2*>,1
= <*b2*> &
Del <*b1,b2*>,2
= <*b1*> &
Del <*b1,b2,b3*>,1
= <*b2,b3*> &
Del <*b1,b2,b3*>,2
= <*b1,b3*> &
Del <*b1,b2,b3*>,3
= <*b1,b2*> )
Lemma40:
for b1 being FinSequence holds
( 1 <= len b1 implies ( b1 = <*(b1 . 1)*> ^ (Del b1,1) & b1 = (Del b1,(len b1)) ^ <*(b1 . (len b1))*> ) )
Lemma41:
for b1 being Nat
for b2 being FinSequence of REAL holds
( b1 in dom b2 implies (Product (Del b2,b1)) * (b2 . b1) = Product b2 )
theorem Th27: :: WSIERP_1:27
theorem Th28: :: WSIERP_1:28
theorem Th29: :: WSIERP_1:29
theorem Th30: :: WSIERP_1:30
theorem Th31: :: WSIERP_1:31
for b
1, b
2 being
Nat holds
not ( ex b
3 being
Rational st b
1 = b
3 |^ b
2 & ( for b
3 being
Integer holds
not b
1 = b
3 |^ b
2 ) )
theorem Th32: :: WSIERP_1:32
for b
1, b
2 being
Nat holds
not ( ex b
3 being
Rational st b
1 = b
3 |^ b
2 & ( for b
3 being
Nat holds
not b
1 = b
3 |^ b
2 ) )
theorem Th33: :: WSIERP_1:33
theorem Th34: :: WSIERP_1:34
for b
1, b
2 being
Nat holds
ex b
3, b
4 being
Integer st b
1 hcf b
2 = (b1 * b3) + (b2 * b4)
theorem Th35: :: WSIERP_1:35
theorem Th36: :: WSIERP_1:36
theorem Th37: :: WSIERP_1:37
theorem Th38: :: WSIERP_1:38
for b
1, b
2 being
Nat holds
not ( b
1 <> 0 & b
2 <> 0 & ( for b
3, b
4 being
Nat holds
not b
1 hcf b
2 = (b1 * b3) - (b2 * b4) ) )
theorem Th39: :: WSIERP_1:39
for b
1, b
2, b
3, b
4 being
Nat holds
not ( b
1 > 0 & b
2 > 0 & b
1 hcf b
2 = 1 & b
3 |^ b
1 = b
4 |^ b
2 & ( for b
5 being
Nat holds
not ( b
3 = b
5 |^ b
2 & b
4 = b
5 |^ b
1 ) ) )
theorem Th40: :: WSIERP_1:40
theorem Th41: :: WSIERP_1:41
for b
1, b
2, b
3, b
4, b
5 being
Integer holds
( b
1 <> 0 & b
2 <> 0 &
(b1 * b3) + (b2 * b4) = b
5 implies for b
6, b
7 being
Integer holds
not (
(b1 * b6) + (b2 * b7) = b
5 & ( for b
8 being
Integer holds
not ( b
6 = b
3 + (b8 * (b2 / (b1 gcd b2))) & b
7 = b
4 - (b8 * (b1 / (b1 gcd b2))) ) ) ) )
theorem Th42: :: WSIERP_1:42
for b
1, b
2, b
3, b
4 being
Nat holds
not ( b
1 hcf b
2 = 1 & b
1 * b
2 = b
3 |^ b
4 & ( for b
5, b
6 being
Nat holds
not ( b
1 = b
5 |^ b
4 & b
2 = b
6 |^ b
4 ) ) )
theorem Th43: :: WSIERP_1:43
theorem Th44: :: WSIERP_1:44
Lemma52:
for b1, b2 being Nat holds
( b1 divides b2 & b2 < b1 implies b2 = 0 )
Lemma53:
for b1, b2 being Integer holds
( b1 divides b2 iff b1 divides abs b2 )
Lemma54:
for b1 being Nat
for b2 being Integer holds
( b1 divides b2 iff b1 divides abs b2 )
Lemma55:
for b1, b2 being Integer holds (b1 * b2) mod b2 = 0
Lemma56:
for b1, b2, b3 being Integer holds
( b1 mod b2 = b3 mod b2 implies (b1 - b3) mod b2 = 0 )
Lemma57:
for b1, b2 being Integer holds
( b1 <> 0 & b2 mod b1 = 0 implies b1 divides b2 )
Lemma58:
for b1 being Integer holds
( ( 1 < b1 implies ( 1 < sqrt b1 & sqrt b1 < b1 ) ) & ( 0 < b1 & b1 < 1 implies ( 0 < sqrt b1 & sqrt b1 < 1 & b1 < sqrt b1 ) ) )
theorem Th45: :: WSIERP_1:45
canceled;
theorem Th46: :: WSIERP_1:46
theorem Th47: :: WSIERP_1:47
theorem Th48: :: WSIERP_1:48