:: Hilbert Positive Propositional Calculus
:: by Adam Grabowski
::
:: Received February 20, 1999
:: Copyright (c) 1999 Association of Mizar Users
:: deftheorem Def1 defines with_VERUM HILBERT1:def 1 :
:: deftheorem Def2 defines with_implication HILBERT1:def 2 :
:: deftheorem Def3 defines with_conjunction HILBERT1:def 3 :
:: deftheorem Def4 defines with_propositional_variables HILBERT1:def 4 :
:: deftheorem Def5 defines HP-closed HILBERT1:def 5 :
Lm1:
for D being set st D is HP-closed holds
not D is empty
:: deftheorem Def6 defines HP-WFF HILBERT1:def 6 :
:: deftheorem defines VERUM HILBERT1:def 7 :
:: deftheorem defines => HILBERT1:def 8 :
:: deftheorem defines '&' HILBERT1:def 9 :
:: deftheorem Def10 defines Hilbert_theory HILBERT1:def 10 :
:: deftheorem Def11 defines CnPos HILBERT1:def 11 :
:: deftheorem defines HP_TAUT HILBERT1:def 12 :
theorem Th1: :: HILBERT1:1
theorem Th2: :: HILBERT1:2
theorem Th3: :: HILBERT1:3
theorem Th4: :: HILBERT1:4
theorem Th5: :: HILBERT1:5
theorem Th6: :: HILBERT1:6
theorem Th7: :: HILBERT1:7
theorem Th8: :: HILBERT1:8
theorem Th9: :: HILBERT1:9
theorem Th10: :: HILBERT1:10
Lm2:
for X being Subset of HP-WFF holds CnPos (CnPos X) c= CnPos X
theorem :: HILBERT1:11
Lm3:
for X being Subset of HP-WFF holds CnPos X is Hilbert_theory
theorem Th12: :: HILBERT1:12
theorem :: HILBERT1:13
theorem Th14: :: HILBERT1:14
theorem Th15: :: HILBERT1:15
theorem :: HILBERT1:16
theorem :: HILBERT1:17
theorem :: HILBERT1:18
theorem Th19: :: HILBERT1:19
theorem Th20: :: HILBERT1:20
theorem Th21: :: HILBERT1:21
theorem Th22: :: HILBERT1:22
theorem Th23: :: HILBERT1:23
Lm4:
for q, r, p, s being Element of HP-WFF holds (((q => r) => (p => r)) => s) => ((p => q) => s) in HP_TAUT
theorem Th24: :: HILBERT1:24
theorem :: HILBERT1:25
theorem Th26: :: HILBERT1:26
theorem Th27: :: HILBERT1:27
theorem :: HILBERT1:28
theorem Th29: :: HILBERT1:29
theorem Th30: :: HILBERT1:30
theorem Th31: :: HILBERT1:31
theorem :: HILBERT1:32
theorem Th33: :: HILBERT1:33
theorem Th34: :: HILBERT1:34
theorem Th35: :: HILBERT1:35
theorem Th36: :: HILBERT1:36
theorem :: HILBERT1:37
theorem :: HILBERT1:38
theorem Th39: :: HILBERT1:39
theorem Th40: :: HILBERT1:40
theorem Th41: :: HILBERT1:41
theorem Th42: :: HILBERT1:42
theorem Th43: :: HILBERT1:43
theorem Th44: :: HILBERT1:44
theorem Th45: :: HILBERT1:45
theorem Th46: :: HILBERT1:46
theorem :: HILBERT1:47
theorem :: HILBERT1:48
Lm5:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => q in HP_TAUT
Lm6:
for p, q, s being Element of HP-WFF holds (((p '&' q) '&' s) '&' ((p '&' q) '&' s)) => (((p '&' q) '&' s) '&' q) in HP_TAUT
Lm7:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (((p '&' q) '&' s) '&' q) in HP_TAUT
Lm8:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (p '&' s) in HP_TAUT
Lm9:
for p, q, s being Element of HP-WFF holds (((p '&' q) '&' s) '&' q) => ((p '&' s) '&' q) in HP_TAUT
Lm10:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((p '&' s) '&' q) in HP_TAUT
Lm11:
for p, s, q being Element of HP-WFF holds ((p '&' s) '&' q) => ((s '&' p) '&' q) in HP_TAUT
Lm12:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((s '&' p) '&' q) in HP_TAUT
Lm13:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => ((s '&' q) '&' p) in HP_TAUT
Lm14:
for p, q, s being Element of HP-WFF holds ((p '&' q) '&' s) => (p '&' (s '&' q)) in HP_TAUT
Lm15:
for p, s, q being Element of HP-WFF holds (p '&' (s '&' q)) => (p '&' (q '&' s)) in HP_TAUT
theorem :: HILBERT1:49
Lm16:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((s '&' q) '&' p) in HP_TAUT
Lm17:
for s, q, p being Element of HP-WFF holds ((s '&' q) '&' p) => ((q '&' s) '&' p) in HP_TAUT
Lm18:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((q '&' s) '&' p) in HP_TAUT
Lm19:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => ((p '&' s) '&' q) in HP_TAUT
Lm20:
for p, q, s being Element of HP-WFF holds (p '&' (q '&' s)) => (p '&' (s '&' q)) in HP_TAUT
theorem :: HILBERT1:50