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euclidean__tactics.dk
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classic :
sttfa.eps (sttfa.forall sttfa.bool (x:(sttfa.eta sttfa.bool) => logic.or x (logic.not x))).
thm NNPP :
sttfa.eps
(sttfa.forall sttfa.bool (p:(sttfa.eta sttfa.bool) => sttfa.impl (logic.not (logic.not p)) p))
:=
p:(sttfa.eta sttfa.bool) =>
H:(sttfa.eps (sttfa.impl (sttfa.impl p logic.false) logic.false)) =>
logic.or__ind
p
(logic.not p)
p
(H0:(sttfa.eps p) => H0)
(NP:(sttfa.eps (logic.not p)) => logic.False__ind p (H NP))
(classic p).
thm Col__or__nCol :
sttfa.eps
(sttfa.forall
euclidean__axioms.Point
(A:(sttfa.eta euclidean__axioms.Point) =>
sttfa.forall
euclidean__axioms.Point
(B:(sttfa.eta euclidean__axioms.Point) =>
sttfa.forall
euclidean__axioms.Point
(C:(sttfa.eta euclidean__axioms.Point) =>
logic.or (euclidean__axioms.Col A B C) (euclidean__axioms.nCol A B C)))))
:=
A:(sttfa.eta euclidean__axioms.Point) =>
B:(sttfa.eta euclidean__axioms.Point) =>
C:(sttfa.eta euclidean__axioms.Point) =>
NNPP
(logic.or
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or (euclidean__axioms.BetS A B C) (euclidean__axioms.BetS A C B))))))
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C))))))))
(H:(sttfa.eps
(logic.not
(logic.or
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))))
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))))) =>
(H0:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))))
logic.false)) =>
(H1:(sttfa.eps
(sttfa.impl
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))
logic.false)) =>
(H2:(sttfa.eps
(sttfa.impl
(euclidean__axioms.neq A B)
(sttfa.impl
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C))))))
logic.false))) =>
(H3:(sttfa.eps (sttfa.impl (euclidean__axioms.eq A B) logic.false)) =>
(H4:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B)))))
logic.false)) =>
(H5:(sttfa.eps
(sttfa.impl
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C))))))
logic.false)) =>
(H6:(sttfa.eps
(sttfa.impl
(euclidean__axioms.neq A C)
(sttfa.impl
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not
(euclidean__axioms.BetS A C B))
(logic.not
(euclidean__axioms.BetS B A C)))))
logic.false))) =>
(H7:(sttfa.eps
(sttfa.impl (euclidean__axioms.eq A C) logic.false)) =>
(H8:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))
logic.false)) =>
(H9:(sttfa.eps
(sttfa.impl
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not
(euclidean__axioms.BetS A B C))
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C)))))
logic.false)) =>
(H10:(sttfa.eps
(sttfa.impl
(euclidean__axioms.neq B C)
(sttfa.impl
(logic.and
(logic.not
(euclidean__axioms.BetS
A
B
C))
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C))))
logic.false))) =>
(H11:(sttfa.eps
(sttfa.impl
(euclidean__axioms.eq B C)
logic.false)) =>
(H12:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.BetS
B
A
C)
(logic.or
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B)))
logic.false)) =>
(H13:(sttfa.eps
(sttfa.impl
(logic.and
(logic.not
(euclidean__axioms.BetS
A
B
C))
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C))))
logic.false)) =>
(H14:(sttfa.eps
(sttfa.impl
(logic.not
(euclidean__axioms.BetS
A
B
C))
(sttfa.impl
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C)))
logic.false))) =>
(H15:(sttfa.eps
(sttfa.impl
(euclidean__axioms.BetS
B
A
C)
logic.false)) =>
(H16:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B))
logic.false)) =>
(H17:(sttfa.eps
(sttfa.impl
(euclidean__axioms.BetS
A
B
C)
logic.false)) =>
(H18:(sttfa.eps
(sttfa.impl
(euclidean__axioms.BetS
A
C
B)
logic.false)) =>
(H19:(sttfa.eps
(sttfa.impl
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C)))
logic.false)) =>
(H20:(sttfa.eps
(sttfa.impl
(logic.not
(euclidean__axioms.BetS
A
C
B))
(sttfa.impl
(logic.not
(euclidean__axioms.BetS
B
A
C))
logic.false))) =>
(H21:(sttfa.eps
(sttfa.impl
(logic.not
(euclidean__axioms.BetS
B
A
C))
logic.false)) =>
(H22:(sttfa.eps
logic.false) =>
logic.False__ind
logic.false
H22)
(H21
H15))
(H20 H18))
(H20:(sttfa.eps
(logic.not
(euclidean__axioms.BetS
A
C
B))) =>
H21:(sttfa.eps
(logic.not
(euclidean__axioms.BetS
B
A
C))) =>
H19
(logic.conj
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C))
H20
H21)))
(H14 H17))
(H18:(sttfa.eps
(euclidean__axioms.BetS
A
C
B)) =>
H16
(logic.or__intror
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B)
H18)))
(H17:(sttfa.eps
(euclidean__axioms.BetS
A
B
C)) =>
H16
(logic.or__introl
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B)
H17)))
(H16:(sttfa.eps
(logic.or
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B))) =>
H12
(logic.or__intror
(euclidean__axioms.BetS
B
A
C)
(logic.or
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B))
H16)))
(H15:(sttfa.eps
(euclidean__axioms.BetS
B
A
C)) =>
H12
(logic.or__introl
(euclidean__axioms.BetS
B
A
C)
(logic.or
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B))
H15)))
(H14:(sttfa.eps
(logic.not
(euclidean__axioms.BetS
A
B
C))) =>
H15:(sttfa.eps
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C)))) =>
H13
(logic.conj
(logic.not
(euclidean__axioms.BetS
A
B
C))
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C)))
H14
H15)))
(H10 H11))
(H12:(sttfa.eps
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B)))) =>
H8
(logic.or__intror
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B)))
H12)))
(H11:(sttfa.eps (euclidean__axioms.eq B C)) =>
H8
(logic.or__introl
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B)))
H11)))
(H10:(sttfa.eps (euclidean__axioms.neq B C)) =>
H11:(sttfa.eps
(logic.and
(logic.not
(euclidean__axioms.BetS A B C))
(logic.and
(logic.not
(euclidean__axioms.BetS A C B))
(logic.not
(euclidean__axioms.BetS B A C))))) =>
H9
(logic.conj
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not
(euclidean__axioms.BetS A C B))
(logic.not
(euclidean__axioms.BetS B A C))))
H10
H11)))
(H6 H7))
(H8:(sttfa.eps
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))) =>
H4
(logic.or__intror
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))
H8)))
(H7:(sttfa.eps (euclidean__axioms.eq A C)) =>
H4
(logic.or__introl
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))
H7)))
(H6:(sttfa.eps (euclidean__axioms.neq A C)) =>
H7:(sttfa.eps
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))) =>
H5
(logic.conj
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))
H6
H7)))
(H2 H3))
(H4:(sttfa.eps
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B)))))) =>
H0
(logic.or__intror
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B)))))
H4)))
(H3:(sttfa.eps (euclidean__axioms.eq A B)) =>
H0
(logic.or__introl
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B)))))
H3)))
(H2:(sttfa.eps (euclidean__axioms.neq A B)) =>
H3:(sttfa.eps
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C))))))) =>
H1
(logic.conj
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C))))))
H2
H3)))
(H1:(sttfa.eps
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))) =>
H
(logic.or__intror
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))))
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))
H1)))
(H0:(sttfa.eps
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))))) =>
H
(logic.or__introl
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))))
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))
H0))).
thm nCol__or__Col :
sttfa.eps
(sttfa.forall
euclidean__axioms.Point
(A:(sttfa.eta euclidean__axioms.Point) =>
sttfa.forall
euclidean__axioms.Point
(B:(sttfa.eta euclidean__axioms.Point) =>
sttfa.forall
euclidean__axioms.Point
(C:(sttfa.eta euclidean__axioms.Point) =>
logic.or (euclidean__axioms.nCol A B C) (euclidean__axioms.Col A B C)))))
:=
A:(sttfa.eta euclidean__axioms.Point) =>
B:(sttfa.eta euclidean__axioms.Point) =>
C:(sttfa.eta euclidean__axioms.Point) =>
NNPP
(logic.or
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or (euclidean__axioms.BetS A B C) (euclidean__axioms.BetS A C B)))))))
(H:(sttfa.eps
(logic.not
(logic.or
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))))))) =>
(H0:(sttfa.eps
(sttfa.impl
(logic.and
(euclidean__axioms.neq A B)
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C)))))))
logic.false)) =>
(H1:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.eq A B)
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))))
logic.false)) =>
(H2:(sttfa.eps
(sttfa.impl
(euclidean__axioms.neq A B)
(sttfa.impl
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C))))))
logic.false))) =>
(H3:(sttfa.eps (sttfa.impl (euclidean__axioms.eq A B) logic.false)) =>
(H4:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.eq A C)
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B)))))
logic.false)) =>
(H5:(sttfa.eps
(sttfa.impl
(logic.and
(euclidean__axioms.neq A C)
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not (euclidean__axioms.BetS A C B))
(logic.not (euclidean__axioms.BetS B A C))))))
logic.false)) =>
(H6:(sttfa.eps
(sttfa.impl
(euclidean__axioms.neq A C)
(sttfa.impl
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not (euclidean__axioms.BetS A B C))
(logic.and
(logic.not
(euclidean__axioms.BetS A C B))
(logic.not
(euclidean__axioms.BetS B A C)))))
logic.false))) =>
(H7:(sttfa.eps
(sttfa.impl (euclidean__axioms.eq A C) logic.false)) =>
(H8:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.eq B C)
(logic.or
(euclidean__axioms.BetS B A C)
(logic.or
(euclidean__axioms.BetS A B C)
(euclidean__axioms.BetS A C B))))
logic.false)) =>
(H9:(sttfa.eps
(sttfa.impl
(logic.and
(euclidean__axioms.neq B C)
(logic.and
(logic.not
(euclidean__axioms.BetS A B C))
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C)))))
logic.false)) =>
(H10:(sttfa.eps
(sttfa.impl
(euclidean__axioms.neq B C)
(sttfa.impl
(logic.and
(logic.not
(euclidean__axioms.BetS
A
B
C))
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C))))
logic.false))) =>
(H11:(sttfa.eps
(sttfa.impl
(euclidean__axioms.eq B C)
logic.false)) =>
(H12:(sttfa.eps
(sttfa.impl
(logic.or
(euclidean__axioms.BetS
B
A
C)
(logic.or
(euclidean__axioms.BetS
A
B
C)
(euclidean__axioms.BetS
A
C
B)))
logic.false)) =>
(H13:(sttfa.eps
(sttfa.impl
(logic.and
(logic.not
(euclidean__axioms.BetS
A
B
C))
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C))))
logic.false)) =>
(H14:(sttfa.eps
(sttfa.impl
(logic.not
(euclidean__axioms.BetS
A
B
C))
(sttfa.impl
(logic.and
(logic.not
(euclidean__axioms.BetS
A
C
B))
(logic.not
(euclidean__axioms.BetS
B
A
C)))
logic.false))) =>
(H15:(sttfa.eps