def Q3_1():
clear()
A = Var("A")
B = Var("B")
p1 = premise(Or(A, B))
return orE(p1, arrowI(assume(A), orIR(assumed(A), Or(B, A)), Arrow(A, Or(B, A))), arrowI(assume(B), orIL(assumed(B), Or(B, A)), Arrow(B, Or(B, A))), Or(B, A))
def Q3_2():
clear()
A = Var("A")
B = Var("B")
p1 = premise(Or(A, B))
p2 = premise(Not(B))
return orE(p1, Arrow(A,A), arrowI(assume(B), FE(notE(assumed(B), p2, false()), A), Arrow(B, A)), A)
def Q3_3():
clear()
A = Var("A")
B = Var("B")
p1 = premise(Or(Not(A), Not(B)))
end = Not(And(A,B))
A_B = And(A,B)
return orE(p1, arrowI(assume(Not(A)), notI(arrowI(assume(A_B), notE(andEL(assumed(A_B), A), assumed(Not(A)), false()), Arrow(A_B, false())), Not(A_B)), Arrow(Not(A), Not(A_B))), arrowI(assume(Not(B)), notI(arrowI(assume(A_B), notE(andER(assumed(A_B), B), assumed(Not(B)), false()), Arrow(A_B, false())), Not(A_B)), Arrow(Not(B), Not(A_B))), end)
def andComm():
clear()
a = Var("A")
b = Var("B")
p1 = premise(And(a, b))
# The proof itself looks different, but it's really just on it's side.
#
# each proof rule is written as rule(support, expression)
# since the andI rule B && A needs two pieces (B and A), we need the two proofs
# the proof for B is andER, and the proof of A is andEL
return andI(andER(p1, b), \
andEL(p1, a), \
And(b,a))
def example():
clear()
p1 = premise(parse("EX x. FA y. P(x,y)"))
a1 = assume(parse("FA y. P(u,y)"))
a2 = assume(Var("v"))
l1 = assumed(parse("FA y. P(u,y)"))
l2 = forallE(l1, "v", parse("P(u,v)"))
l3 = existsI(l2, "u", parse("EX x. P(x,v)"))
l4 = forallI(a2, l3, parse("FA y. EX x. P(x,y)"))
l5 = arrowI(a1, l4, parse("(FA y. P(u,y)) -> (FA y. EX x. P(x,y))"))
l6 = existsE(p1, "u", l5, parse("FA y. EX x. P(x,y)"))
return l6
def main():
try:
clear()
forall_comm().print_proof()
clear()
DM1().print_proof()
clear()
DM2().print_proof()
clear()
DM3().print_proof()
clear()
contra().print_proof()
except (ParseException, LexException) as e:
print(str(e))
except (ProofException) as e:
e.print()
except Exception as e:
print(e)
def main():
try:
clear()
ren_forall().print_proof()
clear()
ren_exists().print_proof()
clear()
exists_comm().print_proof()
except (ProofException) as e:
e.print()
def main():
try:
clear()
orComm().print_proof()
clear()
disSyl().print_proof()
clear()
DM1().print_proof()
clear()
a = Var("a")
b = Var("b")
dlPrem = premise(Not(Or(Not(a), Not(b))))
DL1(dlPrem).print_proof()
clear()
dlPrem = premise(Not(Or(Not(a), Not(b))))
DL2(dlPrem).print_proof()
clear()
DM2().print_proof()
except (ProofException) as e:
e.print()