def exists_comm():
return existsE(
premise(parse("EX x. EX y. P(x,y)")), "c",
arrowI(
assume(parse("EX y. P(c,y)")),
existsE(
assumed(parse("EX y. P(c,y)")), "d",
arrowI(
assume(parse("P(c,d)")),
existsI(
existsI(assumed(parse("P(c,d)")), "c",
parse("EX x. P(x,d)")), "d",
parse("EX y. EX x. P(x,y)")),
parse("P(c,d) -> EX y. EX x. P(x,y)")),
parse("EX y. EX x. P(x,y)")),
parse("(EX y. P(c,y)) -> EX y. EX x. P(x,y)")),
parse("EX y. EX x. P(x,y)"))
def DM3():
return forallI(
assume(Var("a")),
notI(
arrowI(
assume(parse("P(a)")),
notE(existsI(assumed(parse("P(a)")), "a", parse("EX x. P(x)")),
premise(Not(parse("EX x. P(x)"))), false()),
Arrow(parse("P(a)"), false())), Not(parse("P(a)"))),
parse("FA x. ~P(x)"))
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 ren_exists():
return existsE(
premise(parse("EX x. P(x)")), "c",
arrowI(assume(parse("P(c)")),
existsI(assumed(parse("P(c)")), "c", parse("EX z. P(z)")),
parse("P(c) -> EX z. P(z)")), parse("EX z. P(z)"))