def contra():
return arrowI(
assume(parse("EX x.P(x)")),
existsE(
assumed(parse("EX x.P(x)")), "c",
forallE(premise(parse("FA x.P(x) -> Q")), "c", parse("P(c) -> Q")),
parse("Q")), parse("(EX x.P(x)) -> Q"))
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 contra():
return arrowI(
assume(parse("EX x. P(x)")),
arrowE(
existsE(
assumed(parse("EX x. P(x)")), "a",
arrowI(assume(parse("P(a)")), assumed(parse("P(a)")),
Arrow(parse("P(a)"), parse("P(a)"))), parse("P(a)")),
forallE(premise(parse("FA x. (P(x)->Q)")), "a", parse("P(a)->Q")),
Var("Q")), Arrow(parse("EX x. P(x)"), Var("Q")))
def DM2():
return notI(
arrowI(
assume(parse("FA x. P(x)")),
notE(
forallE(assumed(parse("FA x. P(x)")), "a", parse("P(a)")),
existsE(
premise(parse("EX x. ~P(x)")), "a",
arrowI(assume(Not(parse("P(a)"))),
assumed(Not(parse("P(a)"))),
Arrow(Not(parse("P(a)")), Not(parse("P(a)")))),
Not(parse("P(a)"))), false()),
(Arrow(parse("FA x. P(x)"), false()))), Not(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 DM2():
return notI(
arrowI(
assume(parse("FA x. P(x)")),
notE(
forallE(assumed(parse("FA x. P(x)")), "c", parse("P(c)")),
existsE(
premise(parse("EX x. ~P(x)")), "d",
arrowI(
assume(parse("~P(d)")),
FE(
notE(
forallE(assumed(parse("FA x. P(x)")), "d",
parse("P(d)")),
assumed(parse("~P(d)")), false()),
parse("~P(c)")), parse("~P(d) -> ~P(c)")),
parse("~P(c)")), false()),
Arrow(parse("FA x. P(x)"), false())), parse("~FA x. P(x)"))
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)"))