def forall_comm(): return forallI( assume(Var("a")), forallI( assume(Var("b")), forallE( forallE(premise(parse("FA x. FA y. P(x, y)")), "b", parse("FA y. P(b, y)")), "a", parse("P(b, a)")), parse("FA x. P(x, a)")), parse("FA y. FA x. P(x, y)"))
def forall_comm(): return forallI( assume(Var("c")), forallI( assume(Var("d")), forallE( forallE( premise(parse("FA x. FA y. P(x, y)")), "d", parse("FA y. P(d, y)"), ), "c", parse("P(d, c)"), ), parse("FA x. P(x, c)")), parse("FA y. FA x. P(x, y)"))
def ren_forall(): return forallI(assume(Var("x")), forallE( premise(parse("FA x. P(x)")), "x", parse("P(x)"), ), parse("FA z. P(z)"))
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 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 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