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 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 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 DL2(p1):
a = Var("a")
b = Var("b")
return doubleNeg(
notI(
arrowI(
assume(Not(b)),
notE(orIR(assumed(Not(b)), Or(Not(a), Not(b))), p1, false()),
Arrow(Not(b), false())), Not(Not(b))), b)
def disSyl():
A = Var("a")
B = Var("b")
p1 = premise(Or(A, B))
p2 = premise(Not(B))
return orE(
p1, arrowI(assume(A), assumed(A), Arrow(A, A)),
arrowI(assume(B), FE(notE(assumed(B), p2, false()), A), Arrow(B, A)),
A)
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 doubleNeg(p, a):
l1 = LEM(Or(a, Not(a)))
l2 = assume(a)
l3 = assumed(a)
l4 = arrowI(l2, l3, Arrow(a, a))
l5 = assume(Not(a))
l6 = assumed(Not(a))
l7 = notE(l6, p, false())
l8 = FE(l7, a)
l9 = arrowI(l5, l8, Arrow(Not(a), a))
l10 = orE(l1, l4, l9, a)
return l10
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 DM2():
a = Var("a")
b = Var("b")
prem = premise(Not(And(a, b))) #The original premise, ~(A && B)
d1 = premise(
Not(Or(Not(a), Not(b)))
) #d1 is the phrase required to run DL1 and DL2. I ran it as a premise, but it is assumed, NOT a premise
return doubleNeg(
notI(
arrowI(assume(Not(Or(Not(a), Not(b)))),
notE(andI(DL1(d1), DL2(d1), And(a, b)), prem, false()),
Arrow(Not(Or(Not(a), Not(b))), false())),
Not(Not(Or(Not(a), Not(b))))), Or(Not(a), Not(b)))
def DM1():
A = Var("a")
B = Var("b")
p1 = premise(Or(Not(A), Not(B)))
end = Not(And(A, B))
A_B = And(A, B)
#I condensed parts of the proofs so I could substitute in things that made sense to me, otherwise I was going nuts
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)