def orComm():
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 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 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 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 arrTrans():
a = Var("A")
b = Var("B")
c = Var("C")
p1 = premise(Arrow(a, b))
p2 = premise(Arrow(b, c))
a1 = assume(a)
return arrowI(a1, \
arrowE(arrowE(assumed(a), \
p1, \
b), \
p2, \
c), \
Arrow(a,c))
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 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 arrow_expr(tokens):
follow = [TType.TEOF, TType.TRPAREN]
lhs = or_expr(tokens)
if tokens[0].ttype == TType.TARROW:
tokens.pop(0)
rhs = arrow_expr(tokens)
lhs = Arrow(lhs, rhs)
if tokens[0].ttype not in follow:
raise ParseException(tokens[0].pos, follow, tokens[0].val)
return lhs
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)
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)"))