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 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 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 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 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 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 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 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 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_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 Q3_1():
clear()
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 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 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 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 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 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 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)"))
