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 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 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 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 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 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)