def if_A3(F):
# (!B->!A) -> ((!B->A)->B)
# ________ ____________
# F1 F2
try:
F1 = F.left
F2 = F.right
B = F2.right
A = F2.left.right
nB = notFormula(B)
nA = notFormula(A)
if F1.left == nB and F1.right == nA and \
F2 == Node(Node(nB, A), B):
return True
a3 = A3(A, B)
if a3 == F:
return True
except:
# F is variable or does not have left or right Node
pass
return False
def build_T7(F, G):
# (F->G)->((!F->G)->G)
nF = notFormula(F)
nG = notFormula(G)
F1 = Node(F, G)
F1.msg = "t7 hypoth"
F2 = Node(nF, G)
F2.msg = "t7 hypoth"
f3 = build_T5(F, G)
F3 = f3[-1]
F4 = axiom.A3(F, G)
f5 = syl_1(F3, F4)
F5 = f5[-1]
F6 = MP(F1, F5)
f7 = build_T4(nG, F)
F7 = f7[-1]
# (!F->G)->(!F->!!G)
F8 = nF
F9 = Node(nF, G)
F10 = MP(F8, F9)
f11 = build_T2(G)
F11 = f11[-1]
F12 = MP(F10, F11)
f13 = build_deduction([F1, F2, F9], F8, F12, [F8, F9, F10] + f11 + [F12])
F13 = f13[-1]
f14 = build_deduction([], F9, F13, f13)
F14 = f14[-1]
#...
f15 = syl_1(F14, F7)
F15 = f15[-1]
f16 = syl_1(F15, F6)
F16 = f16[-1]
proof = [F1, F2] + f14 + [F4] + f5
f17 = build_deduction([], F1, F16, proof)
proof = f17
return proof
def A3(F, G):
"""
(!G -> !F) -> ( (!G -> F) -> G ) axiom (3)
"""
try:
notG = notFormula(G)
notF = notFormula(F)
ngnf = Node(notG, notF)
ngf = Node(notG, F)
ngf_g = Node(ngf, G)
msg = "A3 for " + str(F) + ", " + str(G)
return Node(ngnf, ngf_g, msg=msg)
except:
return None
def build_T4(F, G):
# (!G->!F)->(F->G)
nF = notFormula(F)
nG = notFormula(G)
F1 = Node(nG, nF)
F2 = F
F3 = axiom.A3(F, G)
F4 = MP(F1, F3)
F5 = axiom.A1(F, nG)
f6 = syl_1(F5, F4)
proof = [F3, F4, F5]
proof.extend(f6)
proof = build_deduction([F1, F], F1, f6[-1], proof)
return proof
def build_T2(F):
nF = notFormula(F) # !F
nnF = notFormula(nF) # !!F
nnnF = notFormula(nnF) # !!!F
F1 = axiom.A3(F, nnF) # (!!!F->!F)->((!!F->F)->!!F)
f2 = build_T1(nF) # build th. 1
F2 = f2[-1] # !!!A->!A
F3 = MP(F2, F1) # ...->!!A
F4 = axiom.A1(F, nnnF) # A->...
f5 = syl_1(F4, F3) # f5[-1] = A->!!A
res = [F1]
res.extend(f2)
res.extend([F3, F4])
res.extend(f5)
return res
def build_T5(F, G):
"""
(F->G)->(!G->!F)
"""
nF = notFormula(F)
nG = notFormula(G)
nnF = notFormula(nF)
nnG = notFormula(nG)
F1 = Node(F, G)
f2 = build_T2(G)
F2 = f2[-1]
F3 = axiom.A3(nG, nF)
f4 = syl_1(F1, F2) # f->!!g
F4 = f4[-1]
f5 = build_T1(F)
F5 = f5[-1]
f6 = syl_1(F5, F4) # !!f->!!g
F6 = f6[-1]
F7 = MP(F6, F3) # (!!f->!g)->!f
F8 = axiom.A1(nG, nnF)
f9 = syl_1(F8, F7)
F9 = f9[-1]
proof = f2 + [F3] + f4 + f5 + f6 + [F7, F8] + f9
proof = build_deduction([F1, nG], F1, F9, proof)
return proof
def build_T3(F, G):
# !f -> (f->G)
nG = notFormula(G)
F1 = notFormula(F)
F2 = F
F3 = axiom.A1(F2, nG)
F4 = MP(F2, F3)
F5 = axiom.A1(F1, nG)
F6 = MP(F1, F5)
F7 = axiom.A3(F, G)
F8 = MP(F6, F7)
F9 = MP(F4, F8)
proof = [F1, F2, F3, F4, F5, F6, F7, F8, F9]
ded1 = build_deduction([F1], F, G, proof)
proof = ded1
ded2 = build_deduction([], F1, ded1[-1], proof)
proof = ded2
return proof
def build_T1(F):
nF = notFormula(F)
F1 = axiom.A3(nF, F)
f2 = build_TL(nF)
F2 = f2[-1]
f3 = syl_2(F1, F2)
F3 = f3[-1]
nnF = notFormula(nF)
F4 = axiom.A1(nnF, nF)
f5 = syl_1(F4, F3)
F5 = f5[-1]
res = [F1]
res.extend(f2)
res.extend(f3)
res.extend([F4])
res.extend(f5)
return res
def build_T6(F, G):
fg = Node(F, G)
fg.msg = "hypoth t6"
t5 = build_T5(fg, G)
T5 = t5[-1]
ng = notFormula(G)
ng.msg = "hypoth t6"
F1 = F
F1.msg = "hypoth t6"
# we need (F->G)->G
F2 = MP(F1, fg)
ded = build_deduction([F1, ng], fg, G, [F1, F2])
F3 = ded[-1]
F4 = MP(F3, T5)
proof = [F1, ng, F2] + ded + [F4]
proof = build_deduction([], F, F4, proof)
return proof
def Calmar_Theorem(F, values):
'''
Calmar's Theorem implementation
:param F: Propositional calculus formula: Variable or Node
:return:
'''
if type(F) == Variable:
v = Variable(F.symbol, _not=True if values[F.symbol] == 1 else 0)
return [v]
if F._not: # F = !G
G = Node(None, None)
G.left = F.left # Copy G, but F = !G
G.right = F.right
if F.calculate(values) == 1:
G.msg = "Calmar's theorem, 1.a) F^(alpha) = G^(alpha) => hypoth |- G^() = F^()"
return [G]
else:
# !f = !!g
t2 = build_T2(G)
T2 = t2[-1]
nng = MP(G, T2)
res = t2
res.append(nng)
return res
else: # F = G->H
res = []
G = F.left
H = F.right
if G.calculate(values) == 0:
nG = notFormula(G)
t3 = build_T3(G, H)
T = t3[-1]
mp = MP(nG, T)
res = t3
res.append(mp)
elif H.calculate(values) == 1:
# h->(g->h)
h = H
hgh = axiom.A1(H, G)
gh = MP(h, hgh)
return [hgh, gh]
else:
# F = !(G->H)
# have G, !H. proof !(G->H)
nH = notFormula(H)
t6 = build_T6(G, H)
T6 = t6[-1]
f1 = MP(G, T6)
f2 = MP(nH, f1)
res = t6
res.extend([f1, f2])
return res
proof.extend(deductC2)
proof.extend(t7)
proof.extend([mp1, mp2])
curLen = len(proof) #for debug
vector += 1
n += 1
return proof
if __name__ == '__main__':
f = Variable('F')
g = Variable('G')
nf = notFormula(f)
nnf = notFormula(nf)
Ad_Theorem(Node(nnf, f))
t4 = build_T4(f, g)
T4 = t4[-1]
t5 = build_T5(f, g)
t6 = build_T6(f, g)
t7 = build_T7(f, g)
T7 = t7[-1]
print('no exception? it is small victory')