def get_proof_term(self, t):
pt = refl(t).on_rhs(binop_conv(to_exponent_form()))
if pt.rhs.arg1.is_comb('exp', 1) and pt.rhs.arg.is_comb('exp', 1):
# Both sides are exponentials
return pt.on_rhs(rewr_conv('real_exp_add', sym=True),
arg_conv(auto.auto_conv(self.conds)))
elif pt.rhs.arg1.is_nat_power() and pt.rhs.arg.is_nat_power():
# Both sides are natural number powers, simply add
return pt.on_rhs(rewr_conv('real_pow_add', sym=True),
arg_conv(nat.nat_conv()))
else:
# First check that x > 0 can be proved. If not, just return
# without change.
x = pt.rhs.arg1.arg1
try:
x_gt_0 = auto.solve(x > 0, self.conds)
except TacticException:
return refl(t)
# Convert both sides to real powers
if pt.rhs.arg1.is_nat_power():
pt = pt.on_rhs(arg1_conv(rewr_conv('rpow_pow', sym=True)))
if pt.rhs.arg.is_nat_power():
pt = pt.on_rhs(arg_conv(rewr_conv('rpow_pow', sym=True)))
pt = pt.on_rhs(rewr_conv('rpow_add', sym=True, conds=[x_gt_0]),
arg_conv(real_eval_conv()))
# Simplify back to nat if possible
if pt.rhs.arg.is_comb('of_nat', 1):
pt = pt.on_rhs(rewr_conv('rpow_pow'),
arg_conv(rewr_conv('nat_of_nat_def', sym=True)))
return pt.on_rhs(from_exponent_form())
def get_proof_term(self, t):
assert t.is_equals() or t.is_less_eq() or t.is_less()\
or t.is_greater_eq() or t.is_greater(), "%s is not an equality term" % t
pt1 = refl(t) # a = b <==> a = b
if t.is_equals():
pt2 = pt1.on_rhs(rewr_conv('int_sub_move_0_r',
sym=True)) # a = b <==> a - b = 0
eq_refl = ProofTerm.reflexive(equals(IntType))
elif t.is_less_eq():
pt2 = pt1.on_rhs(rewr_conv('int_leq'))
eq_refl = ProofTerm.reflexive(less_eq(IntType))
elif t.is_less():
pt2 = pt1.on_rhs(rewr_conv('int_less'))
eq_refl = ProofTerm.reflexive(less(IntType))
elif t.is_greater_eq():
pt2 = pt1.on_rhs(rewr_conv('int_geq'))
eq_refl = ProofTerm.reflexive(greater_eq(IntType))
elif t.is_greater():
pt2 = pt1.on_rhs(rewr_conv('int_gt'))
eq_refl = ProofTerm.reflexive(greater(IntType))
pt3 = simp_full().get_proof_term(
pt2.prop.arg.arg1) # a - b = a + (-1) * b
pt4 = ProofTerm.combination(eq_refl, pt3)
pt5 = ProofTerm.combination(pt4, refl(Const(
'zero', IntType))) # a - b = 0 <==> a + (-1)*b = 0
return pt2.transitive(pt5) # a = b <==> a + (-1) * b = 0
def get_proof_term(self, t):
if not is_real_ineq(t):
return refl(t)
pt_refl = refl(t).on_rhs(real_norm_comparison())
left_expr = pt_refl.rhs.arg1
summands = integer.strip_plus(left_expr)
first = summands[0]
if not left_expr.is_plus() or not first.is_constant():
return pt_refl
first_value = real_eval(first)
if pt_refl.rhs.is_greater_eq():
pt_th = ProofTerm.theorem("real_sub_both_sides_geq")
elif pt_refl.rhs.is_greater():
pt_th = ProofTerm.theorem("real_sub_both_sides_gt")
elif pt_refl.rhs.is_less_eq():
pt_th = ProofTerm.theorem("real_sub_both_sides_leq")
elif pt_refl.rhs.is_less():
pt_th = ProofTerm.theorem("real_sub_both_sides_le")
else:
raise NotImplementedError(str(t))
inst = matcher.first_order_match(pt_th.lhs,
pt_refl.rhs,
inst=matcher.Inst(c=first))
return pt_refl.transitive(pt_th.substitution(inst=inst)).on_rhs(
auto.auto_conv())
def get_proof_term(self, t):
if t.get_type() != RealType:
return refl(t)
simp_t = Real(real_eval(t))
if simp_t == t:
return refl(t)
return ProofTerm('real_eval', Eq(t, simp_t))
def get_proof_term(self, goal):
assert isinstance(goal, Term), "%s should be a hol term" % str(goal)
assert goal.is_less() or goal.is_less_eq() or goal.is_greater() or goal.is_greater_eq(),\
"%s should be an inequality term" % str(goal)
norm_ineq_pt = norm_eq().get_proof_term(goal)
# find the first monomial
first_monomial = norm_ineq_pt.rhs
while not first_monomial.is_times() and not first_monomial.is_number():
first_monomial = first_monomial.arg1
coeff = first_monomial.arg1 if first_monomial.is_times(
) else first_monomial
assert coeff.is_int()
coeff_value = int_eval(coeff)
# normalize
if norm_ineq_pt.rhs.is_greater_eq():
return norm_ineq_pt
elif norm_ineq_pt.rhs.is_greater():
return norm_ineq_pt.on_rhs(rewr_conv('int_great_to_geq'))
else:
if norm_ineq_pt.rhs.is_less():
pt_great = norm_ineq_pt.on_rhs(rewr_conv('int_less_to_geq'))
elif norm_ineq_pt.rhs.is_less_eq():
pt_great = norm_ineq_pt.on_rhs(rewr_conv('int_leq_to_geq'))
pt_norm_lhs = refl(pt_great.rhs.arg1).on_rhs(
omega_simp_full_conv())
return pt_great.transitive(
refl(pt_great.rhs.head).combination(pt_norm_lhs).combination(
refl(pt_great.rhs.arg)))
def get_proof_term(self, t):
if t.get_type() != IntType:
return refl(t)
simp_t = Int(int_eval(t))
if simp_t == t:
return refl(t)
else:
return ProofTerm('int_eval', Eq(t, int_eval(t)))
def get_proof_term(self, goal, prevs):
assert goal.is_equals(), "int_eq_norm, %s is not equation" % goal
# Get normal form on both sides.
pt1 = refl(goal.lhs).on_rhs(norm_eq())
pt2 = refl(goal.rhs).on_rhs(norm_eq())
assert pt1.rhs == pt2.rhs
return pt1.transitive(pt2.symmetric())
def get_proof_term(self, t):
if t.is_not():
if t.arg == true:
return rewr_conv("not_true").get_proof_term(t)
elif t.arg == false:
return rewr_conv("not_false").get_proof_term(t)
else:
return refl(t)
else:
return refl(t)
def get_proof_term(self, goal):
assert goal.is_equals() and goal.lhs.is_compares(
) and goal.rhs.is_compares()
pt1 = refl(goal.lhs).on_rhs(omega_form_conv())
pt2 = refl(goal.rhs).on_rhs(omega_form_conv())
if pt1.rhs != pt2.rhs:
raise ConvException(str(goal))
return pt1.transitive(pt2.symmetric())
def get_proof_term(self, t):
if not t.is_equals():
return refl(t)
a, b = t.args
if not (a.is_number() and b.is_number()):
return refl(t)
if a == b:
return refl(a).on_prop(rewr_conv("eq_true"))
else:
return nat_const_ineq(a, b).on_prop(rewr_conv("eq_false"))
def get_proof_term(self, goal, prevs=[]):
assert goal.is_equals() and goal.lhs.is_compares(
) and goal.rhs.is_compares()
comp1, comp2 = goal.lhs, goal.rhs
pt_refl1, pt_refl2 = refl(goal.lhs).on_rhs(real_norm_comparison()), \
refl(goal.rhs).on_rhs(real_norm_comparison())
assert pt_refl1.rhs == pt_refl2.rhs
return pt_refl1.transitive(pt_refl2.symmetric())
def get_proof_term(self, t):
"""Unfold powers of 2 and 3, leave other powers unexpanded."""
a, p = t.args
# Exponent is 2, apply real_add_ldistrib to avoid folding back.
if p.is_number() and p.dest_number() == 2 and a.is_plus():
return refl(t).on_rhs(rewr_conv('real_pow_2'),
rewr_conv('real_add_ldistrib'))
# Exponent is 3
elif p.is_number() and p.dest_number() == 3 and a.is_plus():
return refl(t).on_rhs(rewr_conv('real_pow_3'),
rewr_conv('real_add_ldistrib'))
return refl(t)
def get_proof_term(self, t):
pt = refl(t)
if t.arg.is_times():
return pt.on_rhs(rewr_conv("mult_assoc", sym=True),
arg1_conv(norm_mult_monomial()), norm_mult_atom())
else:
return pt.on_rhs(norm_mult_atom())
def get_proof_term(self, t):
pt = refl(t)
if t.is_plus():
if t.arg1.is_zero():
return pt.on_rhs(rewr_conv('int_add_0_left'))
elif t.arg.is_zero():
return pt.on_rhs(rewr_conv('int_add_0_right'))
elif t.arg.is_plus(): # t is of form a + (b + c)
return pt.on_rhs(
rewr_conv('int_add_assoc'), # (a + b) + c
arg1_conv(self), # merge terms in b into a
norm_add_monomial()) # merge c into a + b
elif t.arg.is_minus(): # t is of form a + (b - c)
return pt.on_rhs(
arg_conv(norm_add_monomial()), # a + b + (-1) * c
arg1_conv(self),
norm_add_polynomial())
else:
return pt.on_rhs(norm_add_monomial())
elif t.is_minus():
if t.arg.is_plus(): # t is of form a - (b + c)
return pt.on_rhs(
rewr_conv('int_sub_add_distr'), # a - b - c
arg1_conv(self), # merge terms in b into a
norm_add_monomial() # merge c into a - b
)
elif t.arg.is_minus(): # t is of form a - (b - c)
return pt.on_rhs(
rewr_conv('int_sub_sub_distr'), # a - b + c
arg1_conv(self), # merge terms in b into c
norm_add_monomial() # merge c into a - b
)
else:
return pt.on_rhs(norm_add_monomial(), )
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_times(): # t is of form (a * b) * c
cp = compare_atom(t.arg1.arg,
t.arg) # compare b with c by their base
if cp > 0: # if b > c, need to swap b with c
return pt.on_rhs(
swap_mult_r(), # (a * c) * b
arg1_conv(self)) # possibly move c further inside a
elif cp == 0: # if b and c have the same base, combine the exponents
return pt.on_rhs(
rewr_conv('int_mult_assoc', sym=True), # a * (b^e1 * b^e2)
arg_conv(rewr_conv('int_power_add',
sym=True)), # a * (b^(e1 + e2))
arg_conv(arg_conv(int_eval_conv()))) # evaluate e1 + e2
else: # if b < c, atoms already ordered since we assume b is ordered.
return pt
else: # t is of the form a * b
cp = compare_atom(t.arg1, t.arg) # compare a with b by their base
if cp > 0: # if a > b, need to swap a and b
return pt.on_rhs(rewr_conv('int_mult_comm'))
elif cp == 0: # if a and b have the same base, combine the exponents
return pt.on_rhs(rewr_conv('int_power_add', sym=True),
arg_conv(int_eval_conv()))
else:
return pt
def norm_divides(t, pts):
"""Normalization of divides."""
pt = refl(t)
if t.is_number():
return pt.on_rhs(real_eval_conv())
else:
return pt.on_rhs(rewr_conv('real_divide_def'))
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_zero():
return pt.on_rhs(rewr_conv("nat_times_def_1"))
elif t.arg.is_zero():
return pt.on_rhs(rewr_conv("mult_0_right"))
elif t.arg1.is_one():
return pt.on_rhs(rewr_conv("mult_1_left"))
elif t.arg.is_one():
return pt.on_rhs(rewr_conv("mult_1_right"))
elif t.arg1.is_times():
cp = compare_atom(t.arg1.arg, t.arg)
if cp > 0:
return pt.on_rhs(swap_times_r(), arg1_conv(norm_mult_atom()))
elif cp == 0:
if t.arg.is_number() and has_binary_thms():
return pt.on_rhs(rewr_conv("mult_assoc"),
arg_conv(nat_conv()))
else:
return pt
else:
return pt
else:
cp = compare_atom(t.arg1, t.arg)
if cp > 0:
return pt.on_rhs(rewr_conv("mult_comm"))
elif cp == 0:
if t.arg.is_number() and has_binary_thms():
return pt.on_rhs(nat_conv())
else:
return pt
else:
return pt
def get_proof_term(self, t):
pt = refl(t)
if t.is_conj():
return pt.on_rhs(binop_conv(self), norm_conj_conjunction())
elif t.is_disj():
return pt.on_rhs(binop_conv(self), norm_disj_disjunction())
elif t.is_equals():
lhs, rhs = t.lhs, t.rhs
if lhs == rhs:
return pt.on_rhs(rewr_conv('eq_mean_true'))
elif lhs == true: # (true ⟷ P) ⟷ P
return pt.on_rhs(rewr_conv('eq_sym_eq'),
rewr_conv('eq_true', sym=True))
elif rhs == true: # (P ⟷ true) ⟷ P
return pt.on_rhs(rewr_conv('eq_true', sym=True))
elif lhs == false: # (false ⟷ P) ⟷ ¬P
return pt.on_rhs(rewr_conv('eq_sym_eq'),
rewr_conv('eq_false', sym=True))
elif rhs == true: # (P ⟷ false) ⟷ ¬P
return pt.on_rhs(rewr_conv('eq_false', sym=True))
else:
return pt.on_rhs(binop_conv(self))
elif t.is_not() and (t.arg.is_conj() or t.arg.is_disj()
or t.arg.is_not() or t.arg == true
or t.arg == false):
return pt.on_rhs(nnf_conv(), self)
elif t.is_not() and t.arg.is_equals():
return pt.on_rhs(nnf_conv())
else:
return pt
def get_proof_term(self, args, pts):
assert isinstance(args, Term), "rewrite_goal_macro: signature"
goal = args
eq_pt = pts[0]
new_names = get_forall_names(eq_pt.prop)
new_vars, _, _ = strip_all_implies(eq_pt.prop, new_names)
for new_var in new_vars:
eq_pt = eq_pt.forall_elim(new_var)
pts = pts[1:]
cv = then_conv(top_sweep_conv(rewr_conv(eq_pt, sym=self.sym)),
beta_norm_conv())
pt = cv.get_proof_term(goal) # goal = th.prop
pt = pt.symmetric() # th.prop = goal
if pt.prop.lhs.is_reflexive():
pt = pt.equal_elim(refl(pt.prop.lhs.rhs))
else:
pt = pt.equal_elim(pts[0])
pts = pts[1:]
for A in pts:
pt = pt.implies_intr(A.prop).implies_elim(A)
return pt
def get_proof_term(self, t):
pt = refl(t)
if theory.thy.has_theorem('mult_comm'):
# Full conversion, with or without binary numbers
if t.is_number():
return pt
elif t.is_comb('Suc', 1):
return pt.on_rhs(rewr_conv("add_1_right", sym=True),
norm_full())
elif t.is_plus():
return pt.on_rhs(binop_conv(norm_full()),
norm_add_polynomial())
elif t.is_times():
return pt.on_rhs(binop_conv(norm_full()),
norm_mult_polynomial())
else:
return pt
elif theory.thy.has_theorem('add_assoc'):
# Conversion using only AC rules for addition
if t.is_number():
return pt
elif t.is_comb('Suc', 1):
return pt.on_rhs(rewr_conv("add_1_right", sym=True),
norm_full())
elif t.is_plus():
return pt.on_rhs(binop_conv(norm_full()), norm_add_1())
else:
return pt
else:
return pt
def get_proof_term(self, t):
pt = refl(t)
if t.arg.is_plus():
return pt.on_rhs(rewr_conv('real_add_assoc'), arg1_conv(self),
norm_add_monomial())
else:
return pt.on_rhs(norm_add_monomial())
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_zero():
return pt.on_rhs(rewr_conv("real_add_lid"))
elif t.arg.is_zero():
return pt.on_rhs(rewr_conv("real_add_rid"))
elif t.arg1.is_plus():
# Left side has more than one term. Compare last term with a
m1, m2 = dest_monomial(t.arg1.arg), dest_monomial(t.arg)
if m1 == m2:
pt = pt.on_rhs(rewr_conv('real_add_assoc', sym=True),
arg_conv(combine_monomial()))
if pt.rhs.arg.is_zero():
pt = pt.on_rhs(rewr_conv('real_add_rid'))
return pt
elif atom_less(m1, m2):
return pt
else:
pt = pt.on_rhs(swap_add_r(), arg1_conv(self))
if pt.rhs.arg1.is_zero():
pt = pt.on_rhs(rewr_conv('real_add_lid'))
return pt
else:
# Left side is an atom. Compare two sides
m1, m2 = dest_monomial(t.arg1), dest_monomial(t.arg)
if m1 == m2:
return pt.on_rhs(combine_monomial())
elif atom_less(m1, m2):
return pt
else:
return pt.on_rhs(rewr_conv('real_add_comm'))
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_one():
return pt.on_rhs(rewr_conv('real_mul_lid'))
elif t.arg.is_one():
return pt.on_rhs(rewr_conv('real_mul_rid'))
elif t.arg1.is_times():
# Left side has more than one atom. Compare last atom with b
m1, m2 = dest_atom(t.arg1.arg), dest_atom(t.arg)
if m1 == m2:
pt = pt.on_rhs(rewr_conv('real_mult_assoc', sym=True),
arg_conv(combine_atom(self.conds)))
if pt.rhs.arg.is_one():
pt = pt.on_rhs(rewr_conv('real_mul_rid'))
return pt
elif atom_less(m1, m2):
return pt
else:
pt = pt.on_rhs(swap_mult_r(), arg1_conv(self))
if pt.rhs.arg1.is_one():
pt = pt.on_rhs(rewr_conv('real_mul_lid'))
return pt
else:
# Left side is an atom. Compare two sides
m1, m2 = dest_atom(t.arg1), dest_atom(t.arg)
if m1 == m2:
return pt.on_rhs(combine_atom(self.conds))
elif atom_less(m1, m2):
return pt
else:
return pt.on_rhs(rewr_conv('real_mult_comm'))
def get_proof_term(self, t):
pt = refl(t)
if t.arg.is_times():
return pt.on_rhs(rewr_conv('real_mult_assoc'), arg1_conv(self),
norm_mult_atom(self.conds))
else:
return pt.on_rhs(norm_mult_atom(self.conds))
def get_proof_term(self, t):
if not t.is_compares() or not t.arg1.get_type() == IntType:
raise ConvException(str(t))
pt = refl(t)
# first move all terms in rhs to lhs and normalize lhs
if t.is_greater():
pt1 = pt.on_rhs(rewr_conv('int_gt_to_geq'), rewr_conv('int_geq'),
arg1_conv(omega_simp_full_conv()))
elif t.is_greater_eq():
pt1 = pt.on_rhs(rewr_conv('int_geq'),
arg1_conv(omega_simp_full_conv()))
elif t.is_less():
pt1 = pt.on_rhs(rewr_conv('int_less_to_leq'), rewr_conv('int_leq'),
arg1_conv(omega_simp_full_conv()))
elif t.is_less_eq():
pt1 = pt.on_rhs(rewr_conv('int_leq'),
arg1_conv(omega_simp_full_conv()))
else:
raise NotImplementedError
# move all constant term on lhs to rhs
lhs = pt1.rhs.arg1
if lhs.is_number() or lhs.is_times() or lhs.arg.is_times():
return pt1
elif pt1.rhs.is_greater_eq() and lhs.arg.is_number():
return pt1.on_rhs(rewr_conv('int_geq_shift'),
arg_conv(int_eval_conv()))
elif pt1.rhs.is_less_eq() and lhs.arg.is_number():
return pt1.on_rhs(rewr_conv('int_leq_shift'),
arg_conv(int_eval_conv()))
else:
raise NotImplementedError
def get_proof_term(self, t):
pt = refl(t)
is_l_atom = (dest_monomial(t.arg1) == t.arg1)
is_r_atom = (dest_monomial(t.arg) == t.arg)
if is_l_atom and is_r_atom:
return pt.on_rhs(norm_mult_monomial(self.conds))
elif is_l_atom and not is_r_atom:
if t.arg.is_number():
return pt.on_rhs(rewr_conv('real_mult_comm'),
try_conv(rewr_conv('real_mul_rid')),
try_conv(rewr_conv('real_mul_lzero')))
else:
return pt.on_rhs(arg_conv(rewr_conv('real_mult_comm')),
rewr_conv('real_mult_assoc'),
rewr_conv('real_mult_comm'),
arg_conv(norm_mult_monomial(self.conds)))
elif not is_l_atom and is_r_atom:
if t.arg1.is_number():
return pt.on_rhs(try_conv(rewr_conv('real_mul_rid')),
try_conv(rewr_conv('real_mul_lzero')))
else:
return pt.on_rhs(rewr_conv('real_mult_assoc', sym=True),
arg_conv(norm_mult_monomial(self.conds)))
else:
return pt.on_rhs(
binop_conv(to_coeff_form()), # (c_1 * m_1) * (c_2 * m_2)
rewr_conv('real_mult_assoc'), # (c_1 * m_1 * c_2) * m_2
arg1_conv(swap_mult_r()), # (c_1 * c_2 * m_1) * m_2
arg1_conv(arg1_conv(real_eval_conv())), # (c_1c_2 * m_1) * m_2
rewr_conv('real_mult_assoc', sym=True), # c_1c_2 * (m_1 * m_2)
arg_conv(norm_mult_monomial(self.conds)),
from_coeff_form())
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_disj():
return pt.on_rhs(rewr_conv('disj_assoc_eq', sym=True),
arg_conv(self), norm_disj_atom())
else:
return pt.on_rhs(norm_disj_atom())
def norm_uminus(t, pts):
"""Normalization of uminus."""
pt = refl(t)
if t.is_number():
return pt.on_rhs(real_eval_conv())
else:
return pt.on_rhs(rewr_conv('real_poly_neg1'))
def get_proof_term(self, t):
if t.is_comb():
return comb_conv(self.cv).get_proof_term(t)
elif t.is_abs():
return abs_conv(self.cv).get_proof_term(t)
else:
return refl(t)
def get_proof_term(self, t):
pt = refl(t)
if t.arg.is_plus():
return pt.on_rhs(rewr_conv("add_assoc", sym=True),
arg1_conv(norm_add_1()), norm_add_atom_1())
else:
return pt.on_rhs(norm_add_atom_1())