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 interval_union_subset(t):
"""Given t of the form I1 Un I2, return a theorem of the form
I1 Un I2 SUB I.
"""
assert t.is_comb('union', 2), "interval_union_subset"
I1, I2 = t.args
a, b = I1.args
c, d = I2.args
if is_closed_interval(I1) and is_closed_interval(I2):
pt = apply_theorem('closed_interval_union',
inst=Inst(a=a, b=b, c=c, d=d))
return pt.on_prop(
arg_conv(
then_conv(arg1_conv(const_min_conv()),
arg_conv(const_max_conv()))))
elif is_open_interval(I1) and is_ropen_interval(I2):
if eval_hol_expr(c) <= eval_hol_expr(a):
pt = apply_theorem('open_ropen_interval_union1',
auto.auto_solve(real.less_eq(c, a)),
inst=Inst(b=b, d=d))
else:
pt = apply_theorem('open_ropen_interval_union2',
auto.auto_solve(real.less(a, c)),
inst=Inst(b=b, d=d))
return pt.on_prop(arg_conv(arg_conv(const_max_conv())))
else:
raise NotImplementedError
return pt
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)
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 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 is_interval(t), "numseg_conv"
mt, nt = t.args
m, n = mt.dest_number(), nt.dest_number()
pt = refl(t)
if n < m:
less_goal = nat.less(nt, mt)
less_pt = nat.nat_const_less_macro().get_proof_term(less_goal, [])
return pt.on_rhs(rewr_conv("natseg_emptyI", conds=[less_pt]))
else:
le_goal = nat.less_eq(mt, nt)
le_pt = nat.nat_const_less_eq_macro().get_proof_term(le_goal, [])
return pt.on_rhs(rewr_conv("natseg_lrec", conds=[le_pt]),
arg_conv(arg1_conv(nat.nat_conv())),
arg_conv(self))
def get_proof_term(self, t):
pt = refl(t)
if t.is_number():
return pt
elif t.is_comb('Suc', 1):
return pt.on_rhs(arg_conv(self), arg_conv(rewr_of_nat_conv()),
Suc_conv(), rewr_of_nat_conv(sym=True))
elif t.is_plus():
return pt.on_rhs(binop_conv(self), binop_conv(rewr_of_nat_conv()),
add_conv(), rewr_of_nat_conv(sym=True))
elif t.is_times():
return pt.on_rhs(binop_conv(self), binop_conv(rewr_of_nat_conv()),
mult_conv(), rewr_of_nat_conv(sym=True))
else:
raise ConvException("nat_conv")
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.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.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 get_proof_term(self, thy, t):
if is_conj(t.arg1):
return then_conv(
rewr_conv("conj_assoc", sym=True),
arg_conv(norm_conj_assoc_clauses())).get_proof_term(thy, t)
else:
return all_conv().get_proof_term(thy, t)
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, thy, t):
if t.arg1 == zero:
cv = rewr_conv("times_def_1")
elif t.arg == zero:
cv = rewr_conv("mult_0_right")
elif t.arg1 == one:
cv = rewr_conv("mult_1_left")
elif t.arg == one:
cv = rewr_conv("mult_1_right")
elif is_times(t.arg1):
cmp = compare_atom(t.arg1.arg, t.arg)
if cmp == term_ord.GREATER:
cv = then_conv(swap_times_r(), arg1_conv(norm_mult_atom()))
elif cmp == term_ord.EQUAL:
if is_binary(t.arg):
cv = then_conv(rewr_conv("mult_assoc"),
arg_conv(nat_conv()))
else:
cv = all_conv()
else:
cv = all_conv()
else:
cmp = compare_atom(t.arg1, t.arg)
if cmp == term_ord.GREATER:
cv = rewr_conv("mult_comm")
elif cmp == term_ord.EQUAL:
if is_binary(t.arg):
cv = nat_conv()
else:
cv = all_conv()
else:
cv = all_conv()
return cv.get_proof_term(thy, t)
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, thy, t):
if is_plus(t.arg1):
return every_conv(rewr_conv("add_assoc"),
arg_conv(rewr_conv("add_comm")),
rewr_conv("add_assoc",
sym=True)).get_proof_term(thy, t)
else:
return rewr_conv("add_comm").get_proof_term(thy, t)
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_times():
return pt.on_rhs(rewr_conv("mult_assoc"),
arg_conv(rewr_conv("mult_comm")),
rewr_conv("mult_assoc", sym=True))
else:
return pt.on_rhs(rewr_conv("mult_comm"))
def get_proof_term(self, thy, t):
if is_plus(t.arg):
return every_conv(rewr_conv("distrib_l"),
arg1_conv(norm_mult_polynomial()),
arg_conv(norm_mult_poly_monomial()),
norm_add_polynomial()).get_proof_term(thy, t)
else:
return norm_mult_poly_monomial().get_proof_term(thy, t)
def get_proof_term(self, t):
pt = refl(t)
if t.arg.is_plus():
return pt.on_rhs(rewr_conv("distrib_l"),
arg1_conv(norm_mult_polynomial()),
arg_conv(norm_mult_poly_monomial()),
norm_add_polynomial())
else:
return pt.on_rhs(norm_mult_poly_monomial())
def eval_Sem(c, st):
"""Evaluates the effect of program c on state st."""
T = st.get_type()
if c.is_const("Skip"):
return apply_theorem("Sem_Skip", inst=Inst(s=st))
elif c.is_comb("Assign", 2):
a, b = c.args
Ta = a.get_type()
Tb = b.get_type().range_type()
pt = apply_theorem("Sem_Assign", inst=Inst(a=a, b=b, s=st))
return pt.on_arg(arg_conv(norm_cv))
elif c.is_comb("Seq", 2):
c1, c2 = c.args
pt1 = eval_Sem(c1, st)
pt2 = eval_Sem(c2, pt1.prop.arg)
pt = apply_theorem("Sem_seq", pt1, pt2)
return pt.on_arg(function.fun_upd_norm_one_conv())
elif c.is_comb("Cond", 3):
b, c1, c2 = c.args
b_st = beta_norm(b(st))
b_eval = norm_cond_cv.get_proof_term(b_st)
if b_eval.prop.arg == true:
b_res = b_eval.on_prop(rewr_conv("eq_true", sym=True))
pt1 = eval_Sem(c1, st)
return apply_theorem("Sem_if1",
b_res,
pt1,
concl=Sem(T)(c, st, pt1.prop.arg))
else:
b_res = b_eval.on_prop(rewr_conv("eq_false", sym=True))
pt2 = eval_Sem(c2, st)
return apply_theorem("Sem_if2",
b_res,
pt2,
concl=Sem(T)(c, st, pt2.prop.arg))
elif c.is_comb("While", 3):
b, inv, body = c.args
b_st = beta_norm(b(st))
b_eval = norm_cond_cv.get_proof_term(b_st)
if b_eval.prop.arg == true:
b_res = b_eval.on_prop(rewr_conv("eq_true", sym=True))
pt1 = eval_Sem(body, st)
pt2 = eval_Sem(c, pt1.prop.arg)
pt = apply_theorem("Sem_while_loop",
b_res,
pt1,
pt2,
concl=Sem(T)(c, st, pt2.prop.arg),
inst=Inst(s3=pt1.prop.arg))
return pt.on_arg(function.fun_upd_norm_one_conv())
else:
b_res = b_eval.on_prop(rewr_conv("eq_false", sym=True))
return apply_theorem("Sem_while_skip",
b_res,
concl=Sem(T)(c, st, st))
else:
raise NotImplementedError
def get_proof_term(self, t):
pt = refl(t)
if t.is_minus(): # a - c
return pt.on_rhs(
rewr_conv('int_poly_neg2'), # a + (-k) * body
arg_conv(norm_mult_monomial()),
norm_add_monomial())
elif t.arg1.is_plus(): # (a + b) + c
cp = compare_monomial(t.arg1.arg, t.arg) # compare b with c
if cp > 0: # if b > c, need to swap b with c
return pt.on_rhs(
swap_add_r(), # (a + c) + b
arg1_conv(self), # possibly move c further into a
try_conv(rewr_conv('int_add_0_left'))) # 0 +b = 0
elif cp == 0: # if b and c have the same body, combine coefficients
return pt.on_rhs(
rewr_conv('int_add_assoc',
sym=True), # a + (c1 * b + c2 * b)
arg_conv(rewr_conv('int_mul_add_distr_r',
sym=True)), # a + (c1 + c2) * b
arg_conv(arg1_conv(int_eval_conv())), # evaluate c1 + c2
try_conv(arg_conv(rewr_conv('int_mul_0_l'))),
try_conv(rewr_conv('int_add_0_right')))
else: # if b < c, monomials are already sorted
return pt
else: # a + b
if t.arg.is_zero():
return pt.on_rhs(rewr_conv('int_add_0_right'))
if t.arg1.is_zero():
return pt.on_rhs(rewr_conv('int_add_0_left'))
cp = compare_monomial(t.arg1, t.arg) # compare a with b
if cp > 0: # if a > b, need to swap a with b
return pt.on_rhs(rewr_conv('int_add_comm'))
elif cp == 0: # if b and c have the same body, combine coefficients
if t.arg.is_number():
return pt.on_rhs(int_eval_conv())
else:
return pt.on_rhs(
rewr_conv('int_mul_add_distr_r', sym=True),
arg1_conv(int_eval_conv()),
try_conv(rewr_conv('int_mul_0_l')))
else:
return pt
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_plus(): # (a + b) * c
return pt.on_rhs(
rewr_conv('int_mul_add_distr_r'), # a * c + b * c
arg1_conv(self), # process a * c
arg_conv(norm_mult_monomial()), # process b * c
norm_add_polynomial()) # add the results
else:
return pt.on_rhs(norm_mult_monomial())
def get_proof_term(self, t):
if not t.is_disj():
return refl(t)
nnf_pt = nnf_conv().get_proof_term(Not(t))
norm_neg_disj_pt = sort_conj().get_proof_term(nnf_pt.rhs)
nnf_pt_norm = nnf_pt.transitive(norm_neg_disj_pt)
return nnf_pt_norm.on_prop(rewr_conv('neg_iff_both_sides'),
arg1_conv(rewr_conv('double_neg')),
arg_conv(nnf_conv()))
def get_proof_term(self, t):
pt = refl(t)
if t.arg.is_zero():
return pt.on_rhs(rewr_conv("nat_one_def", sym=True))
elif t.arg.is_one():
return pt.on_rhs(rewr_conv("one_Suc"))
elif is_bit0(t.arg):
return pt.on_rhs(rewr_conv("bit0_Suc"))
else:
return pt.on_rhs(rewr_conv("bit1_Suc"), arg_conv(self))
def get_proof_term(self, thy, t):
n = t.arg # remove Suc
if n == zero:
return all_conv().get_proof_term(thy, t)
elif n == one:
return rewr_conv("one_Suc").get_proof_term(thy, t)
elif n.head == bit0:
return rewr_conv("bit0_Suc").get_proof_term(thy, t)
else:
return then_conv(rewr_conv("bit1_Suc"),
arg_conv(self)).get_proof_term(thy, t)
def get_proof_term(self, t):
pt = refl(t)
if t.arg1.is_zero():
return pt.on_rhs(rewr_conv('int_mul_0_l'))
elif t.arg.is_zero():
return pt.on_rhs(rewr_conv('int_mul_0_r'))
elif t.arg.is_plus(): # a * (b + c)
return pt.on_rhs(
rewr_conv('int_mul_add_distr_l'), # a * b + a * c
arg1_conv(self), # process a * b
arg_conv(norm_mult_poly_monomial()), # process a * c
norm_add_polynomial())
elif t.arg.is_minus():
return pt.on_rhs(arg_conv(norm_add_polynomial()),
norm_mult_polynomials())
elif t.arg1.is_minus():
return pt.on_rhs(arg1_conv(norm_add_polynomial()),
norm_mult_polynomials())
else:
return pt.on_rhs(norm_mult_poly_monomial())
def testRewrConvWithAssum(self):
x = Const("x", natT)
y = Const("y", natT)
x_eq_y = Term.mk_equals(x, y)
th = Thm([], Term.mk_implies(x_eq_y, x_eq_y))
cv = arg_conv(rewr_conv(ProofTerm.atom(0, th)))
f = Const("f", TFun(natT, natT))
res = Thm([x_eq_y], Term.mk_equals(f(x), f(y)))
self.assertEqual(cv.eval(thy, f(x)), res)
prf = Proof()
prf.add_item(0, "sorry", th=th)
cv.get_proof_term(thy, f(x)).export(prf=prf)
self.assertEqual(thy.check_proof(prf), res)
def get_proof_term(self, thy, t):
pt = refl(t)
if is_binary(t):
return pt
else:
if t.head == Suc:
return pt.on_rhs(thy, arg_conv(self), Suc_conv())
elif t.head == plus:
return pt.on_rhs(thy, binop_conv(self), add_conv())
elif t.head == times:
return pt.on_rhs(thy, binop_conv(self), mult_conv())
else:
raise ConvException()
def get_proof_term(self, t):
a, p = t.args
# Exponent is an integer: apply rpow_pow
if p.is_number() and p.is_comb('of_nat', 1) and p.arg.is_binary():
return refl(t).on_rhs(
arg_conv(rewr_conv('real_of_nat_id', sym=True)),
rewr_conv('rpow_pow'))
if not (a.is_number() and p.is_number()):
raise ConvException
a, p = a.dest_number(), p.dest_number()
if a <= 0:
raise ConvException
# Case 1: base is a composite number
factors = factorint(a)
keys = list(factors.keys())
if len(keys) > 1 or (len(keys) == 1 and keys[0] != a):
b1 = list(factors.keys())[0]
b2 = a // b1
eq_th = refl(Real(b1) * b2).on_rhs(real_eval_conv())
pt = refl(t).on_rhs(arg1_conv(rewr_conv(eq_th, sym=True)))
pt = pt.on_rhs(rewr_conv('rpow_mul'))
return pt
# Case 2: exponent is not between 0 and 1
if isinstance(p, Fraction) and p.numerator // p.denominator != 0:
div, mod = divmod(p.numerator, p.denominator)
eq_th = refl(Real(div) + Real(mod) / p.denominator).on_rhs(
real_eval_conv())
pt = refl(t).on_rhs(arg_conv(rewr_conv(eq_th, sym=True)))
a_gt_0 = auto.auto_solve(Real(a) > 0)
pt = pt.on_rhs(rewr_conv('rpow_add', conds=[a_gt_0]))
return pt
return refl(t)
def get_proof_term(self, t):
if not (t.is_compares() and t.arg.get_type() == IntType):
raise ConvException("%s is not an integer comparison." % str(t))
pt_refl = refl(t)
if t.is_less():
pt = pt_refl.on_rhs(rewr_conv('int_zero_less'))
elif t.is_less_eq():
pt = pt_refl.on_rhs(rewr_conv('int_zero_less_eq'))
elif t.is_greater():
pt = pt_refl.on_rhs(rewr_conv('int_zero_greater'))
else:
pt = pt_refl.on_rhs(rewr_conv('int_zero_greater_eq'))
return pt.on_rhs(arg_conv(omega_simp_full_conv()))
def get_proof_term(self, t):
pt = refl(t)
if t.is_not():
if t.arg == true:
return pt.on_rhs(rewr_conv('not_true'))
elif t.arg == false:
return pt.on_rhs(rewr_conv('not_false'))
elif t.arg.is_not():
return pt.on_rhs(rewr_conv('double_neg'), self)
elif t.arg.is_conj():
return pt.on_rhs(rewr_conv('de_morgan_thm1'), arg1_conv(self),
arg_conv(self))
elif t.arg.is_disj():
return pt.on_rhs(rewr_conv('de_morgan_thm2'), arg1_conv(self),
arg_conv(self))
elif t.arg.is_equals() and t.arg.lhs.get_type() == BoolType:
return pt.on_rhs(rewr_conv('neg_iff'), binop_conv(self))
else:
return pt
elif t.is_disj() or t.is_conj() or t.is_equals():
return pt.on_rhs(arg1_conv(self), arg_conv(self))
else:
return pt
