from proveit.basiclogic.booleans.axioms import exists_def, not_exists_def
from proveit.basiclogic import Forall, Not, NotEquals, TRUE
from proveit.common import X, P, S, x_etc, Qetc, Px_etc, etc_Qx_etc
# [exists_{..x.. in S | ..Q(..x..)..} P(..x..)] = not(forall_{..x.. in S | ..Q(..x..)..} (P(..x..) != TRUE))
exists_def_spec = exists_def.instantiate().proven()
# notexists_{..x.. in S | ..Q..(..x..)} P(..x..) = not[exists_{..x.. in S
# | ..Q(..x..)..} P(..x..)]
not_exists_def_spec = not_exists_def.instantiate().proven()
# rhs = forall_{..x.. in S | ..Q(..x..)..} (P(..x..) != TRUE)
rhs = Forall(x_etc, NotEquals(Px_etc, TRUE), S, etc_Qx_etc)
# [forall_{..x.. in S | ..Q(..x..)..} (P(..x..) != TRUE)] in BOOLEANS
rhs.deduce_in_bool().proven()
# not(not(forall_{..x.. in S | ..Q(..x..)..} (P(..x..) != TRUE))) =
# forall_{..x.. in S | ..Q(..x..)..} (P(..x..) != TRUE))
double_negated_forall = Not(
Not(rhs)).deduce_double_negation_equiv().derive_reversed().proven()
# notexists_{..x.. in S | ..Q(..x..)..} P(..x..) = forall_{..x.. in S |
# ..Q(..x..)..} (P(..x..) != TRUE))
equiv = not_exists_def_spec.apply_transitivity(
exists_def_spec.substitution(
Not(X), X)).apply_transitivity(double_negated_forall).proven()
# forall_{P, ..Q..} [notexists_{..x.. in S | ..Q(..x..)..} P(..x..) =
# forall_{..x.. in S | ..Q(..x..)..} (P(..x..) != TRUE)]
equiv.generalize((P, Qetc, S)).qed(__file__)
from proveit.basiclogic.booleans.axioms import exists_def
from proveit.basiclogic import Forall, NotEquals, Implies, TRUE, FALSE, derive_stmt_eq_true, In
from proveit.common import X, P, S, x_etc, y_etc, Px_etc, Py_etc, Qetc, etc_Qx_etc, etc_Qy_etc
in_domain = In(x_etc, S) # ..x.. in S
# never_py = [forall_{..y.. in S | ..Q(..y..)..} (P(..y..) != TRUE)]
never_py = Forall(y_etc, NotEquals(Py_etc, TRUE), S, etc_Qy_etc)
# (P(..x..) != TRUE) assuming ..Q(..x..).., never_py
never_py.instantiate({y_etc: x_etc}).proven({etc_Qx_etc, never_py, in_domain})
# (TRUE != TRUE) assuming ..Q(..x..).., P(..x..), never_py
true_not_eq_true = derive_stmt_eq_true(Px_etc).sub_right_side_into(
NotEquals(X, TRUE), X).proven({etc_Qx_etc, Px_etc, never_py, in_domain})
# FALSE assuming ..Q(..x..).., P(..x..), never_py
true_not_eq_true.evaluation().derive_contradiction().derive_conclusion(
).proven({etc_Qx_etc, Px_etc, never_py, in_domain})
# [forall_{..y.. in S | ..Q(..y..)..} (P(..y..) != TRUE)] in BOOLEANS
never_py.deduce_in_bool().proven()
# Not(forall_{..y.. in S | ..Q(..y..)..} (P(..y..) != TRUE) assuming
# ..Q(..x..).., P(..x..)
Implies(never_py, FALSE).derive_via_contradiction().proven(
{etc_Qx_etc, Px_etc, in_domain})
# exists_{..y.. in S | ..Q(..y..)..} P(..y..) assuming Q(..x..), P(..x..)
existence = exists_def.instantiate({
x_etc: y_etc
}).derive_left_via_equality().proven({etc_Qx_etc, Px_etc, in_domain})
# forall_{P, ..Q.., S} forall_{..x.. in S | ..Q(..x..)..} [P(..x..) =>
# exists_{..y.. in S | ..Q(..y..)..} P(..y..)]
Implies(Px_etc, existence).generalize(x_etc, S, etc_Qx_etc).generalize(
(P, Qetc, S)).qed(__file__)