from proveit.basiclogic import BOOLEANS, TRUE, FALSE, in_bool, Implies, And, derive_stmt_eq_true, Equals
from proveit.common import A, P, PofA
from proveit.basiclogic.common import PofTrue, PofFalse
# hypothesis = [P(TRUE) and P(FALSE)]
hypothesis = And(PofTrue, PofFalse)
# in_bool(A=TRUE), in_bool(A=FALSE), in_bool(P(A) = TRUE)
AeqT = Equals(A, TRUE)
AeqF = Equals(A, FALSE)
PofAeqT = Equals(PofA, TRUE)
for eq_expr in (AeqT, AeqF, PofAeqT):
eq_expr.deduce_in_bool()
# P(TRUE), P(FALSE) assuming hypothesis
for case in hypothesis.decompose():
case.proven({hypothesis})
# A=TRUE => P(A)=TRUE assuming hypothesis
Implies(AeqT,
derive_stmt_eq_true(AeqT.sub_left_side_into(PofA,
A))).proven({hypothesis})
# A=FALSE => P(A)=TRUE assuming hypothesis
Implies(AeqF,
derive_stmt_eq_true(AeqF.sub_left_side_into(PofA,
A))).proven({hypothesis})
# P(A) assuming hypothesis, (A in BOOLEANS)
in_bool(A).unfold().derive_common_conclusion(
PofAeqT).derive_via_boolean_equality().proven({hypothesis,
in_bool(A)})
# forall_{P} P(TRUE) and P(FALSE) => forall_{A in BOOLEANS} P(A)
Implies(hypothesis,
PofA.generalize(A, domain=BOOLEANS)).generalize(P).qed(__file__)
from proveit.basiclogic.booleans.theorems import true_is_bool
from proveit.basiclogic import TRUE, in_bool, Implies, Equals
from proveit.common import A, X
# hypothesis = (A=TRUE)
hypothesis = Equals(A, TRUE)
# in_bool(TRUE)
true_is_bool.proven()
# in_bool(A) assuming hypothesis
conclusion = hypothesis.sub_left_side_into(in_bool(X), X).proven({hypothesis})
# forall_{A} A=TRUE => in_bool(A)
Implies(hypothesis, conclusion).generalize(A).qed(__file__)
from proveit.basiclogic.booleans.theorems import not_false
from proveit.basiclogic import Implies, Not, Equals, FALSE
from proveit.common import A, X
# AeqF := (A=F)
AeqF = Equals(A, FALSE)
# Not(FALSE)
not_false
# Not(A) assuming A=FALSE because Not(FALSE)
not_a = AeqF.sub_left_side_into(Not(X), X).proven({AeqF})
Implies(AeqF, not_a).generalize(A).qed(__file__)
