Axioms for the theory of proveit.core_expr_types.conditionals

import proveit
# Prepare this notebook for defining the axioms of a theory:
%axioms_notebook # Keep this at the top following 'import proveit'.
from proveit import Conditional, ConditionalSet, ExprRange, IndexedVar
from proveit import a, b, c, i, m, n, Q, R
from proveit.core_expr_types import a_1_to_m, c_1_to_n
from proveit.logic import Implies, Iff, Forall, Equals, TRUE, FALSE, Or
from proveit.numbers import one, Natural

%begin axioms

Defining axioms for theory 'proveit.core_expr_types.conditionals'
Subsequent end-of-cell assignments will define axioms
%end_axioms will finalize the definitions

A Conditional is defined to evaluate to its value when the condition is TRUE:

true_condition_reduction = Forall(a, Equals(Conditional(a, TRUE), a))

true_condition_reduction:

A condition may be substituted with a logically equivalent condition. (Since a Conditional is not an Operation, this must be a distinct axiom from Operation substitution axioms).

condition_replacement = \
    Forall((a, Q, R), Equals(Conditional(a, Q),
                             Conditional(a, R)).with_wrap_before_operator(),
          conditions=[Iff(Q, R)])

condition_replacement:

The condition is either true or not true but otherwise it doesn't matter if it is a Boolean. Therefore, a condition of $Q$ is the same as a condition of $Q=\top$

condition__as__condition_eq_true = \
    Forall((a, Q), Equals(Conditional(a, Q), 
                          Conditional(a, Equals(Q, TRUE))).with_wrap_before_operator())

condition__as__condition_eq_true:

If two values are equal when the condition is satisfied, one may replace the other within the Conditional.

conditional_substitution = \
    Forall((a, b, Q), Equals(Conditional(a, Q),
                             Conditional(b, Q)).with_wrap_before_operator(),
          conditions=[Implies(Q, Equals(a, b))])

conditional_substitution:

If one and only one Conditional in a ConditionalSet is True, equate the ConditionalSet to the Conditional.

# singular_truth_reduction = \
#     Forall((m, n), 
#            Forall((a, b, c), 
#                   Equals(ConditionalSet(var_range(a, one, m), b, var_range(c, one, n)), b), 
#                   conditions=[Equals(Or(var_range(a, one, m)), FALSE), Equals(Or(var_range(c, one, n)), FALSE)]), 
# domain=Natural)

true_case_reduction = \
    Forall((m, n), 
           Forall((a_1_to_m, b, c_1_to_n), 
                  Equals(ConditionalSet(ExprRange(i, Conditional(IndexedVar(a, i), FALSE), one, m), b, 
                                        ExprRange(i, Conditional(IndexedVar(c, i), FALSE), one, n)), b)), 
                  domain=Natural)

true_case_reduction:

%end axioms

These axioms may now be imported from the theory package: proveit.core_expr_types.conditionals