Axioms for the theory of proveit.logic.booleans.conjunction

import proveit
# Prepare this notebook for defining the axioms of a theory:
%axioms_notebook # Keep this at the top following 'import proveit'.
from proveit.logic import Equals, And, TRUE, FALSE, Forall, in_bool
from proveit import A, B, m, n
from proveit.core_expr_types import A_1_to_m
from proveit.numbers import Natural

%begin axioms

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

Truth table definitions:

and_t_t = Equals(And(TRUE, TRUE), TRUE)

and_t_t:

and_t_f = Equals(And(TRUE, FALSE), FALSE)

and_t_f:

and_f_t = Equals(And(FALSE, TRUE), FALSE)

and_f_t:

and_f_f = Equals(And(FALSE, FALSE), FALSE)

and_f_f:

Conjunction is only well-defined when each input is a Boolean:

left_in_bool = Forall((A, B), in_bool(A), conditions=[in_bool(And(A, B))]) 

left_in_bool:

right_in_bool = Forall((A, B), in_bool(B), conditions=[in_bool(And(A, B))]) 

right_in_bool:

Definition of multi-operand conjunction:

empty_conjunction = And() # base case

empty_conjunction:

multi_conjunction_def = \
    Forall(m, Forall((A_1_to_m, B), 
                     Equals(And(A_1_to_m, B), And(And(A_1_to_m), B)).with_wrap_after_operator()),
           domain=Natural)

multi_conjunction_def:

%end axioms

These axioms may now be imported from the theory package: proveit.logic.booleans.conjunction