Expression of type Implies

from the theory of proveit.logic.booleans.disjunction

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import A, B
from proveit.logic import And, Boolean, Forall, Implies, Not, Or

# build up the expression from sub-expressions
sub_expr1 = Or(A, B)
sub_expr2 = Not(And(Not(A), Not(B)))
expr = Implies(Forall(instance_param_or_params = [A], instance_expr = Forall(instance_param_or_params = [B], instance_expr = sub_expr1, domain = Boolean, condition = sub_expr2), domain = Boolean), Forall(instance_param_or_params = [A, B], instance_expr = sub_expr1, domain = Boolean, condition = sub_expr2)).with_wrapping_at(1)

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\begin{array}{c} \begin{array}{l} \left[\forall_{A \in \mathbb{B}}~\left[\forall_{B \in \mathbb{B}~|~\lnot \left((\lnot A) \land (\lnot B)\right)}~\left(A \lor B\right)\right]\right] \\  \Rightarrow \left[\forall_{A, B \in \mathbb{B}~|~\lnot \left((\lnot A) \land (\lnot B)\right)}~\left(A \lor B\right)\right] \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(1)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 13 operand: 7
4	Operation	operator: 13 operand: 8
5	ExprTuple	7
6	ExprTuple	8
7	Lambda	parameter: 39 body: 9
8	Lambda	parameters: 23 body: 10
9	Conditional	value: 11 condition: 17
10	Conditional	value: 20 condition: 12
11	Operation	operator: 13 operand: 16
12	Operation	operator: 32 operands: 15
13	Literal
14	ExprTuple	16
15	ExprTuple	17, 25, 26
16	Lambda	parameter: 40 body: 18
17	Operation	operator: 27 operands: 19
18	Conditional	value: 20 condition: 21
19	ExprTuple	39, 30
20	Operation	operator: 22 operands: 23
21	Operation	operator: 32 operands: 24
22	Literal
23	ExprTuple	39, 40
24	ExprTuple	25, 26
25	Operation	operator: 27 operands: 28
26	Operation	operator: 37 operand: 31
27	Literal
28	ExprTuple	40, 30
29	ExprTuple	31
30	Literal
31	Operation	operator: 32 operands: 33
32	Literal
33	ExprTuple	34, 35
34	Operation	operator: 37 operand: 39
35	Operation	operator: 37 operand: 40
36	ExprTuple	39
37	Literal
38	ExprTuple	40
39	Variable
40	Variable