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 Boolean, Forall, Implies, InSet, Or

# build up the expression from sub-expressions
sub_expr1 = InSet(Or(A, B), Boolean)
expr = Implies(Forall(instance_param_or_params = [A], instance_expr = Forall(instance_param_or_params = [B], instance_expr = sub_expr1, domain = Boolean), domain = Boolean), Forall(instance_param_or_params = [A, B], instance_expr = sub_expr1, domain = Boolean)).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}}~\left(\left(A \lor B\right) \in \mathbb{B}\right)\right]\right] \\  \Rightarrow \left[\forall_{A, B \in \mathbb{B}}~\left(\left(A \lor B\right) \in \mathbb{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: 14 operand: 7
4	Operation	operator: 14 operand: 8
5	ExprTuple	7
6	ExprTuple	8
7	Lambda	parameter: 32 body: 10
8	Lambda	parameters: 31 body: 11
9	ExprTuple	32
10	Conditional	value: 12 condition: 19
11	Conditional	value: 23 condition: 13
12	Operation	operator: 14 operand: 18
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18
16	Literal
17	ExprTuple	19, 24
18	Lambda	parameter: 33 body: 21
19	Operation	operator: 26 operands: 22
20	ExprTuple	33
21	Conditional	value: 23 condition: 24
22	ExprTuple	32, 29
23	Operation	operator: 26 operands: 25
24	Operation	operator: 26 operands: 27
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	33, 29
28	Operation	operator: 30 operands: 31
29	Literal
30	Literal
31	ExprTuple	32, 33
32	Variable
33	Variable