Expression of type Implies

from the theory of proveit.logic.booleans

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, PofA
from proveit.logic import And, Boolean, Equals, FALSE, Forall, Implies, PofFalse, PofTrue, TRUE

# build up the expression from sub-expressions
expr = Implies(And(Equals(PofTrue, FALSE), Equals(PofFalse, TRUE)), Equals(Forall(instance_param_or_params = [A], instance_expr = PofA, domain = Boolean), FALSE))

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())

\left(\left(P\left(\top\right) = \bot\right) \land \left(P\left(\bot\right) = \top\right)\right) \Rightarrow \left(\left[\forall_{A \in \mathbb{B}}~P\left(A\right)\right] = \bot\right)

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.	()	()	('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: 5 operands: 6
4	Operation	operator: 12 operands: 7
5	Literal
6	ExprTuple	8, 9
7	ExprTuple	10, 23
8	Operation	operator: 12 operands: 11
9	Operation	operator: 12 operands: 13
10	Operation	operator: 14 operand: 18
11	ExprTuple	16, 23
12	Literal
13	ExprTuple	17, 22
14	Literal
15	ExprTuple	18
16	Operation	operator: 26 operand: 22
17	Operation	operator: 26 operand: 23
18	Lambda	parameter: 30 body: 21
19	ExprTuple	22
20	ExprTuple	23
21	Conditional	value: 24 condition: 25
22	Literal
23	Literal
24	Operation	operator: 26 operand: 30
25	Operation	operator: 28 operands: 29
26	Variable
27	ExprTuple	30
28	Literal
29	ExprTuple	30, 31
30	Variable
31	Literal