Expression of type Implies

from the theory of proveit.logic.booleans.quantification.existence

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.logic import Implies, Not
from proveit.logic.booleans.quantification import general_exists_Px_st_Qx, general_forall__Py_not_T__st_Qy

# build up the expression from sub-expressions
expr = Implies(general_exists_Px_st_Qx, Not(general_forall__Py_not_T__st_Qy)).with_wrapping_at(2)

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[\exists_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ (\lnot \left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(P\left(y_{1}, y_{2}, \ldots, y_{n}\right) \neq \top\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.	()	(2)	('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 operand: 9
4	Operation	operator: 7 operand: 10
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10
9	Lambda	parameters: 17 body: 11
10	Operation	operator: 12 operand: 16
11	Conditional	value: 14 condition: 15
12	Literal
13	ExprTuple	16
14	Operation	operator: 30 operands: 17
15	Operation	operator: 25 operands: 17
16	Lambda	parameters: 31 body: 18
17	ExprTuple	19
18	Conditional	value: 20 condition: 21
19	ExprRange	lambda_map: 22 start_index: 34 end_index: 35
20	Operation	operator: 23 operands: 24
21	Operation	operator: 25 operands: 31
22	Lambda	parameter: 39 body: 26
23	Literal
24	ExprTuple	27, 28
25	Variable
26	IndexedVar	variable: 29 index: 39
27	Operation	operator: 30 operands: 31
28	Literal
29	Variable
30	Variable
31	ExprTuple	32
32	ExprRange	lambda_map: 33 start_index: 34 end_index: 35
33	Lambda	parameter: 39 body: 36
34	Literal
35	Variable
36	IndexedVar	variable: 37 index: 39
37	Variable
38	ExprTuple	39
39	Variable