Expression of type Forall

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 import P, Q
from proveit.logic import Equals, Forall
from proveit.logic.booleans.quantification import general_forall__Py_not_T__st_Qy, general_notexists_Px_st_Qx

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [P, Q], instance_expr = Equals(general_notexists_Px_st_Qx, 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())

\forall_{P, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\nexists_{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] =  \\ \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}\right)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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