Expression of type Conditional

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 Conditional, P, Q, n
from proveit.logic import Equals, Forall, InSet, Not
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy, general_notexists_Px_st_Qx
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [P, Q], instance_expr = Equals(general_notexists_Px_st_Qx, Not(general_exists_Py_st_Qy)).with_wrapping_at(2)), InSet(n, NaturalPos))

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\{\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] =  \\ (\lnot \left[\exists_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right]) \end{array} \end{array}\right) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	39, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	31, 32
10	Operation	operator: 11 operands: 12
11	Literal
12	ExprTuple	13, 14
13	Operation	operator: 15 operand: 24
14	Operation	operator: 16 operand: 18
15	Literal
16	Literal
17	ExprTuple	18
18	Operation	operator: 20 operand: 24
19	Conditional	value: 22 condition: 23
20	Literal
21	ExprTuple	24
22	Operation	operator: 31 operands: 25
23	Operation	operator: 32 operands: 25
24	Lambda	parameters: 33 body: 26
25	ExprTuple	27
26	Conditional	value: 28 condition: 29
27	ExprRange	lambda_map: 30 start_index: 38 end_index: 39
28	Operation	operator: 31 operands: 33
29	Operation	operator: 32 operands: 33
30	Lambda	parameter: 43 body: 34
31	Variable
32	Variable
33	ExprTuple	35
34	IndexedVar	variable: 36 index: 43
35	ExprRange	lambda_map: 37 start_index: 38 end_index: 39
36	Variable
37	Lambda	parameter: 43 body: 40
38	Literal
39	Variable
40	IndexedVar	variable: 41 index: 43
41	Variable
42	ExprTuple	43
43	Variable