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_Px_st_Qx, general_forall__Py_not_T__st_Qy
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_exists_Px_st_Qx, Not(general_forall__Py_not_T__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[\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] =  \\ (\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}\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: 21 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	44, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	39, 34
9	Operation	operator: 10 operands: 11
10	Literal
11	ExprTuple	12, 13
12	Operation	operator: 14 operand: 18
13	Operation	operator: 16 operand: 19
14	Literal
15	ExprTuple	18
16	Literal
17	ExprTuple	19
18	Lambda	parameters: 26 body: 20
19	Operation	operator: 21 operand: 25
20	Conditional	value: 23 condition: 24
21	Literal
22	ExprTuple	25
23	Operation	operator: 39 operands: 26
24	Operation	operator: 34 operands: 26
25	Lambda	parameters: 40 body: 27
26	ExprTuple	28
27	Conditional	value: 29 condition: 30
28	ExprRange	lambda_map: 31 start_index: 43 end_index: 44
29	Operation	operator: 32 operands: 33
30	Operation	operator: 34 operands: 40
31	Lambda	parameter: 48 body: 35
32	Literal
33	ExprTuple	36, 37
34	Variable
35	IndexedVar	variable: 38 index: 48
36	Operation	operator: 39 operands: 40
37	Literal
38	Variable
39	Variable
40	ExprTuple	41
41	ExprRange	lambda_map: 42 start_index: 43 end_index: 44
42	Lambda	parameter: 48 body: 45
43	Literal
44	Variable
45	IndexedVar	variable: 46 index: 48
46	Variable
47	ExprTuple	48
48	Variable