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 Forall, Implies, InSet, Not
from proveit.logic.booleans.quantification import general_exists_notPx_st_Qx, general_forall_Py_if_Qy
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [P, Q], instance_expr = Implies(general_exists_notPx_st_Qx, Not(general_forall_Py_if_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)}~(\lnot 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)}~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: 20 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	40, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	32, 33
9	Operation	operator: 10 operands: 11
10	Literal
11	ExprTuple	12, 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 25 operand: 18
14	Literal
15	ExprTuple	17
16	ExprTuple	18
17	Lambda	parameters: 31 body: 19
18	Operation	operator: 20 operand: 24
19	Conditional	value: 22 condition: 23
20	Literal
21	ExprTuple	24
22	Operation	operator: 25 operand: 28
23	Operation	operator: 33 operands: 31
24	Lambda	parameters: 34 body: 27
25	Literal
26	ExprTuple	28
27	Conditional	value: 29 condition: 30
28	Operation	operator: 32 operands: 31
29	Operation	operator: 32 operands: 34
30	Operation	operator: 33 operands: 34
31	ExprTuple	35
32	Variable
33	Variable
34	ExprTuple	36
35	ExprRange	lambda_map: 37 start_index: 39 end_index: 40
36	ExprRange	lambda_map: 38 start_index: 39 end_index: 40
37	Lambda	parameter: 46 body: 41
38	Lambda	parameter: 46 body: 42
39	Literal
40	Variable
41	IndexedVar	variable: 43 index: 46
42	IndexedVar	variable: 44 index: 46
43	Variable
44	Variable
45	ExprTuple	46
46	Variable