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.core_expr_types import P__y_1_to_n, Q__y_1_to_n, y_1_to_n
from proveit.logic import Forall, InSet
from proveit.logic.booleans.quantification import general_exists_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 = Forall(instance_param_or_params = [y_1_to_n], instance_expr = general_exists_Px_st_Qx, conditions = [P__y_1_to_n, Q__y_1_to_n])), 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[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~P\left(y_{1}, y_{2}, \ldots, y_{n}\right), Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\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]\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: 10 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	36, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	28, 29
9	Operation	operator: 10 operand: 12
10	Literal
11	ExprTuple	12
12	Lambda	parameters: 24 body: 13
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operand: 20
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20
18	Literal
19	ExprTuple	21, 22
20	Lambda	parameters: 30 body: 23
21	Operation	operator: 28 operands: 24
22	Operation	operator: 29 operands: 24
23	Conditional	value: 25 condition: 26
24	ExprTuple	27
25	Operation	operator: 28 operands: 30
26	Operation	operator: 29 operands: 30
27	ExprRange	lambda_map: 31 start_index: 35 end_index: 36
28	Variable
29	Variable
30	ExprTuple	32
31	Lambda	parameter: 41 body: 33
32	ExprRange	lambda_map: 34 start_index: 35 end_index: 36
33	IndexedVar	variable: 37 index: 41
34	Lambda	parameter: 41 body: 38
35	Literal
36	Variable
37	Variable
38	IndexedVar	variable: 39 index: 41
39	Variable
40	ExprTuple	41
41	Variable