Expression of type Conditional

from the theory of proveit.logic.sets.comprehension

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, Q, f, n, x
from proveit.core_expr_types import Q__y_1_to_n, S_1_to_n, f__y_1_to_n, y_1_to_n
from proveit.logic import Equals, Exists, Forall, Implies, InSet
from proveit.logic.sets import general_comprehension_fyn
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [S_1_to_n, Q, f, x], instance_expr = Implies(Exists(instance_param_or_params = [y_1_to_n], instance_expr = Equals(x, f__y_1_to_n), domains = [S_1_to_n], condition = Q__y_1_to_n), InSet(x, general_comprehension_fyn)).with_wrapping_at(1)), 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_{S_{1}, S_{2}, \ldots, S_{n}, Q, f, x}~\left(\begin{array}{c} \begin{array}{l} \left[\exists_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(x = f\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right)\right] \\  \Rightarrow \left(x \in \left\{f\left(y_{1}, y_{2}, \ldots, y_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right\}_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{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: 6
2	Operation	operator: 42 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	46, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	10, 38, 32, 29
9	Operation	operator: 11 operands: 12
10	ExprRange	lambda_map: 13 start_index: 45 end_index: 46
11	Literal
12	ExprTuple	14, 15
13	Lambda	parameter: 52 body: 47
14	Operation	operator: 16 operand: 19
15	Operation	operator: 42 operands: 18
16	Literal
17	ExprTuple	19
18	ExprTuple	29, 20
19	Lambda	parameters: 39 body: 21
20	Operation	operator: 22 operand: 25
21	Conditional	value: 24 condition: 31
22	Literal
23	ExprTuple	25
24	Operation	operator: 26 operands: 27
25	Lambda	parameters: 39 body: 28
26	Literal
27	ExprTuple	29, 30
28	Conditional	value: 30 condition: 31
29	Variable
30	Operation	operator: 32 operands: 39
31	Operation	operator: 33 operands: 34
32	Variable
33	Literal
34	ExprTuple	35, 36
35	ExprRange	lambda_map: 37 start_index: 45 end_index: 46
36	Operation	operator: 38 operands: 39
37	Lambda	parameter: 52 body: 40
38	Variable
39	ExprTuple	41
40	Operation	operator: 42 operands: 43
41	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
42	Literal
43	ExprTuple	48, 47
44	Lambda	parameter: 52 body: 48
45	Literal
46	Variable
47	IndexedVar	variable: 49 index: 52
48	IndexedVar	variable: 50 index: 52
49	Variable
50	Variable
51	ExprTuple	52
52	Variable