Expression of type Lambda

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, Lambda, 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, InSet
from proveit.logic.sets import general_comprehension_fyn
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [S_1_to_n, Q, f, x], instance_expr = Equals(InSet(x, general_comprehension_fyn), 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)).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())

n \mapsto \left\{\forall_{S_{1}, S_{2}, \ldots, S_{n}, Q, f, x}~\left(\begin{array}{c} \begin{array}{l} \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) \\  = \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] \end{array} \end{array}\right) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 47 body: 2
1	ExprTuple	47
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 43 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	47, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	12, 39, 35, 31
11	Operation	operator: 28 operands: 13
12	ExprRange	lambda_map: 14 start_index: 46 end_index: 47
13	ExprTuple	15, 16
14	Lambda	parameter: 53 body: 48
15	Operation	operator: 43 operands: 17
16	Operation	operator: 18 operand: 21
17	ExprTuple	31, 20
18	Literal
19	ExprTuple	21
20	Operation	operator: 22 operand: 25
21	Lambda	parameters: 40 body: 24
22	Literal
23	ExprTuple	25
24	Conditional	value: 26 condition: 30
25	Lambda	parameters: 40 body: 27
26	Operation	operator: 28 operands: 29
27	Conditional	value: 32 condition: 30
28	Literal
29	ExprTuple	31, 32
30	Operation	operator: 33 operands: 34
31	Variable
32	Operation	operator: 35 operands: 40
33	Literal
34	ExprTuple	36, 37
35	Variable
36	ExprRange	lambda_map: 38 start_index: 46 end_index: 47
37	Operation	operator: 39 operands: 40
38	Lambda	parameter: 53 body: 41
39	Variable
40	ExprTuple	42
41	Operation	operator: 43 operands: 44
42	ExprRange	lambda_map: 45 start_index: 46 end_index: 47
43	Literal
44	ExprTuple	49, 48
45	Lambda	parameter: 53 body: 49
46	Literal
47	Variable
48	IndexedVar	variable: 50 index: 53
49	IndexedVar	variable: 51 index: 53
50	Variable
51	Variable
52	ExprTuple	53
53	Variable