Expression of type Forall

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 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 = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [S_1_to_n, Q, f, x], instance_expr = Implies(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)), domain = 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())

\forall_{n \in \mathbb{N}^+}~\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) \\  \Rightarrow \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)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 50 body: 4
3	ExprTuple	50
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 46 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	50, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	14, 42, 38, 34
13	Operation	operator: 15 operands: 16
14	ExprRange	lambda_map: 17 start_index: 49 end_index: 50
15	Literal
16	ExprTuple	18, 19
17	Lambda	parameter: 56 body: 51
18	Operation	operator: 46 operands: 20
19	Operation	operator: 21 operand: 24
20	ExprTuple	34, 23
21	Literal
22	ExprTuple	24
23	Operation	operator: 25 operand: 28
24	Lambda	parameters: 43 body: 27
25	Literal
26	ExprTuple	28
27	Conditional	value: 29 condition: 33
28	Lambda	parameters: 43 body: 30
29	Operation	operator: 31 operands: 32
30	Conditional	value: 35 condition: 33
31	Literal
32	ExprTuple	34, 35
33	Operation	operator: 36 operands: 37
34	Variable
35	Operation	operator: 38 operands: 43
36	Literal
37	ExprTuple	39, 40
38	Variable
39	ExprRange	lambda_map: 41 start_index: 49 end_index: 50
40	Operation	operator: 42 operands: 43
41	Lambda	parameter: 56 body: 44
42	Variable
43	ExprTuple	45
44	Operation	operator: 46 operands: 47
45	ExprRange	lambda_map: 48 start_index: 49 end_index: 50
46	Literal
47	ExprTuple	52, 51
48	Lambda	parameter: 56 body: 52
49	Literal
50	Variable
51	IndexedVar	variable: 53 index: 56
52	IndexedVar	variable: 54 index: 56
53	Variable
54	Variable
55	ExprTuple	56
56	Variable