Theorem equiv_sums of type Forall

from the theory of proveit.numbers.summation

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.numbers.summation import equiv_sums

# check that the built expression is the same as the stored expression
assert equiv_sums.expr == stored_expr
assert equiv_sums.expr._style_id == stored_expr._style_id
print("Passed sanity check: equiv_sums matches stored_expr")

Passed sanity check: equiv_sums 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_{f, g, R, S~|~\left\{g\left(y\right)\right\}_{y \in S} = R, |S| = n, |R| = n}~\left(\left[\forall_{a \in R}~\left(f\left(a\right) \in \mathbb{C}\right)\right] \Rightarrow \left(\left[\sum_{x \in R}~f\left(x\right)\right] = \left[\sum_{y \in S}~f\left(g\left(y\right)\right)\right]\right)\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: 24 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 37 body: 4
3	ExprTuple	37
4	Conditional	value: 5 condition: 6
5	Operation	operator: 24 operand: 9
6	Operation	operator: 68 operands: 8
7	ExprTuple	9
8	ExprTuple	37, 10
9	Lambda	parameters: 11 body: 12
10	Literal
11	ExprTuple	66, 75, 72, 74
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22, 23
19	Operation	operator: 24 operand: 31
20	Operation	operator: 29 operands: 26
21	Operation	operator: 29 operands: 27
22	Operation	operator: 29 operands: 28
23	Operation	operator: 29 operands: 30
24	Literal
25	ExprTuple	31
26	ExprTuple	32, 33
27	ExprTuple	34, 72
28	ExprTuple	35, 37
29	Literal
30	ExprTuple	36, 37
31	Lambda	parameter: 70 body: 38
32	Operation	operator: 40 operand: 49
33	Operation	operator: 40 operand: 50
34	Operation	operator: 42 operand: 51
35	Operation	operator: 45 operand: 74
36	Operation	operator: 45 operand: 72
37	Variable
38	Conditional	value: 47 condition: 48
39	ExprTuple	49
40	Literal
41	ExprTuple	50
42	Literal
43	ExprTuple	51
44	ExprTuple	74
45	Literal
46	ExprTuple	72
47	Operation	operator: 68 operands: 52
48	Operation	operator: 68 operands: 53
49	Lambda	parameter: 71 body: 54
50	Lambda	parameter: 77 body: 55
51	Lambda	parameter: 77 body: 56
52	ExprTuple	57, 58
53	ExprTuple	70, 72
54	Conditional	value: 59 condition: 60
55	Conditional	value: 61 condition: 62
56	Conditional	value: 73 condition: 62
57	Operation	operator: 66 operand: 70
58	Literal
59	Operation	operator: 66 operand: 71
60	Operation	operator: 68 operands: 65
61	Operation	operator: 66 operand: 73
62	Operation	operator: 68 operands: 69
63	ExprTuple	70
64	ExprTuple	71
65	ExprTuple	71, 72
66	Variable
67	ExprTuple	73
68	Literal
69	ExprTuple	77, 74
70	Variable
71	Variable
72	Variable
73	Operation	operator: 75 operand: 77
74	Variable
75	Variable
76	ExprTuple	77
77	Variable