Expression of type Forall

from the theory of proveit.numbers.summation

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 Function, R, S, a, f, fa, fx, g, gy, n, x, y
from proveit.logic import Card, Equals, Forall, Implies, InSet, SetOfAll
from proveit.numbers import Complex, Sum

# build up the expression from sub-expressions
sub_expr1 = [y]
expr = Forall(instance_param_or_params = [f, g, R, S], instance_expr = Implies(Forall(instance_param_or_params = [a], instance_expr = InSet(fa, Complex), domain = R), Equals(Sum(index_or_indices = [x], summand = fx, domain = R), Sum(index_or_indices = sub_expr1, summand = Function(f, [gy]), domain = S))), conditions = [Equals(SetOfAll(instance_param_or_params = sub_expr1, instance_element = gy, domain = S), R), Equals(Card(S), n), Equals(Card(R), n)])

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_{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)

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: 16 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	58, 67, 64, 66
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	13, 14, 15
11	Operation	operator: 16 operand: 23
12	Operation	operator: 21 operands: 18
13	Operation	operator: 21 operands: 19
14	Operation	operator: 21 operands: 20
15	Operation	operator: 21 operands: 22
16	Literal
17	ExprTuple	23
18	ExprTuple	24, 25
19	ExprTuple	26, 64
20	ExprTuple	27, 29
21	Literal
22	ExprTuple	28, 29
23	Lambda	parameter: 62 body: 30
24	Operation	operator: 32 operand: 41
25	Operation	operator: 32 operand: 42
26	Operation	operator: 34 operand: 43
27	Operation	operator: 37 operand: 66
28	Operation	operator: 37 operand: 64
29	Variable
30	Conditional	value: 39 condition: 40
31	ExprTuple	41
32	Literal
33	ExprTuple	42
34	Literal
35	ExprTuple	43
36	ExprTuple	66
37	Literal
38	ExprTuple	64
39	Operation	operator: 60 operands: 44
40	Operation	operator: 60 operands: 45
41	Lambda	parameter: 63 body: 46
42	Lambda	parameter: 69 body: 47
43	Lambda	parameter: 69 body: 48
44	ExprTuple	49, 50
45	ExprTuple	62, 64
46	Conditional	value: 51 condition: 52
47	Conditional	value: 53 condition: 54
48	Conditional	value: 65 condition: 54
49	Operation	operator: 58 operand: 62
50	Literal
51	Operation	operator: 58 operand: 63
52	Operation	operator: 60 operands: 57
53	Operation	operator: 58 operand: 65
54	Operation	operator: 60 operands: 61
55	ExprTuple	62
56	ExprTuple	63
57	ExprTuple	63, 64
58	Variable
59	ExprTuple	65
60	Literal
61	ExprTuple	69, 66
62	Variable
63	Variable
64	Variable
65	Operation	operator: 67 operand: 69
66	Variable
67	Variable
68	ExprTuple	69
69	Variable