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 c, d, fj, gj, j, x
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Integer, Interval, Mult, Sum

# build up the expression from sub-expressions
sub_expr1 = [j]
sub_expr2 = Interval(c, d)
expr = Forall(instance_param_or_params = [x], instance_expr = Forall(instance_param_or_params = [c, d], instance_expr = Equals(Sum(index_or_indices = sub_expr1, summand = Mult(Mult(x, gj), fj), domain = sub_expr2), Mult(x, Sum(index_or_indices = sub_expr1, summand = Mult(gj, fj), domain = sub_expr2))), domain = Integer), domain = Complex)

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_{x \in \mathbb{C}}~\left[\forall_{c, d \in \mathbb{Z}}~\left(\left(\sum_{j = c}^{d} \left(\left(x \cdot g\left(j\right)\right) \cdot f\left(j\right)\right)\right) = \left(x \cdot \left(\sum_{j = c}^{d} \left(g\left(j\right) \cdot f\left(j\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 45 body: 4
3	ExprTuple	45
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 43 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	45, 11
10	Lambda	parameters: 53 body: 12
11	Literal
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
19	Operation	operator: 31 operand: 27
20	Operation	operator: 41 operands: 24
21	Operation	operator: 43 operands: 25
22	Operation	operator: 43 operands: 26
23	ExprTuple	27
24	ExprTuple	45, 28
25	ExprTuple	55, 29
26	ExprTuple	56, 29
27	Lambda	parameter: 54 body: 30
28	Operation	operator: 31 operand: 34
29	Literal
30	Conditional	value: 33 condition: 39
31	Literal
32	ExprTuple	34
33	Operation	operator: 41 operands: 35
34	Lambda	parameter: 54 body: 36
35	ExprTuple	37, 47
36	Conditional	value: 38 condition: 39
37	Operation	operator: 41 operands: 40
38	Operation	operator: 41 operands: 42
39	Operation	operator: 43 operands: 44
40	ExprTuple	45, 46
41	Literal
42	ExprTuple	46, 47
43	Literal
44	ExprTuple	54, 48
45	Variable
46	Operation	operator: 49 operand: 54
47	Operation	operator: 50 operand: 54
48	Operation	operator: 52 operands: 53
49	Variable
50	Variable
51	ExprTuple	54
52	Literal
53	ExprTuple	55, 56
54	Variable
55	Variable
56	Variable