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, f, fj, g, 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 = [f, g], instance_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_{f, g}~\left[\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]\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: 11 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	54, 53
4	Operation	operator: 11 operand: 6
5	ExprTuple	6
6	Lambda	parameter: 49 body: 8
7	ExprTuple	49
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operand: 14
10	Operation	operator: 47 operands: 13
11	Literal
12	ExprTuple	14
13	ExprTuple	49, 15
14	Lambda	parameters: 57 body: 16
15	Literal
16	Conditional	value: 17 condition: 18
17	Operation	operator: 19 operands: 20
18	Operation	operator: 21 operands: 22
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	25, 26
23	Operation	operator: 35 operand: 31
24	Operation	operator: 45 operands: 28
25	Operation	operator: 47 operands: 29
26	Operation	operator: 47 operands: 30
27	ExprTuple	31
28	ExprTuple	49, 32
29	ExprTuple	59, 33
30	ExprTuple	60, 33
31	Lambda	parameter: 58 body: 34
32	Operation	operator: 35 operand: 38
33	Literal
34	Conditional	value: 37 condition: 43
35	Literal
36	ExprTuple	38
37	Operation	operator: 45 operands: 39
38	Lambda	parameter: 58 body: 40
39	ExprTuple	41, 51
40	Conditional	value: 42 condition: 43
41	Operation	operator: 45 operands: 44
42	Operation	operator: 45 operands: 46
43	Operation	operator: 47 operands: 48
44	ExprTuple	49, 50
45	Literal
46	ExprTuple	50, 51
47	Literal
48	ExprTuple	58, 52
49	Variable
50	Operation	operator: 53 operand: 58
51	Operation	operator: 54 operand: 58
52	Operation	operator: 56 operands: 57
53	Variable
54	Variable
55	ExprTuple	58
56	Literal
57	ExprTuple	59, 60
58	Variable
59	Variable
60	Variable