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, a, b, c, f, fx, x
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Integer, Interval, Sum, subtract

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [f], instance_expr = Forall(instance_param_or_params = [a, b, c], instance_expr = Equals(Sum(index_or_indices = sub_expr1, summand = fx, domain = Interval(a, b)), Sum(index_or_indices = sub_expr1, summand = Function(f, [subtract(x, c)]), domain = Interval(Add(a, c), Add(b, c)))), domain = Integer))

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