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 a, b, c, fx, x
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Integer, Interval, Less, LessEq, Sum, one

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

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_{a, b, c \in \mathbb{Z}~|~a \leq b, b < c}~\left(\left(\sum_{x = a}^{c} f\left(x\right)\right) = \left(\left(\sum_{x = a}^{b} f\left(x\right)\right) + \left(\sum_{x = b + 1}^{c} f\left(x\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	57, 62, 59
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15, 16, 17, 18
12	Operation	operator: 33 operand: 27
13	Operation	operator: 60 operands: 20
14	Operation	operator: 49 operands: 21
15	Operation	operator: 49 operands: 22
16	Operation	operator: 49 operands: 23
17	Operation	operator: 24 operands: 54
18	Operation	operator: 25 operands: 26
19	ExprTuple	27
20	ExprTuple	28, 29
21	ExprTuple	57, 30
22	ExprTuple	62, 30
23	ExprTuple	59, 30
24	Literal
25	Literal
26	ExprTuple	62, 59
27	Lambda	parameter: 52 body: 31
28	Operation	operator: 33 operand: 36
29	Operation	operator: 33 operand: 37
30	Literal
31	Conditional	value: 43 condition: 35
32	ExprTuple	36
33	Literal
34	ExprTuple	37
35	Operation	operator: 49 operands: 38
36	Lambda	parameter: 52 body: 39
37	Lambda	parameter: 52 body: 40
38	ExprTuple	52, 41
39	Conditional	value: 43 condition: 42
40	Conditional	value: 43 condition: 44
41	Operation	operator: 55 operands: 45
42	Operation	operator: 49 operands: 46
43	Operation	operator: 47 operand: 52
44	Operation	operator: 49 operands: 50
45	ExprTuple	57, 59
46	ExprTuple	52, 51
47	Variable
48	ExprTuple	52
49	Literal
50	ExprTuple	52, 53
51	Operation	operator: 55 operands: 54
52	Variable
53	Operation	operator: 55 operands: 56
54	ExprTuple	57, 62
55	Literal
56	ExprTuple	58, 59
57	Variable
58	Operation	operator: 60 operands: 61
59	Variable
60	Literal
61	ExprTuple	62, 63
62	Variable
63	Literal