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 And, Equals, Forall
from proveit.numbers import Add, Integer, Interval, 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, condition = And(LessEq(a, b), LessEq(b, c)).with_total_ordering_style())

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