Expression of type Forall

from the theory of proveit.linear_algebra.addition

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, i, v, vb, vi
from proveit.linear_algebra import VecAdd, VecSum
from proveit.logic import Equals, Forall
from proveit.numbers import Integer, Interval, Less, one, subtract

# build up the expression from sub-expressions
sub_expr1 = [i]
expr = Forall(instance_param_or_params = [v], instance_expr = Forall(instance_param_or_params = [a, b], instance_expr = Equals(VecSum(index_or_indices = sub_expr1, summand = vi, domain = Interval(a, b)), VecAdd(VecSum(index_or_indices = sub_expr1, summand = vi, domain = Interval(a, subtract(b, one))), vb)), domain = Integer, condition = Less(a, b)))

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