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, c, i, v, vi
from proveit.linear_algebra import VecAdd, VecSum
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Integer, Interval, Less, LessEq, one

# 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, c], instance_expr = Equals(VecSum(index_or_indices = sub_expr1, summand = vi, domain = Interval(a, c)), VecAdd(VecSum(index_or_indices = sub_expr1, summand = vi, domain = Interval(a, b)), VecSum(index_or_indices = sub_expr1, summand = vi, 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_{v}~\left[\forall_{a, b, c \in \mathbb{Z}~|~a \leq b, b < c}~\left(\left(\sum_{i=a}^{c} v\left(i\right)\right) = \left(\left(\sum_{i=a}^{b} v\left(i\right)\right) + \left(\sum_{i=b + 1}^{c} v\left(i\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: 5 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 52 body: 4
3	ExprTuple	52
4	Operation	operator: 5 operand: 7
5	Literal
6	ExprTuple	7
7	Lambda	parameters: 8 body: 9
8	ExprTuple	62, 67, 64
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, 21, 22
16	Operation	operator: 38 operand: 32
17	Operation	operator: 24 operands: 25
18	Operation	operator: 54 operands: 26
19	Operation	operator: 54 operands: 27
20	Operation	operator: 54 operands: 28
21	Operation	operator: 29 operands: 59
22	Operation	operator: 30 operands: 31
23	ExprTuple	32
24	Literal
25	ExprTuple	33, 34
26	ExprTuple	62, 35
27	ExprTuple	67, 35
28	ExprTuple	64, 35
29	Literal
30	Literal
31	ExprTuple	67, 64
32	Lambda	parameter: 57 body: 36
33	Operation	operator: 38 operand: 41
34	Operation	operator: 38 operand: 42
35	Literal
36	Conditional	value: 48 condition: 40
37	ExprTuple	41
38	Literal
39	ExprTuple	42
40	Operation	operator: 54 operands: 43
41	Lambda	parameter: 57 body: 44
42	Lambda	parameter: 57 body: 45
43	ExprTuple	57, 46
44	Conditional	value: 48 condition: 47
45	Conditional	value: 48 condition: 49
46	Operation	operator: 60 operands: 50
47	Operation	operator: 54 operands: 51
48	Operation	operator: 52 operand: 57
49	Operation	operator: 54 operands: 55
50	ExprTuple	62, 64
51	ExprTuple	57, 56
52	Variable
53	ExprTuple	57
54	Literal
55	ExprTuple	57, 58
56	Operation	operator: 60 operands: 59
57	Variable
58	Operation	operator: 60 operands: 61
59	ExprTuple	62, 67
60	Literal
61	ExprTuple	63, 64
62	Variable
63	Operation	operator: 65 operands: 66
64	Variable
65	Literal
66	ExprTuple	67, 68
67	Variable
68	Literal