Expression of type Conditional

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 Conditional, K, V, c, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, d_1_to_k
from proveit.linear_algebra import VecAdd, VecSpaces
from proveit.logic import Equals, Forall, InClass
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [i, j, k], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c, d_1_to_k], instance_expr = Equals(VecAdd(a_1_to_i, b_1_to_j, c, d_1_to_k), VecAdd(a_1_to_i, c, b_1_to_j, d_1_to_k)).with_wrapping_at(2), domain = V), domain = Natural), InClass(V, VecSpaces(K)))

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())

\left\{\forall_{i, j, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, c, d_{1}, d_{2}, \ldots, d_{k} \in V}~\left(\begin{array}{c} \begin{array}{l} \left(a_{1} +  a_{2} +  \ldots +  a_{i}+ b_{1} +  b_{2} +  \ldots +  b_{j} + c+ d_{1} +  d_{2} +  \ldots +  d_{k}\right) =  \\ \left(a_{1} +  a_{2} +  \ldots +  a_{i} + c+ b_{1} +  b_{2} +  \ldots +  b_{j}+ d_{1} +  d_{2} +  \ldots +  d_{k}\right) \end{array} \end{array}\right)\right] \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 15 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	67, 7
6	Lambda	parameters: 8 body: 9
7	Operation	operator: 10 operand: 14
8	ExprTuple	54, 56, 59
9	Conditional	value: 12 condition: 13
10	Literal
11	ExprTuple	14
12	Operation	operator: 15 operand: 18
13	Operation	operator: 31 operands: 17
14	Variable
15	Literal
16	ExprTuple	18
17	ExprTuple	19, 20, 21
18	Lambda	parameters: 39 body: 22
19	Operation	operator: 62 operands: 23
20	Operation	operator: 62 operands: 24
21	Operation	operator: 62 operands: 25
22	Conditional	value: 26 condition: 27
23	ExprTuple	54, 28
24	ExprTuple	56, 28
25	ExprTuple	59, 28
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 32
28	Literal
29	Literal
30	ExprTuple	33, 34
31	Literal
32	ExprTuple	35, 36, 37, 38
33	Operation	operator: 40 operands: 39
34	Operation	operator: 40 operands: 41
35	ExprRange	lambda_map: 42 start_index: 58 end_index: 54
36	ExprRange	lambda_map: 43 start_index: 58 end_index: 56
37	Operation	operator: 62 operands: 44
38	ExprRange	lambda_map: 45 start_index: 58 end_index: 59
39	ExprTuple	46, 47, 51, 48
40	Literal
41	ExprTuple	46, 51, 47, 48
42	Lambda	parameter: 72 body: 49
43	Lambda	parameter: 72 body: 50
44	ExprTuple	51, 67
45	Lambda	parameter: 72 body: 52
46	ExprRange	lambda_map: 53 start_index: 58 end_index: 54
47	ExprRange	lambda_map: 55 start_index: 58 end_index: 56
48	ExprRange	lambda_map: 57 start_index: 58 end_index: 59
49	Operation	operator: 62 operands: 60
50	Operation	operator: 62 operands: 61
51	Variable
52	Operation	operator: 62 operands: 63
53	Lambda	parameter: 72 body: 64
54	Variable
55	Lambda	parameter: 72 body: 65
56	Variable
57	Lambda	parameter: 72 body: 66
58	Literal
59	Variable
60	ExprTuple	64, 67
61	ExprTuple	65, 67
62	Literal
63	ExprTuple	66, 67
64	IndexedVar	variable: 68 index: 72
65	IndexedVar	variable: 69 index: 72
66	IndexedVar	variable: 70 index: 72
67	Variable
68	Variable
69	Variable
70	Variable
71	ExprTuple	72
72	Variable