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 K, V, b, i, j, k
from proveit.core_expr_types import a_1_to_i, c_1_to_j, d_1_to_k
from proveit.linear_algebra import VecAdd, VecSpaces
from proveit.logic import Equals, Forall
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [i, j, k], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_j, d_1_to_k], instance_expr = Equals(VecAdd(a_1_to_i, b, c_1_to_j, d_1_to_k), VecAdd(a_1_to_i, c_1_to_j, b, d_1_to_k)).with_wrapping_at(2), domain = V), domain = Natural), domain = 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())

\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{i, j, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b, c_{1}, c_{2}, \ldots, c_{j}, 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+ c_{1} +  c_{2} +  \ldots +  c_{j}+ d_{1} +  d_{2} +  \ldots +  d_{k}\right) =  \\ \left(a_{1} +  a_{2} +  \ldots +  a_{i}+ c_{1} +  c_{2} +  \ldots +  c_{j} + b+ d_{1} +  d_{2} +  \ldots +  d_{k}\right) \end{array} \end{array}\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: 19 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 71 body: 4
3	ExprTuple	71
4	Conditional	value: 5 condition: 6
5	Operation	operator: 19 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	71, 11
10	Lambda	parameters: 12 body: 13
11	Operation	operator: 14 operand: 18
12	ExprTuple	58, 60, 63
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	18
16	Operation	operator: 19 operand: 22
17	Operation	operator: 35 operands: 21
18	Variable
19	Literal
20	ExprTuple	22
21	ExprTuple	23, 24, 25
22	Lambda	parameters: 43 body: 26
23	Operation	operator: 66 operands: 27
24	Operation	operator: 66 operands: 28
25	Operation	operator: 66 operands: 29
26	Conditional	value: 30 condition: 31
27	ExprTuple	58, 32
28	ExprTuple	60, 32
29	ExprTuple	63, 32
30	Operation	operator: 33 operands: 34
31	Operation	operator: 35 operands: 36
32	Literal
33	Literal
34	ExprTuple	37, 38
35	Literal
36	ExprTuple	39, 40, 41, 42
37	Operation	operator: 44 operands: 43
38	Operation	operator: 44 operands: 45
39	ExprRange	lambda_map: 46 start_index: 62 end_index: 58
40	Operation	operator: 66 operands: 47
41	ExprRange	lambda_map: 48 start_index: 62 end_index: 60
42	ExprRange	lambda_map: 49 start_index: 62 end_index: 63
43	ExprTuple	50, 54, 51, 52
44	Literal
45	ExprTuple	50, 51, 54, 52
46	Lambda	parameter: 76 body: 53
47	ExprTuple	54, 71
48	Lambda	parameter: 76 body: 55
49	Lambda	parameter: 76 body: 56
50	ExprRange	lambda_map: 57 start_index: 62 end_index: 58
51	ExprRange	lambda_map: 59 start_index: 62 end_index: 60
52	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
53	Operation	operator: 66 operands: 64
54	Variable
55	Operation	operator: 66 operands: 65
56	Operation	operator: 66 operands: 67
57	Lambda	parameter: 76 body: 68
58	Variable
59	Lambda	parameter: 76 body: 69
60	Variable
61	Lambda	parameter: 76 body: 70
62	Literal
63	Variable
64	ExprTuple	68, 71
65	ExprTuple	69, 71
66	Literal
67	ExprTuple	70, 71
68	IndexedVar	variable: 72 index: 76
69	IndexedVar	variable: 73 index: 76
70	IndexedVar	variable: 74 index: 76
71	Variable
72	Variable
73	Variable
74	Variable
75	ExprTuple	76
76	Variable