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, 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
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [K], instance_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_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), 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_{K}~\left[\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_{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]\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: 22 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 21 body: 3
3	Operation	operator: 22 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 74 body: 7
6	ExprTuple	74
7	Conditional	value: 8 condition: 9
8	Operation	operator: 22 operand: 13
9	Operation	operator: 11 operands: 12
10	ExprTuple	13
11	Literal
12	ExprTuple	74, 14
13	Lambda	parameters: 15 body: 16
14	Operation	operator: 17 operand: 21
15	ExprTuple	61, 63, 66
16	Conditional	value: 19 condition: 20
17	Literal
18	ExprTuple	21
19	Operation	operator: 22 operand: 25
20	Operation	operator: 38 operands: 24
21	Variable
22	Literal
23	ExprTuple	25
24	ExprTuple	26, 27, 28
25	Lambda	parameters: 46 body: 29
26	Operation	operator: 69 operands: 30
27	Operation	operator: 69 operands: 31
28	Operation	operator: 69 operands: 32
29	Conditional	value: 33 condition: 34
30	ExprTuple	61, 35
31	ExprTuple	63, 35
32	ExprTuple	66, 35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 39
35	Literal
36	Literal
37	ExprTuple	40, 41
38	Literal
39	ExprTuple	42, 43, 44, 45
40	Operation	operator: 47 operands: 46
41	Operation	operator: 47 operands: 48
42	ExprRange	lambda_map: 49 start_index: 65 end_index: 61
43	ExprRange	lambda_map: 50 start_index: 65 end_index: 63
44	Operation	operator: 69 operands: 51
45	ExprRange	lambda_map: 52 start_index: 65 end_index: 66
46	ExprTuple	53, 54, 58, 55
47	Literal
48	ExprTuple	53, 58, 54, 55
49	Lambda	parameter: 79 body: 56
50	Lambda	parameter: 79 body: 57
51	ExprTuple	58, 74
52	Lambda	parameter: 79 body: 59
53	ExprRange	lambda_map: 60 start_index: 65 end_index: 61
54	ExprRange	lambda_map: 62 start_index: 65 end_index: 63
55	ExprRange	lambda_map: 64 start_index: 65 end_index: 66
56	Operation	operator: 69 operands: 67
57	Operation	operator: 69 operands: 68
58	Variable
59	Operation	operator: 69 operands: 70
60	Lambda	parameter: 79 body: 71
61	Variable
62	Lambda	parameter: 79 body: 72
63	Variable
64	Lambda	parameter: 79 body: 73
65	Literal
66	Variable
67	ExprTuple	71, 74
68	ExprTuple	72, 74
69	Literal
70	ExprTuple	73, 74
71	IndexedVar	variable: 75 index: 79
72	IndexedVar	variable: 76 index: 79
73	IndexedVar	variable: 77 index: 79
74	Variable
75	Variable
76	Variable
77	Variable
78	ExprTuple	79
79	Variable