Expression of type Forall

from the theory of proveit.linear_algebra.tensors

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, i, j, k
from proveit.core_expr_types import A_1_to_i, B_1_to_j, C_1_to_k
from proveit.linear_algebra import TensorProd, 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 = [i, j, k], instance_expr = Forall(instance_param_or_params = [A_1_to_i, B_1_to_j, C_1_to_k], instance_expr = Equals(TensorProd(A_1_to_i, TensorProd(B_1_to_j), C_1_to_k), TensorProd(A_1_to_i, B_1_to_j, C_1_to_k)).with_wrapping_at(1), domain = VecSpaces(K)).with_wrapping(), domain = Natural)

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