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 = [K], 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_1_to_k], instance_expr = Equals(TensorProd(A_1_to_i, B_1_to_j, C_1_to_k), TensorProd(A_1_to_i, TensorProd(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_{K}~\left[\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} B_{1} {\otimes}  B_{2} {\otimes}  \ldots {\otimes}  B_{j}{\otimes} C_{1} {\otimes}  C_{2} {\otimes}  \ldots {\otimes}  C_{k}\right) \\  = \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) \end{array} \end{array}\right)\end{array}\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: 10 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 67 body: 3
3	Operation	operator: 10 operand: 5
4	ExprTuple	5
5	Lambda	parameters: 6 body: 7
6	ExprTuple	46, 61, 50
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operand: 13
9	Operation	operator: 27 operands: 12
10	Literal
11	ExprTuple	13
12	ExprTuple	14, 15, 16
13	Lambda	parameters: 34 body: 17
14	Operation	operator: 20 operands: 18
15	Operation	operator: 20 operands: 19
16	Operation	operator: 20 operands: 21
17	Conditional	value: 22 condition: 23
18	ExprTuple	46, 24
19	ExprTuple	61, 24
20	Literal
21	ExprTuple	50, 24
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operands: 28
24	Literal
25	Literal
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	31, 32, 33
29	Operation	operator: 47 operands: 34
30	Operation	operator: 47 operands: 35
31	ExprRange	lambda_map: 36 start_index: 60 end_index: 46
32	ExprRange	lambda_map: 37 start_index: 60 end_index: 61
33	ExprRange	lambda_map: 38 start_index: 60 end_index: 50
34	ExprTuple	39, 55, 41
35	ExprTuple	39, 40, 41
36	Lambda	parameter: 70 body: 42
37	Lambda	parameter: 70 body: 43
38	Lambda	parameter: 70 body: 44
39	ExprRange	lambda_map: 45 start_index: 60 end_index: 46
40	Operation	operator: 47 operands: 48
41	ExprRange	lambda_map: 49 start_index: 60 end_index: 50
42	Operation	operator: 53 operands: 51
43	Operation	operator: 53 operands: 52
44	Operation	operator: 53 operands: 54
45	Lambda	parameter: 70 body: 56
46	Variable
47	Literal
48	ExprTuple	55
49	Lambda	parameter: 70 body: 57
50	Variable
51	ExprTuple	56, 58
52	ExprTuple	66, 58
53	Literal
54	ExprTuple	57, 58
55	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
56	IndexedVar	variable: 62 index: 70
57	IndexedVar	variable: 63 index: 70
58	Operation	operator: 64 operand: 67
59	Lambda	parameter: 70 body: 66
60	Literal
61	Variable
62	Variable
63	Variable
64	Literal
65	ExprTuple	67
66	IndexedVar	variable: 68 index: 70
67	Variable
68	Variable
69	ExprTuple	70
70	Variable