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 V, b
from proveit.core_expr_types import U_1_to_i, W_1_to_k, a_1_to_i, c_1_to_k, d_1_to_i, e_1_to_k
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals, Forall, Implies

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a_1_to_i, c_1_to_k, d_1_to_i, e_1_to_k], instance_expr = Forall(instance_param_or_params = [b], instance_expr = Implies(Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, e_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, b, e_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V), domains = [U_1_to_i, W_1_to_k, U_1_to_i, W_1_to_k]).with_wrapping()

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())

\begin{array}{l}\forall_{\left(a_{1} \in U_{1}\right), \left(a_{2} \in U_{2}\right), \ldots, \left(a_{i} \in U_{i}\right),\left(c_{1} \in W_{1}\right), \left(c_{2} \in W_{2}\right), \ldots, \left(c_{k} \in W_{k}\right),\left(d_{1} \in U_{1}\right), \left(d_{2} \in U_{2}\right), \ldots, \left(d_{i} \in U_{i}\right),\left(e_{1} \in W_{1}\right), \left(e_{2} \in W_{2}\right), \ldots, \left(e_{k} \in W_{k}\right)}~\\
\left[\forall_{b \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i} {\otimes} b{\otimes} e_{1} {\otimes}  e_{2} {\otimes}  \ldots {\otimes}  e_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\right]\end{array}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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	55, 56, 57, 59
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12, 13, 14, 15
11	Lambda	parameter: 58 body: 17
12	ExprRange	lambda_map: 18 start_index: 65 end_index: 63
13	ExprRange	lambda_map: 19 start_index: 65 end_index: 66
14	ExprRange	lambda_map: 20 start_index: 65 end_index: 63
15	ExprRange	lambda_map: 21 start_index: 65 end_index: 66
16	ExprTuple	58
17	Conditional	value: 22 condition: 23
18	Lambda	parameter: 76 body: 24
19	Lambda	parameter: 76 body: 25
20	Lambda	parameter: 76 body: 26
21	Lambda	parameter: 76 body: 27
22	Operation	operator: 28 operands: 29
23	Operation	operator: 34 operands: 30
24	Operation	operator: 34 operands: 31
25	Operation	operator: 34 operands: 32
26	Operation	operator: 34 operands: 33
27	Operation	operator: 34 operands: 35
28	Literal
29	ExprTuple	36, 37
30	ExprTuple	58, 38
31	ExprTuple	67, 39
32	ExprTuple	68, 40
33	ExprTuple	69, 39
34	Literal
35	ExprTuple	70, 40
36	Operation	operator: 42 operands: 41
37	Operation	operator: 42 operands: 43
38	Variable
39	IndexedVar	variable: 44 index: 76
40	IndexedVar	variable: 45 index: 76
41	ExprTuple	46, 47
42	Literal
43	ExprTuple	48, 49
44	Variable
45	Variable
46	Operation	operator: 53 operands: 50
47	Operation	operator: 53 operands: 51
48	Operation	operator: 53 operands: 52
49	Operation	operator: 53 operands: 54
50	ExprTuple	55, 56
51	ExprTuple	57, 59
52	ExprTuple	55, 58, 56
53	Literal
54	ExprTuple	57, 58, 59
55	ExprRange	lambda_map: 60 start_index: 65 end_index: 63
56	ExprRange	lambda_map: 61 start_index: 65 end_index: 66
57	ExprRange	lambda_map: 62 start_index: 65 end_index: 63
58	Variable
59	ExprRange	lambda_map: 64 start_index: 65 end_index: 66
60	Lambda	parameter: 76 body: 67
61	Lambda	parameter: 76 body: 68
62	Lambda	parameter: 76 body: 69
63	Variable
64	Lambda	parameter: 76 body: 70
65	Literal
66	Variable
67	IndexedVar	variable: 71 index: 76
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