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, e
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, f_1_to_k
from proveit.linear_algebra import TensorProd, VecZero
from proveit.logic import Equals, Forall, Implies, NotEquals

# 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, f_1_to_k], instance_expr = Forall(instance_param_or_params = [b, e], instance_expr = Implies(Equals(TensorProd(a_1_to_i, b, c_1_to_k), TensorProd(d_1_to_i, e, f_1_to_k)).with_wrapping_at(2), Equals(TensorProd(a_1_to_i, c_1_to_k), TensorProd(d_1_to_i, f_1_to_k)).with_wrapping_at(2)).with_wrapping_at(2), domain = V, conditions = [Equals(b, e), NotEquals(b, VecZero(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(f_{1} \in W_{1}\right), \left(f_{2} \in W_{2}\right), \ldots, \left(f_{k} \in W_{k}\right)}~\\
\left[\forall_{b, e \in V~|~b = e, b \neq \vec{0}\left(V\right)}~\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} 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} e{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(d_{1} {\otimes}  d_{2} {\otimes}  \ldots {\otimes}  d_{i}{\otimes} f_{1} {\otimes}  f_{2} {\otimes}  \ldots {\otimes}  f_{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	67, 68, 69, 70
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 28 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12, 13, 14
10	Lambda	parameters: 48 body: 15
11	ExprRange	lambda_map: 16 start_index: 77 end_index: 75
12	ExprRange	lambda_map: 17 start_index: 77 end_index: 78
13	ExprRange	lambda_map: 18 start_index: 77 end_index: 75
14	ExprRange	lambda_map: 19 start_index: 77 end_index: 78
15	Conditional	value: 20 condition: 21
16	Lambda	parameter: 88 body: 22
17	Lambda	parameter: 88 body: 23
18	Lambda	parameter: 88 body: 24
19	Lambda	parameter: 88 body: 25
20	Operation	operator: 26 operands: 27
21	Operation	operator: 28 operands: 29
22	Operation	operator: 45 operands: 30
23	Operation	operator: 45 operands: 31
24	Operation	operator: 45 operands: 32
25	Operation	operator: 45 operands: 33
26	Literal
27	ExprTuple	34, 35
28	Literal
29	ExprTuple	36, 37, 38, 39
30	ExprTuple	79, 40
31	ExprTuple	80, 41
32	ExprTuple	81, 40
33	ExprTuple	82, 41
34	Operation	operator: 47 operands: 42
35	Operation	operator: 47 operands: 43
36	Operation	operator: 45 operands: 44
37	Operation	operator: 45 operands: 46
38	Operation	operator: 47 operands: 48
39	Operation	operator: 49 operands: 50
40	IndexedVar	variable: 51 index: 88
41	IndexedVar	variable: 52 index: 88
42	ExprTuple	53, 54
43	ExprTuple	55, 56
44	ExprTuple	65, 71
45	Literal
46	ExprTuple	66, 71
47	Literal
48	ExprTuple	65, 66
49	Literal
50	ExprTuple	65, 57
51	Variable
52	Variable
53	Operation	operator: 61 operands: 58
54	Operation	operator: 61 operands: 59
55	Operation	operator: 61 operands: 60
56	Operation	operator: 61 operands: 62
57	Operation	operator: 63 operand: 71
58	ExprTuple	67, 65, 68
59	ExprTuple	69, 66, 70
60	ExprTuple	67, 68
61	Literal
62	ExprTuple	69, 70
63	Literal
64	ExprTuple	71
65	Variable
66	Variable
67	ExprRange	lambda_map: 72 start_index: 77 end_index: 75
68	ExprRange	lambda_map: 73 start_index: 77 end_index: 78
69	ExprRange	lambda_map: 74 start_index: 77 end_index: 75
70	ExprRange	lambda_map: 76 start_index: 77 end_index: 78
71	Variable
72	Lambda	parameter: 88 body: 79
73	Lambda	parameter: 88 body: 80
74	Lambda	parameter: 88 body: 81
75	Variable
76	Lambda	parameter: 88 body: 82
77	Literal
78	Variable
79	IndexedVar	variable: 83 index: 88
80	IndexedVar	variable: 84 index: 88
81	IndexedVar	variable: 85 index: 88
82	IndexedVar	variable: 86 index: 88
83	Variable
84	Variable
85	Variable
86	Variable
87	ExprTuple	88
88	Variable