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, 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, VecSpaces
from proveit.logic import Equals, Forall, Implies

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [U_1_to_i, V, W_1_to_k], instance_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(), domain = VecSpaces(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_{U_{1}, U_{2}, \ldots, U_{i}, V, W_{1}, W_{2}, \ldots, W_{k} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\left[\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}\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: 26 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5, 64, 6
4	Conditional	value: 7 condition: 8
5	ExprRange	lambda_map: 9 start_index: 91 end_index: 89
6	ExprRange	lambda_map: 10 start_index: 91 end_index: 92
7	Operation	operator: 26 operand: 13
8	Operation	operator: 28 operands: 12
9	Lambda	parameter: 102 body: 65
10	Lambda	parameter: 102 body: 66
11	ExprTuple	13
12	ExprTuple	14, 15, 16
13	Lambda	parameters: 17 body: 18
14	ExprRange	lambda_map: 19 start_index: 91 end_index: 89
15	Operation	operator: 31 operands: 20
16	ExprRange	lambda_map: 21 start_index: 91 end_index: 92
17	ExprTuple	81, 82, 83, 85
18	Conditional	value: 22 condition: 23
19	Lambda	parameter: 102 body: 24
20	ExprTuple	64, 38
21	Lambda	parameter: 102 body: 25
22	Operation	operator: 26 operand: 33
23	Operation	operator: 28 operands: 29
24	Operation	operator: 31 operands: 30
25	Operation	operator: 31 operands: 32
26	Literal
27	ExprTuple	33
28	Literal
29	ExprTuple	34, 35, 36, 37
30	ExprTuple	65, 38
31	Literal
32	ExprTuple	66, 38
33	Lambda	parameter: 84 body: 40
34	ExprRange	lambda_map: 41 start_index: 91 end_index: 89
35	ExprRange	lambda_map: 42 start_index: 91 end_index: 92
36	ExprRange	lambda_map: 43 start_index: 91 end_index: 89
37	ExprRange	lambda_map: 44 start_index: 91 end_index: 92
38	Operation	operator: 45 operand: 53
39	ExprTuple	84
40	Conditional	value: 47 condition: 48
41	Lambda	parameter: 102 body: 49
42	Lambda	parameter: 102 body: 50
43	Lambda	parameter: 102 body: 51
44	Lambda	parameter: 102 body: 52
45	Literal
46	ExprTuple	53
47	Operation	operator: 54 operands: 55
48	Operation	operator: 60 operands: 56
49	Operation	operator: 60 operands: 57
50	Operation	operator: 60 operands: 58
51	Operation	operator: 60 operands: 59
52	Operation	operator: 60 operands: 61
53	Variable
54	Literal
55	ExprTuple	62, 63
56	ExprTuple	84, 64
57	ExprTuple	93, 65
58	ExprTuple	94, 66
59	ExprTuple	95, 65
60	Literal
61	ExprTuple	96, 66
62	Operation	operator: 68 operands: 67
63	Operation	operator: 68 operands: 69
64	Variable
65	IndexedVar	variable: 70 index: 102
66	IndexedVar	variable: 71 index: 102
67	ExprTuple	72, 73
68	Literal
69	ExprTuple	74, 75
70	Variable
71	Variable
72	Operation	operator: 79 operands: 76
73	Operation	operator: 79 operands: 77
74	Operation	operator: 79 operands: 78
75	Operation	operator: 79 operands: 80
76	ExprTuple	81, 82
77	ExprTuple	83, 85
78	ExprTuple	81, 84, 82
79	Literal
80	ExprTuple	83, 84, 85
81	ExprRange	lambda_map: 86 start_index: 91 end_index: 89
82	ExprRange	lambda_map: 87 start_index: 91 end_index: 92
83	ExprRange	lambda_map: 88 start_index: 91 end_index: 89
84	Variable
85	ExprRange	lambda_map: 90 start_index: 91 end_index: 92
86	Lambda	parameter: 102 body: 93
87	Lambda	parameter: 102 body: 94
88	Lambda	parameter: 102 body: 95
89	Variable
90	Lambda	parameter: 102 body: 96
91	Literal
92	Variable
93	IndexedVar	variable: 97 index: 102
94	IndexedVar	variable: 98 index: 102
95	IndexedVar	variable: 99 index: 102
96	IndexedVar	variable: 100 index: 102
97	Variable
98	Variable
99	Variable
100	Variable
101	ExprTuple	102
102	Variable