Expression of type Lambda ¶

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, Lambda, i, j, k
from proveit.core_expr_types import U_1_to_i, V_1_to_j, W_1_to_k, 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
sub_expr1 = [U_1_to_i, V_1_to_j, W_1_to_k]
expr = Lambda(K, Forall(instance_param_or_params = [i, j, k], instance_expr = Forall(instance_param_or_params = sub_expr1, 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), domains = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), domain = Natural))

# 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())

K \mapsto \left[\forall_{i, j, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V_{1}, V_{2}, \ldots, V_{j}, 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(b_{1} \in V_{1}\right), \left(b_{2} \in V_{2}\right), \ldots, \left(b_{j} \in V_{j}\right),\left(c_{1} \in W_{1}\right), \left(c_{2} \in W_{2}\right), \ldots, \left(c_{k} \in W_{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]\end{array}\right]\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Lambda	parameter: 71 body: 1
1	Operation	operator: 28 operand: 3
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	75, 90, 77
5	Conditional	value: 6 condition: 7
6	Operation	operator: 28 operand: 10
7	Operation	operator: 46 operands: 9
8	ExprTuple	10
9	ExprTuple	11, 12, 13
10	Lambda	parameters: 14 body: 15
11	Operation	operator: 80 operands: 16
12	Operation	operator: 80 operands: 17
13	Operation	operator: 80 operands: 18
14	ExprTuple	19, 20, 21
15	Conditional	value: 22 condition: 23
16	ExprTuple	75, 24
17	ExprTuple	90, 24
18	ExprTuple	77, 24
19	ExprRange	lambda_map: 25 start_index: 89 end_index: 75
20	ExprRange	lambda_map: 26 start_index: 89 end_index: 90
21	ExprRange	lambda_map: 27 start_index: 89 end_index: 77
22	Operation	operator: 28 operand: 31
23	Operation	operator: 46 operands: 30
24	Literal
25	Lambda	parameter: 99 body: 84
26	Lambda	parameter: 99 body: 85
27	Lambda	parameter: 99 body: 87
28	Literal
29	ExprTuple	31
30	ExprTuple	32, 33, 34
31	Lambda	parameters: 59 body: 35
32	ExprRange	lambda_map: 36 start_index: 89 end_index: 75
33	ExprRange	lambda_map: 37 start_index: 89 end_index: 90
34	ExprRange	lambda_map: 38 start_index: 89 end_index: 77
35	Conditional	value: 39 condition: 40
36	Lambda	parameter: 99 body: 41
37	Lambda	parameter: 99 body: 42
38	Lambda	parameter: 99 body: 43
39	Operation	operator: 44 operands: 45
40	Operation	operator: 46 operands: 47
41	Operation	operator: 50 operands: 48
42	Operation	operator: 50 operands: 49
43	Operation	operator: 50 operands: 51
44	Literal
45	ExprTuple	52, 53
46	Literal
47	ExprTuple	54, 55, 56
48	ExprTuple	84, 57
49	ExprTuple	85, 57
50	Literal
51	ExprTuple	87, 57
52	Operation	operator: 72 operands: 58
53	Operation	operator: 72 operands: 59
54	ExprRange	lambda_map: 60 start_index: 89 end_index: 75
55	ExprRange	lambda_map: 61 start_index: 89 end_index: 90
56	ExprRange	lambda_map: 62 start_index: 89 end_index: 77
57	Operation	operator: 63 operand: 71
58	ExprTuple	66, 65, 67
59	ExprTuple	66, 82, 67
60	Lambda	parameter: 99 body: 68
61	Lambda	parameter: 99 body: 69
62	Lambda	parameter: 99 body: 70
63	Literal
64	ExprTuple	71
65	Operation	operator: 72 operands: 73
66	ExprRange	lambda_map: 74 start_index: 89 end_index: 75
67	ExprRange	lambda_map: 76 start_index: 89 end_index: 77
68	Operation	operator: 80 operands: 78
69	Operation	operator: 80 operands: 79
70	Operation	operator: 80 operands: 81
71	Variable
72	Literal
73	ExprTuple	82
74	Lambda	parameter: 99 body: 83
75	Variable
76	Lambda	parameter: 99 body: 86
77	Variable
78	ExprTuple	83, 84
79	ExprTuple	96, 85
80	Literal
81	ExprTuple	86, 87
82	ExprRange	lambda_map: 88 start_index: 89 end_index: 90
83	IndexedVar	variable: 91 index: 99
84	IndexedVar	variable: 92 index: 99
85	IndexedVar	variable: 93 index: 99
86	IndexedVar	variable: 94 index: 99
87	IndexedVar	variable: 95 index: 99
88	Lambda	parameter: 99 body: 96
89	Literal
90	Variable
91	Variable
92	Variable
93	Variable
94	Variable
95	Variable
96	IndexedVar	variable: 97 index: 99
97	Variable
98	ExprTuple	99
99	Variable

Expression of type Lambda¶

from the theory of proveit.linear_algebra.tensors¶

Expression of type Lambda ¶

from the theory of proveit.linear_algebra.tensors ¶