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 Conditional, 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 And, Equals, Forall, InSet
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([i, j, k], Conditional(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(), And(InSet(i, Natural), InSet(j, Natural), InSet(k, 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())

\left(i, j, k\right) \mapsto \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} \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N} ,  k \in \mathbb{N}\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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