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, V, alpha, b, i, k
from proveit.core_expr_types import U_1_to_i, W_1_to_k, a_1_to_i, c_1_to_k
from proveit.linear_algebra import ScalarMult, 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, W_1_to_k]
expr = Lambda(K, Forall(instance_param_or_params = [alpha], instance_expr = Forall(instance_param_or_params = [i, k], instance_expr = Forall(instance_param_or_params = sub_expr1, instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_k], instance_expr = Equals(TensorProd(a_1_to_i, ScalarMult(alpha, b), c_1_to_k), ScalarMult(alpha, TensorProd(a_1_to_i, b, c_1_to_k))).with_wrapping_at(1), domains = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), domain = Natural), domain = K))

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())

K \mapsto \left[\forall_{\alpha \in K}~\left[\forall_{i, k \in \mathbb{N}}~\left[\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), b \in V,\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(\alpha \cdot b\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \\  = \left(\alpha \cdot \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)\right) \end{array} \end{array}\right)\end{array}\right]\end{array}\right]\right]\right]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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