Expression of type Conditional

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, 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, InSet
from proveit.numbers import Natural

# build up the expression from sub-expressions
sub_expr1 = [U_1_to_i, V, W_1_to_k]
expr = Conditional(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), InSet(alpha, 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())

\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] \textrm{ if } \alpha \in K\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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