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, i, j, k
from proveit.core_expr_types import Q__b_1_to_j, a_1_to_i, b_1_to_j, c_1_to_k, f__b_1_to_j
from proveit.linear_algebra import TensorProd, VecSpaces, VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import Natural, NaturalPos

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = TensorProd(a_1_to_i, f__b_1_to_j, c_1_to_k)
expr = Conditional(Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr2, V), condition = Q__b_1_to_j), Equals(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, condition = Q__b_1_to_j), TensorProd(a_1_to_i, vec_summation_b1toj_fQ, c_1_to_k)).with_wrapping_at(1)).with_wrapping_at(2)), domain = VecSpaces(K)), And(InSet(i, Natural), InSet(j, NaturalPos), InSet(k, Natural)))

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_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k}}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right] \\  = \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\right] \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N}^+ ,  k \in \mathbb{N}\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: 33 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	7, 8, 9
6	Lambda	parameter: 51 body: 11
7	Operation	operator: 46 operands: 12
8	Operation	operator: 46 operands: 13
9	Operation	operator: 46 operands: 14
10	ExprTuple	51
11	Conditional	value: 15 condition: 16
12	ExprTuple	62, 18
13	ExprTuple	75, 17
14	ExprTuple	64, 18
15	Operation	operator: 33 operand: 22
16	Operation	operator: 20 operands: 21
17	Literal
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	51, 23
22	Lambda	parameters: 24 body: 25
23	Operation	operator: 26 operand: 30
24	ExprTuple	57, 58
25	Operation	operator: 28 operands: 29
26	Literal
27	ExprTuple	30
28	Literal
29	ExprTuple	31, 32
30	Variable
31	Operation	operator: 33 operand: 37
32	Operation	operator: 35 operands: 36
33	Literal
34	ExprTuple	37
35	Literal
36	ExprTuple	38, 39
37	Lambda	parameters: 67 body: 40
38	Operation	operator: 49 operand: 44
39	Operation	operator: 54 operands: 42
40	Conditional	value: 43 condition: 60
41	ExprTuple	44
42	ExprTuple	57, 45, 58
43	Operation	operator: 46 operands: 47
44	Lambda	parameters: 67 body: 48
45	Operation	operator: 49 operand: 53
46	Literal
47	ExprTuple	52, 51
48	Conditional	value: 52 condition: 60
49	Literal
50	ExprTuple	53
51	Variable
52	Operation	operator: 54 operands: 55
53	Lambda	parameters: 67 body: 56
54	Literal
55	ExprTuple	57, 59, 58
56	Conditional	value: 59 condition: 60
57	ExprRange	lambda_map: 61 start_index: 74 end_index: 62
58	ExprRange	lambda_map: 63 start_index: 74 end_index: 64
59	Operation	operator: 65 operands: 67
60	Operation	operator: 66 operands: 67
61	Lambda	parameter: 79 body: 68
62	Variable
63	Lambda	parameter: 79 body: 69
64	Variable
65	Variable
66	Variable
67	ExprTuple	70
68	IndexedVar	variable: 71 index: 79
69	IndexedVar	variable: 72 index: 79
70	ExprRange	lambda_map: 73 start_index: 74 end_index: 75
71	Variable
72	Variable
73	Lambda	parameter: 79 body: 76
74	Literal
75	Variable
76	IndexedVar	variable: 77 index: 79
77	Variable
78	ExprTuple	79
79	Variable