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, Function, K, V, s
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 ScalarMult, TensorProd, VecSpaces, VecSum
from proveit.logic import Equals, Forall, Implies, InClass, InSet

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(s, sub_expr1)
sub_expr3 = ScalarMult(sub_expr2, f__b_1_to_j)
expr = Conditional(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(TensorProd(a_1_to_i, sub_expr3, c_1_to_k), V), condition = Q__b_1_to_j), Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, TensorProd(a_1_to_i, f__b_1_to_j, c_1_to_k)), condition = Q__b_1_to_j), TensorProd(a_1_to_i, VecSum(index_or_indices = sub_expr1, summand = sub_expr3, condition = Q__b_1_to_j), c_1_to_k)).with_wrapping_at(1)).with_wrapping_at(2)).with_wrapping(), InClass(V, VecSpaces(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\{\begin{array}{l}\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} \left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\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(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot \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)\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)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array} \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\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: 17 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	36, 7
6	Lambda	parameters: 8 body: 9
7	Operation	operator: 10 operand: 14
8	ExprTuple	50, 51
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	14
12	Literal
13	ExprTuple	15, 16
14	Variable
15	Operation	operator: 17 operand: 21
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21
19	Literal
20	ExprTuple	22, 23
21	Lambda	parameters: 60 body: 24
22	Operation	operator: 33 operand: 28
23	Operation	operator: 45 operands: 26
24	Conditional	value: 27 condition: 44
25	ExprTuple	28
26	ExprTuple	50, 29, 51
27	Operation	operator: 30 operands: 31
28	Lambda	parameters: 60 body: 32
29	Operation	operator: 33 operand: 38
30	Literal
31	ExprTuple	35, 36
32	Conditional	value: 37 condition: 44
33	Literal
34	ExprTuple	38
35	Operation	operator: 45 operands: 39
36	Variable
37	Operation	operator: 47 operands: 40
38	Lambda	parameters: 60 body: 41
39	ExprTuple	50, 43, 51
40	ExprTuple	52, 42
41	Conditional	value: 43 condition: 44
42	Operation	operator: 45 operands: 46
43	Operation	operator: 47 operands: 48
44	Operation	operator: 49 operands: 60
45	Literal
46	ExprTuple	50, 53, 51
47	Literal
48	ExprTuple	52, 53
49	Variable
50	ExprRange	lambda_map: 54 start_index: 67 end_index: 55
51	ExprRange	lambda_map: 56 start_index: 67 end_index: 57
52	Operation	operator: 58 operands: 60
53	Operation	operator: 59 operands: 60
54	Lambda	parameter: 72 body: 61
55	Variable
56	Lambda	parameter: 72 body: 62
57	Variable
58	Variable
59	Variable
60	ExprTuple	63
61	IndexedVar	variable: 64 index: 72
62	IndexedVar	variable: 65 index: 72
63	ExprRange	lambda_map: 66 start_index: 67 end_index: 68
64	Variable
65	Variable
66	Lambda	parameter: 72 body: 69
67	Literal
68	Variable
69	IndexedVar	variable: 70 index: 72
70	Variable
71	ExprTuple	72
72	Variable