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, ExprRange, IndexedVar, K, V, Variable, b, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.linear_algebra import TensorProd, VecAdd, VecSpaces
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import Natural, one

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