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 ExprRange, IndexedVar, K, Lambda, 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 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 = Lambda(K, Forall(instance_param_or_params = [i, j, k], instance_expr = 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(sub_expr2, VecAdd(ExprRange(sub_expr1, TensorProd(a_1_to_i, IndexedVar(b, sub_expr1), c_1_to_k), one, j))).with_wrapping_at(2)).with_wrapping_at(2)).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), domain = 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())

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