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 Conditional, 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 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 = Lambda([i, j, k], 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(i, j, k\right) \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

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