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, K, Lambda, 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 = Lambda([i, j, k], 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))))

# 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\{\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()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Lambda	parameters: 1 body: 2
1	ExprTuple	64, 77, 66
2	Conditional	value: 3 condition: 4
3	Operation	operator: 35 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	9, 10, 11
8	Lambda	parameter: 53 body: 13
9	Operation	operator: 48 operands: 14
10	Operation	operator: 48 operands: 15
11	Operation	operator: 48 operands: 16
12	ExprTuple	53
13	Conditional	value: 17 condition: 18
14	ExprTuple	64, 20
15	ExprTuple	77, 19
16	ExprTuple	66, 20
17	Operation	operator: 35 operand: 24
18	Operation	operator: 22 operands: 23
19	Literal
20	Literal
21	ExprTuple	24
22	Literal
23	ExprTuple	53, 25
24	Lambda	parameters: 26 body: 27
25	Operation	operator: 28 operand: 32
26	ExprTuple	59, 60
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32
30	Literal
31	ExprTuple	33, 34
32	Variable
33	Operation	operator: 35 operand: 39
34	Operation	operator: 37 operands: 38
35	Literal
36	ExprTuple	39
37	Literal
38	ExprTuple	40, 41
39	Lambda	parameters: 69 body: 42
40	Operation	operator: 51 operand: 46
41	Operation	operator: 56 operands: 44
42	Conditional	value: 45 condition: 62
43	ExprTuple	46
44	ExprTuple	59, 47, 60
45	Operation	operator: 48 operands: 49
46	Lambda	parameters: 69 body: 50
47	Operation	operator: 51 operand: 55
48	Literal
49	ExprTuple	54, 53
50	Conditional	value: 54 condition: 62
51	Literal
52	ExprTuple	55
53	Variable
54	Operation	operator: 56 operands: 57
55	Lambda	parameters: 69 body: 58
56	Literal
57	ExprTuple	59, 61, 60
58	Conditional	value: 61 condition: 62
59	ExprRange	lambda_map: 63 start_index: 76 end_index: 64
60	ExprRange	lambda_map: 65 start_index: 76 end_index: 66
61	Operation	operator: 67 operands: 69
62	Operation	operator: 68 operands: 69
63	Lambda	parameter: 81 body: 70
64	Variable
65	Lambda	parameter: 81 body: 71
66	Variable
67	Variable
68	Variable
69	ExprTuple	72
70	IndexedVar	variable: 73 index: 81
71	IndexedVar	variable: 74 index: 81
72	ExprRange	lambda_map: 75 start_index: 76 end_index: 77
73	Variable
74	Variable
75	Lambda	parameter: 81 body: 78
76	Literal
77	Variable
78	IndexedVar	variable: 79 index: 81
79	Variable
80	ExprTuple	81
81	Variable

Expression of type Lambda¶

from the theory of proveit.linear_algebra.tensors¶

Expression of type Lambda ¶

from the theory of proveit.linear_algebra.tensors ¶