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, Variable, a, i
from proveit.core_expr_types import V_1_to_i, a_1_to_i
from proveit.linear_algebra import InnerProdSpaces, Norm, TensorProd
from proveit.logic import Equals, Forall
from proveit.numbers import Mult, NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [V_1_to_i]
expr = Lambda(K, Forall(instance_param_or_params = [i], instance_expr = Forall(instance_param_or_params = sub_expr2, instance_expr = Forall(instance_param_or_params = [a_1_to_i], instance_expr = Equals(Norm(TensorProd(a_1_to_i)), Mult(ExprRange(sub_expr1, Norm(IndexedVar(a, sub_expr1)), one, i))), domains = sub_expr2).with_wrapping(), domain = InnerProdSpaces(K)).with_wrapping(), domain = NaturalPos))

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 \in \mathbb{N}^+}~\left[\begin{array}{l}\forall_{V_{1}, V_{2}, \ldots, V_{i} \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\\
\left[\begin{array}{l}\forall_{\left(a_{1} \in V_{1}\right), \left(a_{2} \in V_{2}\right), \ldots, \left(a_{i} \in V_{i}\right)}~\\
\left(\left \|a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}\right \| = \left(\left \|a_{1}\right \| \cdot  \left \|a_{2}\right \| \cdot  \ldots \cdot  \left \|a_{i}\right \|\right)\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: 47 body: 1
1	Operation	operator: 18 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 58 body: 5
4	ExprTuple	58
5	Conditional	value: 6 condition: 7
6	Operation	operator: 18 operand: 10
7	Operation	operator: 51 operands: 9
8	ExprTuple	10
9	ExprTuple	58, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	14
13	Conditional	value: 15 condition: 16
14	ExprRange	lambda_map: 17 start_index: 57 end_index: 58
15	Operation	operator: 18 operand: 21
16	Operation	operator: 30 operands: 20
17	Lambda	parameter: 65 body: 55
18	Literal
19	ExprTuple	21
20	ExprTuple	22
21	Lambda	parameters: 49 body: 23
22	ExprRange	lambda_map: 24 start_index: 57 end_index: 58
23	Conditional	value: 25 condition: 26
24	Lambda	parameter: 65 body: 27
25	Operation	operator: 28 operands: 29
26	Operation	operator: 30 operands: 31
27	Operation	operator: 32 operands: 33
28	Literal
29	ExprTuple	34, 35
30	Literal
31	ExprTuple	36
32	Literal
33	ExprTuple	55, 37
34	Operation	operator: 59 operand: 44
35	Operation	operator: 39 operands: 40
36	ExprRange	lambda_map: 41 start_index: 57 end_index: 58
37	Operation	operator: 42 operand: 47
38	ExprTuple	44
39	Literal
40	ExprTuple	45
41	Lambda	parameter: 65 body: 46
42	Literal
43	ExprTuple	47
44	Operation	operator: 48 operands: 49
45	ExprRange	lambda_map: 50 start_index: 57 end_index: 58
46	Operation	operator: 51 operands: 52
47	Variable
48	Literal
49	ExprTuple	53
50	Lambda	parameter: 65 body: 54
51	Literal
52	ExprTuple	62, 55
53	ExprRange	lambda_map: 56 start_index: 57 end_index: 58
54	Operation	operator: 59 operand: 62
55	IndexedVar	variable: 61 index: 65
56	Lambda	parameter: 65 body: 62
57	Literal
58	Variable
59	Literal
60	ExprTuple	62
61	Variable
62	IndexedVar	variable: 63 index: 65
63	Variable
64	ExprTuple	65
65	Variable