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, 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, InSet
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(i, Conditional(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(), InSet(i, 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())

i \mapsto \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} \textrm{ if } i \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	parameter: 55 body: 2
1	ExprTuple	55
2	Conditional	value: 3 condition: 4
3	Operation	operator: 15 operand: 7
4	Operation	operator: 48 operands: 6
5	ExprTuple	7
6	ExprTuple	55, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	11
10	Conditional	value: 12 condition: 13
11	ExprRange	lambda_map: 14 start_index: 54 end_index: 55
12	Operation	operator: 15 operand: 18
13	Operation	operator: 27 operands: 17
14	Lambda	parameter: 62 body: 52
15	Literal
16	ExprTuple	18
17	ExprTuple	19
18	Lambda	parameters: 46 body: 20
19	ExprRange	lambda_map: 21 start_index: 54 end_index: 55
20	Conditional	value: 22 condition: 23
21	Lambda	parameter: 62 body: 24
22	Operation	operator: 25 operands: 26
23	Operation	operator: 27 operands: 28
24	Operation	operator: 29 operands: 30
25	Literal
26	ExprTuple	31, 32
27	Literal
28	ExprTuple	33
29	Literal
30	ExprTuple	52, 34
31	Operation	operator: 56 operand: 41
32	Operation	operator: 36 operands: 37
33	ExprRange	lambda_map: 38 start_index: 54 end_index: 55
34	Operation	operator: 39 operand: 44
35	ExprTuple	41
36	Literal
37	ExprTuple	42
38	Lambda	parameter: 62 body: 43
39	Literal
40	ExprTuple	44
41	Operation	operator: 45 operands: 46
42	ExprRange	lambda_map: 47 start_index: 54 end_index: 55
43	Operation	operator: 48 operands: 49
44	Variable
45	Literal
46	ExprTuple	50
47	Lambda	parameter: 62 body: 51
48	Literal
49	ExprTuple	59, 52
50	ExprRange	lambda_map: 53 start_index: 54 end_index: 55
51	Operation	operator: 56 operand: 59
52	IndexedVar	variable: 58 index: 62
53	Lambda	parameter: 62 body: 59
54	Literal
55	Variable
56	Literal
57	ExprTuple	59
58	Variable
59	IndexedVar	variable: 60 index: 62
60	Variable
61	ExprTuple	62
62	Variable