Expression of type Forall

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, 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 = 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())

\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]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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