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 And, Equals, Forall
from proveit.numbers import 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 = [K], instance_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)), one), domains = sub_expr2, condition = And(ExprRange(sub_expr1, Equals(Norm(IndexedVar(a, sub_expr1)), one), one, i))).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_{K}~\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}\right \| = 1\right) \land  \left(\left \|a_{2}\right \| = 1\right) \land  \ldots \land  \left(\left \|a_{i}\right \| = 1\right)}~\\
\left(\left \|a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i}\right \| = 1\right)\end{array}\right]\end{array}\right]\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: 20 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 47 body: 3
3	Operation	operator: 20 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 57 body: 7
6	ExprTuple	57
7	Conditional	value: 8 condition: 9
8	Operation	operator: 20 operand: 12
9	Operation	operator: 50 operands: 11
10	ExprTuple	12
11	ExprTuple	57, 13
12	Lambda	parameters: 14 body: 15
13	Literal
14	ExprTuple	16
15	Conditional	value: 17 condition: 18
16	ExprRange	lambda_map: 19 start_index: 62 end_index: 57
17	Operation	operator: 20 operand: 23
18	Operation	operator: 40 operands: 22
19	Lambda	parameter: 68 body: 54
20	Literal
21	ExprTuple	23
22	ExprTuple	24
23	Lambda	parameters: 49 body: 25
24	ExprRange	lambda_map: 26 start_index: 62 end_index: 57
25	Conditional	value: 27 condition: 28
26	Lambda	parameter: 68 body: 29
27	Operation	operator: 59 operands: 30
28	Operation	operator: 40 operands: 31
29	Operation	operator: 32 operands: 33
30	ExprTuple	34, 62
31	ExprTuple	35, 36
32	Literal
33	ExprTuple	54, 37
34	Operation	operator: 63 operand: 44
35	ExprRange	lambda_map: 39 start_index: 62 end_index: 57
36	Operation	operator: 40 operands: 41
37	Operation	operator: 42 operand: 47
38	ExprTuple	44
39	Lambda	parameter: 68 body: 45
40	Literal
41	ExprTuple	46
42	Literal
43	ExprTuple	47
44	Operation	operator: 48 operands: 49
45	Operation	operator: 50 operands: 51
46	ExprRange	lambda_map: 52 start_index: 62 end_index: 57
47	Variable
48	Literal
49	ExprTuple	53
50	Literal
51	ExprTuple	65, 54
52	Lambda	parameter: 68 body: 55
53	ExprRange	lambda_map: 56 start_index: 62 end_index: 57
54	IndexedVar	variable: 58 index: 68
55	Operation	operator: 59 operands: 60
56	Lambda	parameter: 68 body: 65
57	Variable
58	Variable
59	Literal
60	ExprTuple	61, 62
61	Operation	operator: 63 operand: 65
62	Literal
63	Literal
64	ExprTuple	65
65	IndexedVar	variable: 66 index: 68
66	Variable
67	ExprTuple	68
68	Variable