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 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 = 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()

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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 9 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5
4	Conditional	value: 6 condition: 7
5	ExprRange	lambda_map: 8 start_index: 51 end_index: 46
6	Operation	operator: 9 operand: 12
7	Operation	operator: 29 operands: 11
8	Lambda	parameter: 57 body: 43
9	Literal
10	ExprTuple	12
11	ExprTuple	13
12	Lambda	parameters: 38 body: 14
13	ExprRange	lambda_map: 15 start_index: 51 end_index: 46
14	Conditional	value: 16 condition: 17
15	Lambda	parameter: 57 body: 18
16	Operation	operator: 48 operands: 19
17	Operation	operator: 29 operands: 20
18	Operation	operator: 21 operands: 22
19	ExprTuple	23, 51
20	ExprTuple	24, 25
21	Literal
22	ExprTuple	43, 26
23	Operation	operator: 52 operand: 33
24	ExprRange	lambda_map: 28 start_index: 51 end_index: 46
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operand: 36
27	ExprTuple	33
28	Lambda	parameter: 57 body: 34
29	Literal
30	ExprTuple	35
31	Literal
32	ExprTuple	36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 39 operands: 40
35	ExprRange	lambda_map: 41 start_index: 51 end_index: 46
36	Variable
37	Literal
38	ExprTuple	42
39	Literal
40	ExprTuple	54, 43
41	Lambda	parameter: 57 body: 44
42	ExprRange	lambda_map: 45 start_index: 51 end_index: 46
43	IndexedVar	variable: 47 index: 57
44	Operation	operator: 48 operands: 49
45	Lambda	parameter: 57 body: 54
46	Variable
47	Variable
48	Literal
49	ExprTuple	50, 51
50	Operation	operator: 52 operand: 54
51	Literal
52	Literal
53	ExprTuple	54
54	IndexedVar	variable: 55 index: 57
55	Variable
56	ExprTuple	57
57	Variable