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

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
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 = [V_1_to_i]).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_{\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}

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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 22 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11, 12
9	Literal
10	ExprTuple	13
11	Operation	operator: 32 operand: 18
12	Operation	operator: 15 operands: 16
13	ExprRange	lambda_map: 17 start_index: 30 end_index: 31
14	ExprTuple	18
15	Literal
16	ExprTuple	19
17	Lambda	parameter: 38 body: 20
18	Operation	operator: 21 operands: 22
19	ExprRange	lambda_map: 23 start_index: 30 end_index: 31
20	Operation	operator: 24 operands: 25
21	Literal
22	ExprTuple	26
23	Lambda	parameter: 38 body: 27
24	Literal
25	ExprTuple	35, 28
26	ExprRange	lambda_map: 29 start_index: 30 end_index: 31
27	Operation	operator: 32 operand: 35
28	IndexedVar	variable: 34 index: 38
29	Lambda	parameter: 38 body: 35
30	Literal
31	Variable
32	Literal
33	ExprTuple	35
34	Variable
35	IndexedVar	variable: 36 index: 38
36	Variable
37	ExprTuple	38
38	Variable