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, m, n
from proveit.core_expr_types import n_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import CartExp, Forall, SubsetEq
from proveit.numbers import Mult, NaturalPos, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [n_1_to_m], instance_expr = SubsetEq(TensorProd(ExprRange(sub_expr1, CartExp(K, IndexedVar(n, sub_expr1)), one, m)), CartExp(K, Mult(n_1_to_m))), domain = NaturalPos), 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_{m \in \mathbb{N}^+}~\left[\forall_{n_{1}, n_{2}, \ldots, n_{m} \in \mathbb{N}^+}~\left(\left(K^{n_{1}} {\otimes}  K^{n_{2}} {\otimes}  \ldots {\otimes}  K^{n_{m}}\right) \subseteq K^{n_{1} \cdot  n_{2} \cdot  \ldots \cdot  n_{m}}\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 40 body: 4
3	ExprTuple	40
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 31 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	40, 35
10	Lambda	parameters: 30 body: 11
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operands: 15
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 36 operands: 23
20	ExprRange	lambda_map: 24 start_index: 39 end_index: 40
21	Literal
22	ExprTuple	25
23	ExprTuple	41, 26
24	Lambda	parameter: 45 body: 27
25	ExprRange	lambda_map: 28 start_index: 39 end_index: 40
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operands: 32
28	Lambda	parameter: 45 body: 33
29	Literal
30	ExprTuple	34
31	Literal
32	ExprTuple	42, 35
33	Operation	operator: 36 operands: 37
34	ExprRange	lambda_map: 38 start_index: 39 end_index: 40
35	Literal
36	Literal
37	ExprTuple	41, 42
38	Lambda	parameter: 45 body: 42
39	Literal
40	Variable
41	Variable
42	IndexedVar	variable: 43 index: 45
43	Variable
44	ExprTuple	45
45	Variable