Expression of type Implies

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 beta, i, x, y
from proveit.linear_algebra import ScalarMult, TensorProd, VecSum
from proveit.logic import CartExp, Equals, Forall, Implies, InSet
from proveit.numbers import Interval, Real, four, three, two

# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = ScalarMult(beta, x)
sub_expr3 = ScalarMult(beta, y)
sub_expr4 = CartExp(Real, three)
sub_expr5 = Interval(two, four)
sub_expr6 = TensorProd(sub_expr2, sub_expr3)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(sub_expr6, TensorProd(sub_expr4, sub_expr4)), domain = sub_expr5), Equals(TensorProd(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = sub_expr5), sub_expr3), VecSum(index_or_indices = sub_expr1, summand = sub_expr6, domain = sub_expr5)).with_wrapping_at(1)).with_wrapping_at(2)

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}{c} \begin{array}{l} \left[\forall_{i \in \{2~\ldotp \ldotp~4\}}~\left(\left(\left(\beta \cdot x\right) {\otimes} \left(\beta \cdot y\right)\right) \in \left(\mathbb{R}^{3} {\otimes} \mathbb{R}^{3}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\left(\sum_{i=2}^{4} \left(\beta \cdot x\right)\right) {\otimes} \left(\beta \cdot y\right)\right) \\  = \left(\sum_{i=2}^{4} \left(\left(\beta \cdot x\right) {\otimes} \left(\beta \cdot y\right)\right)\right) \end{array} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 43 body: 12
10	Operation	operator: 28 operands: 13
11	Operation	operator: 19 operand: 17
12	Conditional	value: 15 condition: 31
13	ExprTuple	16, 33
14	ExprTuple	17
15	Operation	operator: 36 operands: 18
16	Operation	operator: 19 operand: 23
17	Lambda	parameter: 43 body: 21
18	ExprTuple	24, 22
19	Literal
20	ExprTuple	23
21	Conditional	value: 24 condition: 31
22	Operation	operator: 28 operands: 25
23	Lambda	parameter: 43 body: 27
24	Operation	operator: 28 operands: 29
25	ExprTuple	30, 30
26	ExprTuple	43
27	Conditional	value: 32 condition: 31
28	Literal
29	ExprTuple	32, 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 36 operands: 37
32	Operation	operator: 39 operands: 38
33	Operation	operator: 39 operands: 40
34	Literal
35	ExprTuple	41, 42
36	Literal
37	ExprTuple	43, 44
38	ExprTuple	46, 45
39	Literal
40	ExprTuple	46, 47
41	Literal
42	Literal
43	Variable
44	Operation	operator: 48 operands: 49
45	Variable
46	Variable
47	Variable
48	Literal
49	ExprTuple	50, 51
50	Literal
51	Literal