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 fi, gamma, i, x, y, z
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 = CartExp(Real, three)
sub_expr3 = Interval(two, four)
sub_expr4 = TensorProd(y, fi)
sub_expr5 = ScalarMult(gamma, sub_expr4)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(TensorProd(x, sub_expr5, z), TensorProd(sub_expr2, sub_expr2, sub_expr2, sub_expr2)), domain = sub_expr3), Equals(TensorProd(x, VecSum(index_or_indices = sub_expr1, summand = sub_expr5, domain = sub_expr3), z), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(gamma, TensorProd(x, sub_expr4, z)), domain = sub_expr3)).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(x {\otimes} \left(\gamma \cdot \left(y {\otimes} f\left(i\right)\right)\right) {\otimes} z\right) \in \left(\mathbb{R}^{3} {\otimes} \mathbb{R}^{3} {\otimes} \mathbb{R}^{3} {\otimes} \mathbb{R}^{3}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(x {\otimes} \left(\sum_{i=2}^{4} \left(\gamma \cdot \left(y {\otimes} f\left(i\right)\right)\right)\right) {\otimes} z\right) \\  = \left(\sum_{i=2}^{4} \left(\gamma \cdot \left(x {\otimes} \left(y {\otimes} f\left(i\right)\right) {\otimes} z\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: 58 body: 12
10	Operation	operator: 50 operands: 13
11	Operation	operator: 19 operand: 17
12	Conditional	value: 15 condition: 32
13	ExprTuple	45, 16, 47
14	ExprTuple	17
15	Operation	operator: 38 operands: 18
16	Operation	operator: 19 operand: 24
17	Lambda	parameter: 58 body: 21
18	ExprTuple	22, 23
19	Literal
20	ExprTuple	24
21	Conditional	value: 25 condition: 32
22	Operation	operator: 50 operands: 26
23	Operation	operator: 50 operands: 27
24	Lambda	parameter: 58 body: 28
25	Operation	operator: 36 operands: 29
26	ExprTuple	45, 31, 47
27	ExprTuple	30, 30, 30, 30
28	Conditional	value: 31 condition: 32
29	ExprTuple	43, 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 36 operands: 37
32	Operation	operator: 38 operands: 39
33	Operation	operator: 50 operands: 40
34	Literal
35	ExprTuple	41, 42
36	Literal
37	ExprTuple	43, 46
38	Literal
39	ExprTuple	58, 44
40	ExprTuple	45, 46, 47
41	Literal
42	Literal
43	Variable
44	Operation	operator: 48 operands: 49
45	Variable
46	Operation	operator: 50 operands: 51
47	Variable
48	Literal
49	ExprTuple	52, 53
50	Literal
51	ExprTuple	54, 55
52	Literal
53	Literal
54	Variable
55	Operation	operator: 56 operand: 58
56	Variable
57	ExprTuple	58
58	Variable