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 Function, V, s
from proveit.core_expr_types import Q__b_1_to_j, a_1_to_i, b_1_to_j, c_1_to_k, f__b_1_to_j
from proveit.linear_algebra import ScalarMult, TensorProd, VecSum
from proveit.logic import Equals, Forall, Implies, InSet

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(s, sub_expr1)
sub_expr3 = ScalarMult(sub_expr2, f__b_1_to_j)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(TensorProd(a_1_to_i, sub_expr3, c_1_to_k), V), condition = Q__b_1_to_j), Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, TensorProd(a_1_to_i, f__b_1_to_j, c_1_to_k)), condition = Q__b_1_to_j), TensorProd(a_1_to_i, VecSum(index_or_indices = sub_expr1, summand = sub_expr3, condition = Q__b_1_to_j), c_1_to_k)).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_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right)\right] \\  = \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\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	parameters: 48 body: 12
10	Operation	operator: 21 operand: 16
11	Operation	operator: 33 operands: 14
12	Conditional	value: 15 condition: 32
13	ExprTuple	16
14	ExprTuple	38, 17, 39
15	Operation	operator: 18 operands: 19
16	Lambda	parameters: 48 body: 20
17	Operation	operator: 21 operand: 26
18	Literal
19	ExprTuple	23, 24
20	Conditional	value: 25 condition: 32
21	Literal
22	ExprTuple	26
23	Operation	operator: 33 operands: 27
24	Variable
25	Operation	operator: 35 operands: 28
26	Lambda	parameters: 48 body: 29
27	ExprTuple	38, 31, 39
28	ExprTuple	40, 30
29	Conditional	value: 31 condition: 32
30	Operation	operator: 33 operands: 34
31	Operation	operator: 35 operands: 36
32	Operation	operator: 37 operands: 48
33	Literal
34	ExprTuple	38, 41, 39
35	Literal
36	ExprTuple	40, 41
37	Variable
38	ExprRange	lambda_map: 42 start_index: 55 end_index: 43
39	ExprRange	lambda_map: 44 start_index: 55 end_index: 45
40	Operation	operator: 46 operands: 48
41	Operation	operator: 47 operands: 48
42	Lambda	parameter: 60 body: 49
43	Variable
44	Lambda	parameter: 60 body: 50
45	Variable
46	Variable
47	Variable
48	ExprTuple	51
49	IndexedVar	variable: 52 index: 60
50	IndexedVar	variable: 53 index: 60
51	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
52	Variable
53	Variable
54	Lambda	parameter: 60 body: 57
55	Literal
56	Variable
57	IndexedVar	variable: 58 index: 60
58	Variable
59	ExprTuple	60
60	Variable