Expression of type Implies

from the theory of proveit.physics.quantum.algebra

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.core_expr_types import A_1_to_m, C_1_to_n, Q__b_1_to_j, b_1_to_j, f__b_1_to_j
from proveit.linear_algebra import VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Implies, InClass
from proveit.physics.quantum import Qmult, QmultCodomain

# build up the expression from sub-expressions
sub_expr1 = Qmult(A_1_to_m, vec_summation_b1toj_fQ, C_1_to_n)
expr = Implies(InClass(sub_expr1, QmultCodomain), Equals(sub_expr1, VecSum(index_or_indices = [b_1_to_j], summand = Qmult(A_1_to_m, f__b_1_to_j, C_1_to_n), condition = Q__b_1_to_j)).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(\left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right) \underset{{\scriptscriptstyle c}}{\in} \mathcal{Q^*}\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{m} \thinspace f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\thinspace C_{1} \thinspace  C_{2} \thinspace  \ldots \thinspace  C_{n}\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 operands: 6
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	10, 9
7	Literal
8	ExprTuple	10, 11
9	Literal
10	Operation	operator: 22 operands: 12
11	Operation	operator: 16 operand: 15
12	ExprTuple	25, 14, 27
13	ExprTuple	15
14	Operation	operator: 16 operand: 19
15	Lambda	parameters: 32 body: 18
16	Literal
17	ExprTuple	19
18	Conditional	value: 20 condition: 24
19	Lambda	parameters: 32 body: 21
20	Operation	operator: 22 operands: 23
21	Conditional	value: 26 condition: 24
22	Literal
23	ExprTuple	25, 26, 27
24	Operation	operator: 28 operands: 32
25	ExprRange	lambda_map: 29 start_index: 40 end_index: 30
26	Operation	operator: 31 operands: 32
27	ExprRange	lambda_map: 33 start_index: 40 end_index: 34
28	Variable
29	Lambda	parameter: 46 body: 35
30	Variable
31	Variable
32	ExprTuple	36
33	Lambda	parameter: 46 body: 37
34	Variable
35	IndexedVar	variable: 38 index: 46
36	ExprRange	lambda_map: 39 start_index: 40 end_index: 41
37	IndexedVar	variable: 42 index: 46
38	Variable
39	Lambda	parameter: 46 body: 43
40	Literal
41	Variable
42	Variable
43	IndexedVar	variable: 44 index: 46
44	Variable
45	ExprTuple	46
46	Variable