Theorem qmult_distribution_over_summation of type Forall

from the theory of proveit.physics.quantum.algebra

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.physics.quantum.algebra import qmult_distribution_over_summation

# check that the built expression is the same as the stored expression
assert qmult_distribution_over_summation.expr == stored_expr
assert qmult_distribution_over_summation.expr._style_id == stored_expr._style_id
print("Passed sanity check: qmult_distribution_over_summation matches stored_expr")

Passed sanity check: qmult_distribution_over_summation matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{j \in \mathbb{N}^+, m \in \mathbb{N}, n \in \mathbb{N}}~\left[\forall_{A_{1}, A_{2}, \ldots, A_{m}, C_{1}, C_{2}, \ldots, C_{n}, f, Q}~\left(\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}\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	parameters: 3 body: 4
3	ExprTuple	63, 52, 56
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12, 13, 14
11	Lambda	parameters: 15 body: 16
12	Operation	operator: 19 operands: 17
13	Operation	operator: 19 operands: 18
14	Operation	operator: 19 operands: 20
15	ExprTuple	47, 49, 53, 50
16	Operation	operator: 21 operands: 22
17	ExprTuple	63, 23
18	ExprTuple	52, 24
19	Literal
20	ExprTuple	56, 24
21	Literal
22	ExprTuple	25, 26
23	Literal
24	Literal
25	Operation	operator: 27 operands: 28
26	Operation	operator: 29 operands: 30
27	Literal
28	ExprTuple	32, 31
29	Literal
30	ExprTuple	32, 33
31	Literal
32	Operation	operator: 44 operands: 34
33	Operation	operator: 38 operand: 37
34	ExprTuple	47, 36, 49
35	ExprTuple	37
36	Operation	operator: 38 operand: 41
37	Lambda	parameters: 54 body: 40
38	Literal
39	ExprTuple	41
40	Conditional	value: 42 condition: 46
41	Lambda	parameters: 54 body: 43
42	Operation	operator: 44 operands: 45
43	Conditional	value: 48 condition: 46
44	Literal
45	ExprTuple	47, 48, 49
46	Operation	operator: 50 operands: 54
47	ExprRange	lambda_map: 51 start_index: 62 end_index: 52
48	Operation	operator: 53 operands: 54
49	ExprRange	lambda_map: 55 start_index: 62 end_index: 56
50	Variable
51	Lambda	parameter: 68 body: 57
52	Variable
53	Variable
54	ExprTuple	58
55	Lambda	parameter: 68 body: 59
56	Variable
57	IndexedVar	variable: 60 index: 68
58	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
59	IndexedVar	variable: 64 index: 68
60	Variable
61	Lambda	parameter: 68 body: 65
62	Literal
63	Variable
64	Variable
65	IndexedVar	variable: 66 index: 68
66	Variable
67	ExprTuple	68
68	Variable