Expression of type Conditional

from the theory of proveit.physics.quantum.circuits

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 A, Conditional, ExprRange, IndexedVar, N, Variable, VertExprArray, m, n
from proveit.core_expr_types import A_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import And, InSet
from proveit.numbers import Add, Interval, Natural, one, subtract, zero
from proveit.physics.quantum.circuits import Input, MultiQubitElem, N_m, Qcircuit, QcircuitEquiv, each_Nk_is_partial_sum

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr3 = Variable("_c", latex_format = r"{_{-}c}")
expr = Conditional(QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, ExprRange(sub_expr2, MultiQubitElem(element = Input(state = IndexedVar(A, sub_expr1), part = sub_expr2), targets = Interval(Add(IndexedVar(N, subtract(sub_expr1, one)), one), IndexedVar(N, sub_expr1))), one, IndexedVar(n, sub_expr1)).with_wrapping_at(2,6), one, m)])), Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr3, ExprRange(sub_expr1, MultiQubitElem(element = Input(state = TensorProd(A_1_to_m), part = sub_expr1), targets = Interval(one, N_m)), Add(IndexedVar(N, subtract(sub_expr3, one)), one), IndexedVar(N, sub_expr3)).with_wrapping_at(2,6), one, m)]))), And(ExprRange(sub_expr2, InSet(IndexedVar(N, sub_expr2), Natural), zero, m), each_Nk_is_partial_sum))

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())

\left\{\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{A_{1}~\mbox{part}~1~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{A_{1}~\mbox{part}~2~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1}~\mbox{part}~n_{1}~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} & \qw \\
\qin{A_{2}~\mbox{part}~1~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{A_{2}~\mbox{part}~2~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{2}~\mbox{part}~n_{2}~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} & \qw \\
\qin{\begin{array}{c}\vdots\\ \vdots\end{array}} & \qw \\
\qin{A_{m}~\mbox{part}~1~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{m}~\mbox{part}~2~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{m}~\mbox{part}~n_{m}~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} & \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\begin{array}{c}\vdots\\ \vdots\end{array}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw \\
\qin{\vdots} & \qw \\
\qin{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} & \qw
} \end{array}\right) \textrm{ if } \left(N_{0} \in \mathbb{N}\right) ,  \left(N_{1} \in \mathbb{N}\right) ,  \ldots ,  \left(N_{m} \in \mathbb{N}\right) ,  \left(N_{0} = 0\right)\land \left(N_{1} = \left(N_{1 - 1} + n_{1}\right)\right) \land  \left(N_{2} = \left(N_{2 - 1} + n_{2}\right)\right) \land  \ldots \land  \left(N_{m} = \left(N_{m - 1} + n_{m}\right)\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operands: 4
2	Operation	operator: 14 operands: 5
3	Literal
4	ExprTuple	6, 7
5	ExprTuple	8, 9
6	Operation	operator: 11 operand: 16
7	Operation	operator: 11 operand: 17
8	ExprRange	lambda_map: 13 start_index: 42 end_index: 89
9	Operation	operator: 14 operands: 15
10	ExprTuple	16
11	Literal
12	ExprTuple	17
13	Lambda	parameter: 98 body: 18
14	Literal
15	ExprTuple	19, 20
16	ExprTuple	21
17	ExprTuple	22
18	Operation	operator: 23 operands: 24
19	Operation	operator: 35 operands: 25
20	ExprRange	lambda_map: 26 start_index: 101 end_index: 89
21	ExprRange	lambda_map: 27 start_index: 101 end_index: 89
22	ExprRange	lambda_map: 28 start_index: 101 end_index: 89
23	Literal
24	ExprTuple	43, 29
25	ExprTuple	30, 42
26	Lambda	parameter: 98 body: 31
27	Lambda	parameter: 96 body: 32
28	Lambda	parameter: 83 body: 33
29	Literal
30	IndexedVar	variable: 86 index: 42
31	Operation	operator: 35 operands: 36
32	ExprRange	lambda_map: 37 start_index: 101 end_index: 38
33	ExprRange	lambda_map: 39 start_index: 40 end_index: 41
34	ExprTuple	42
35	Literal
36	ExprTuple	43, 44
37	Lambda	parameter: 98 body: 45
38	IndexedVar	variable: 62 index: 96
39	Lambda	parameter: 96 body: 46
40	Operation	operator: 92 operands: 47
41	IndexedVar	variable: 86 index: 83
42	Literal
43	IndexedVar	variable: 86 index: 98
44	Operation	operator: 92 operands: 49
45	Operation	operator: 51 operands: 50
46	Operation	operator: 51 operands: 52
47	ExprTuple	53, 101
48	ExprTuple	83
49	ExprTuple	54, 55
50	NamedExprs	element: 56 targets: 57
51	Literal
52	NamedExprs	element: 58 targets: 59
53	IndexedVar	variable: 86 index: 69
54	IndexedVar	variable: 86 index: 70
55	IndexedVar	variable: 62 index: 98
56	Operation	operator: 65 operands: 63
57	Operation	operator: 67 operands: 64
58	Operation	operator: 65 operands: 66
59	Operation	operator: 67 operands: 68
60	ExprTuple	69
61	ExprTuple	70
62	Variable
63	NamedExprs	state: 71 part: 98
64	ExprTuple	72, 73
65	Literal
66	NamedExprs	state: 74 part: 96
67	Literal
68	ExprTuple	101, 75
69	Operation	operator: 92 operands: 76
70	Operation	operator: 92 operands: 77
71	IndexedVar	variable: 94 index: 96
72	Operation	operator: 92 operands: 78
73	IndexedVar	variable: 86 index: 96
74	Operation	operator: 80 operands: 81
75	IndexedVar	variable: 86 index: 89
76	ExprTuple	83, 97
77	ExprTuple	98, 97
78	ExprTuple	84, 101
79	ExprTuple	96
80	Literal
81	ExprTuple	85
82	ExprTuple	89
83	Variable
84	IndexedVar	variable: 86 index: 90
85	ExprRange	lambda_map: 88 start_index: 101 end_index: 89
86	Variable
87	ExprTuple	90
88	Lambda	parameter: 98 body: 91
89	Variable
90	Operation	operator: 92 operands: 93
91	IndexedVar	variable: 94 index: 98
92	Literal
93	ExprTuple	96, 97
94	Variable
95	ExprTuple	98
96	Variable
97	Operation	operator: 99 operand: 101
98	Variable
99	Literal
100	ExprTuple	101
101	Literal