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, CartExp, Forall, InSet
from proveit.numbers import Add, Complex, Exp, Interval, Natural, one, subtract, two
from proveit.physics.quantum.circuits import MultiQubitElem, N_0_to_m, N_m, Output, 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(Forall(instance_param_or_params = [N_0_to_m], instance_expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, ExprRange(sub_expr2, MultiQubitElem(element = Output(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 = Output(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)]))), domain = Natural, condition = each_Nk_is_partial_sum).with_wrapping(), And(ExprRange(sub_expr2, InSet(IndexedVar(A, sub_expr2), CartExp(Complex, Exp(two, IndexedVar(n, sub_expr2)))), one, m)))

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\{\begin{array}{l}\forall_{N_{0}, N_{1}, \ldots, N_{m} \in \mathbb{N}~|~\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)}~\\
\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qout{A_{1}~\mbox{part}~1~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} \\
& \qout{A_{1}~\mbox{part}~2~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} \\
& \qout{\vdots} \\
& \qout{A_{1}~\mbox{part}~n_{1}~\mbox{on}~\{N_{1 - 1} + 1~\ldotp \ldotp~N_{1}\}} \\
& \qout{A_{2}~\mbox{part}~1~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} \\
& \qout{A_{2}~\mbox{part}~2~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} \\
& \qout{\vdots} \\
& \qout{A_{2}~\mbox{part}~n_{2}~\mbox{on}~\{N_{2 - 1} + 1~\ldotp \ldotp~N_{2}\}} \\
& \qout{\begin{array}{c}\vdots\\ \vdots\end{array}} \\
& \qout{A_{m}~\mbox{part}~1~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{m}~\mbox{part}~2~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{m}~\mbox{part}~n_{m}~\mbox{on}~\{N_{m - 1} + 1~\ldotp \ldotp~N_{m}\}}
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{1}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2 - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{2}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\begin{array}{c}\vdots\\ \vdots\end{array}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 1~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m - 1} + 2~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}} \\
& \qout{\vdots} \\
& \qout{A_{1} {\otimes}  A_{2} {\otimes}  \ldots {\otimes}  A_{m}~\mbox{part}~N_{m}~\mbox{on}~\{1~\ldotp \ldotp~N_{m}\}}
} \end{array}\right)\right)\end{array} \textrm{ if } \left(A_{1} \in \mathbb{C}^{2^{n_{1}}}\right) \land  \left(A_{2} \in \mathbb{C}^{2^{n_{2}}}\right) \land  \ldots \land  \left(A_{m} \in \mathbb{C}^{2^{n_{m}}}\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 operand: 6
2	Operation	operator: 29 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7
6	Lambda	parameters: 8 body: 9
7	ExprRange	lambda_map: 10 start_index: 123 end_index: 111
8	ExprTuple	11
9	Conditional	value: 12 condition: 13
10	Lambda	parameter: 120 body: 14
11	ExprRange	lambda_map: 15 start_index: 64 end_index: 111
12	Operation	operator: 16 operands: 17
13	Operation	operator: 29 operands: 18
14	Operation	operator: 42 operands: 19
15	Lambda	parameter: 120 body: 65
16	Literal
17	ExprTuple	20, 21
18	ExprTuple	22, 23
19	ExprTuple	113, 24
20	Operation	operator: 26 operand: 33
21	Operation	operator: 26 operand: 34
22	ExprRange	lambda_map: 28 start_index: 64 end_index: 111
23	Operation	operator: 29 operands: 30
24	Operation	operator: 31 operands: 32
25	ExprTuple	33
26	Literal
27	ExprTuple	34
28	Lambda	parameter: 120 body: 35
29	Literal
30	ExprTuple	36, 37
31	Literal
32	ExprTuple	38, 39
33	ExprTuple	40
34	ExprTuple	41
35	Operation	operator: 42 operands: 43
36	Operation	operator: 57 operands: 44
37	ExprRange	lambda_map: 45 start_index: 123 end_index: 111
38	Literal
39	Operation	operator: 46 operands: 47
40	ExprRange	lambda_map: 48 start_index: 123 end_index: 111
41	ExprRange	lambda_map: 49 start_index: 123 end_index: 111
42	Literal
43	ExprTuple	65, 50
44	ExprTuple	51, 64
45	Lambda	parameter: 120 body: 52
46	Literal
47	ExprTuple	53, 77
48	Lambda	parameter: 118 body: 54
49	Lambda	parameter: 105 body: 55
50	Literal
51	IndexedVar	variable: 108 index: 64
52	Operation	operator: 57 operands: 58
53	Literal
54	ExprRange	lambda_map: 59 start_index: 123 end_index: 60
55	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
56	ExprTuple	64
57	Literal
58	ExprTuple	65, 66
59	Lambda	parameter: 120 body: 67
60	IndexedVar	variable: 84 index: 118
61	Lambda	parameter: 118 body: 68
62	Operation	operator: 114 operands: 69
63	IndexedVar	variable: 108 index: 105
64	Literal
65	IndexedVar	variable: 108 index: 120
66	Operation	operator: 114 operands: 71
67	Operation	operator: 73 operands: 72
68	Operation	operator: 73 operands: 74
69	ExprTuple	75, 123
70	ExprTuple	105
71	ExprTuple	76, 77
72	NamedExprs	element: 78 targets: 79
73	Literal
74	NamedExprs	element: 80 targets: 81
75	IndexedVar	variable: 108 index: 91
76	IndexedVar	variable: 108 index: 92
77	IndexedVar	variable: 84 index: 120
78	Operation	operator: 87 operands: 85
79	Operation	operator: 89 operands: 86
80	Operation	operator: 87 operands: 88
81	Operation	operator: 89 operands: 90
82	ExprTuple	91
83	ExprTuple	92
84	Variable
85	NamedExprs	state: 93 part: 120
86	ExprTuple	94, 95
87	Literal
88	NamedExprs	state: 96 part: 118
89	Literal
90	ExprTuple	123, 97
91	Operation	operator: 114 operands: 98
92	Operation	operator: 114 operands: 99
93	IndexedVar	variable: 116 index: 118
94	Operation	operator: 114 operands: 100
95	IndexedVar	variable: 108 index: 118
96	Operation	operator: 102 operands: 103
97	IndexedVar	variable: 108 index: 111
98	ExprTuple	105, 119
99	ExprTuple	120, 119
100	ExprTuple	106, 123
101	ExprTuple	118
102	Literal
103	ExprTuple	107
104	ExprTuple	111
105	Variable
106	IndexedVar	variable: 108 index: 112
107	ExprRange	lambda_map: 110 start_index: 123 end_index: 111
108	Variable
109	ExprTuple	112
110	Lambda	parameter: 120 body: 113
111	Variable
112	Operation	operator: 114 operands: 115
113	IndexedVar	variable: 116 index: 120
114	Literal
115	ExprTuple	118, 119
116	Variable
117	ExprTuple	120
118	Variable
119	Operation	operator: 121 operand: 123
120	Variable
121	Literal
122	ExprTuple	123
123	Literal