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, NaturalPos, 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}")
sub_expr4 = IndexedVar(n, sub_expr2)
expr = Conditional(Forall(instance_param_or_params = [A_1_to_m], instance_expr = 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(), domains = [ExprRange(sub_expr2, CartExp(Complex, Exp(two, sub_expr4)), one, m)]).with_wrapping(), And(ExprRange(sub_expr2, InSet(sub_expr4, NaturalPos), 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_{\left(A_{1} \in \mathbb{C}^{2^{n_{1}}}\right), \left(A_{2} \in \mathbb{C}^{2^{n_{2}}}\right), \ldots, \left(A_{m} \in \mathbb{C}^{2^{n_{m}}}\right)}~\\
\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}\right]\end{array} \textrm{ if } \left(n_{1} \in \mathbb{N}^+\right) \land  \left(n_{2} \in \mathbb{N}^+\right) \land  \ldots \land  \left(n_{m} \in \mathbb{N}^+\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: 12 operand: 5
2	Operation	operator: 40 operands: 4
3	ExprTuple	5
4	ExprTuple	6
5	Lambda	parameters: 114 body: 7
6	ExprRange	lambda_map: 8 start_index: 134 end_index: 122
7	Conditional	value: 9 condition: 10
8	Lambda	parameter: 131 body: 11
9	Operation	operator: 12 operand: 16
10	Operation	operator: 40 operands: 14
11	Operation	operator: 53 operands: 15
12	Literal
13	ExprTuple	16
14	ExprTuple	17
15	ExprTuple	88, 18
16	Lambda	parameters: 19 body: 20
17	ExprRange	lambda_map: 21 start_index: 134 end_index: 122
18	Literal
19	ExprTuple	22
20	Conditional	value: 23 condition: 24
21	Lambda	parameter: 131 body: 25
22	ExprRange	lambda_map: 26 start_index: 75 end_index: 122
23	Operation	operator: 27 operands: 28
24	Operation	operator: 40 operands: 29
25	Operation	operator: 53 operands: 30
26	Lambda	parameter: 131 body: 76
27	Literal
28	ExprTuple	31, 32
29	ExprTuple	33, 34
30	ExprTuple	124, 35
31	Operation	operator: 37 operand: 44
32	Operation	operator: 37 operand: 45
33	ExprRange	lambda_map: 39 start_index: 75 end_index: 122
34	Operation	operator: 40 operands: 41
35	Operation	operator: 42 operands: 43
36	ExprTuple	44
37	Literal
38	ExprTuple	45
39	Lambda	parameter: 131 body: 46
40	Literal
41	ExprTuple	47, 48
42	Literal
43	ExprTuple	49, 50
44	ExprTuple	51
45	ExprTuple	52
46	Operation	operator: 53 operands: 54
47	Operation	operator: 68 operands: 55
48	ExprRange	lambda_map: 56 start_index: 134 end_index: 122
49	Literal
50	Operation	operator: 57 operands: 58
51	ExprRange	lambda_map: 59 start_index: 134 end_index: 122
52	ExprRange	lambda_map: 60 start_index: 134 end_index: 122
53	Literal
54	ExprTuple	76, 61
55	ExprTuple	62, 75
56	Lambda	parameter: 131 body: 63
57	Literal
58	ExprTuple	64, 88
59	Lambda	parameter: 129 body: 65
60	Lambda	parameter: 116 body: 66
61	Literal
62	IndexedVar	variable: 119 index: 75
63	Operation	operator: 68 operands: 69
64	Literal
65	ExprRange	lambda_map: 70 start_index: 134 end_index: 71
66	ExprRange	lambda_map: 72 start_index: 73 end_index: 74
67	ExprTuple	75
68	Literal
69	ExprTuple	76, 77
70	Lambda	parameter: 131 body: 78
71	IndexedVar	variable: 95 index: 129
72	Lambda	parameter: 129 body: 79
73	Operation	operator: 125 operands: 80
74	IndexedVar	variable: 119 index: 116
75	Literal
76	IndexedVar	variable: 119 index: 131
77	Operation	operator: 125 operands: 82
78	Operation	operator: 84 operands: 83
79	Operation	operator: 84 operands: 85
80	ExprTuple	86, 134
81	ExprTuple	116
82	ExprTuple	87, 88
83	NamedExprs	element: 89 targets: 90
84	Literal
85	NamedExprs	element: 91 targets: 92
86	IndexedVar	variable: 119 index: 102
87	IndexedVar	variable: 119 index: 103
88	IndexedVar	variable: 95 index: 131
89	Operation	operator: 98 operands: 96
90	Operation	operator: 100 operands: 97
91	Operation	operator: 98 operands: 99
92	Operation	operator: 100 operands: 101
93	ExprTuple	102
94	ExprTuple	103
95	Variable
96	NamedExprs	state: 104 part: 131
97	ExprTuple	105, 106
98	Literal
99	NamedExprs	state: 107 part: 129
100	Literal
101	ExprTuple	134, 108
102	Operation	operator: 125 operands: 109
103	Operation	operator: 125 operands: 110
104	IndexedVar	variable: 127 index: 129
105	Operation	operator: 125 operands: 111
106	IndexedVar	variable: 119 index: 129
107	Operation	operator: 113 operands: 114
108	IndexedVar	variable: 119 index: 122
109	ExprTuple	116, 130
110	ExprTuple	131, 130
111	ExprTuple	117, 134
112	ExprTuple	129
113	Literal
114	ExprTuple	118
115	ExprTuple	122
116	Variable
117	IndexedVar	variable: 119 index: 123
118	ExprRange	lambda_map: 121 start_index: 134 end_index: 122
119	Variable
120	ExprTuple	123
121	Lambda	parameter: 131 body: 124
122	Variable
123	Operation	operator: 125 operands: 126
124	IndexedVar	variable: 127 index: 131
125	Literal
126	ExprTuple	129, 130
127	Variable
128	ExprTuple	131
129	Variable
130	Operation	operator: 132 operand: 134
131	Variable
132	Literal
133	ExprTuple	134
134	Literal