Expression of type ExprTuple

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, ExprTuple, IndexedVar, Lambda, 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 = ExprTuple(Lambda([A_1_to_m], 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(\left(A_{1}, A_{2}, \ldots, A_{m}\right) \mapsto \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..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameters: 105 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 31 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9
8	Lambda	parameters: 10 body: 11
9	ExprRange	lambda_map: 12 start_index: 125 end_index: 113
10	ExprTuple	13
11	Conditional	value: 14 condition: 15
12	Lambda	parameter: 122 body: 16
13	ExprRange	lambda_map: 17 start_index: 66 end_index: 113
14	Operation	operator: 18 operands: 19
15	Operation	operator: 31 operands: 20
16	Operation	operator: 44 operands: 21
17	Lambda	parameter: 122 body: 67
18	Literal
19	ExprTuple	22, 23
20	ExprTuple	24, 25
21	ExprTuple	115, 26
22	Operation	operator: 28 operand: 35
23	Operation	operator: 28 operand: 36
24	ExprRange	lambda_map: 30 start_index: 66 end_index: 113
25	Operation	operator: 31 operands: 32
26	Operation	operator: 33 operands: 34
27	ExprTuple	35
28	Literal
29	ExprTuple	36
30	Lambda	parameter: 122 body: 37
31	Literal
32	ExprTuple	38, 39
33	Literal
34	ExprTuple	40, 41
35	ExprTuple	42
36	ExprTuple	43
37	Operation	operator: 44 operands: 45
38	Operation	operator: 59 operands: 46
39	ExprRange	lambda_map: 47 start_index: 125 end_index: 113
40	Literal
41	Operation	operator: 48 operands: 49
42	ExprRange	lambda_map: 50 start_index: 125 end_index: 113
43	ExprRange	lambda_map: 51 start_index: 125 end_index: 113
44	Literal
45	ExprTuple	67, 52
46	ExprTuple	53, 66
47	Lambda	parameter: 122 body: 54
48	Literal
49	ExprTuple	55, 79
50	Lambda	parameter: 120 body: 56
51	Lambda	parameter: 107 body: 57
52	Literal
53	IndexedVar	variable: 110 index: 66
54	Operation	operator: 59 operands: 60
55	Literal
56	ExprRange	lambda_map: 61 start_index: 125 end_index: 62
57	ExprRange	lambda_map: 63 start_index: 64 end_index: 65
58	ExprTuple	66
59	Literal
60	ExprTuple	67, 68
61	Lambda	parameter: 122 body: 69
62	IndexedVar	variable: 86 index: 120
63	Lambda	parameter: 120 body: 70
64	Operation	operator: 116 operands: 71
65	IndexedVar	variable: 110 index: 107
66	Literal
67	IndexedVar	variable: 110 index: 122
68	Operation	operator: 116 operands: 73
69	Operation	operator: 75 operands: 74
70	Operation	operator: 75 operands: 76
71	ExprTuple	77, 125
72	ExprTuple	107
73	ExprTuple	78, 79
74	NamedExprs	element: 80 targets: 81
75	Literal
76	NamedExprs	element: 82 targets: 83
77	IndexedVar	variable: 110 index: 93
78	IndexedVar	variable: 110 index: 94
79	IndexedVar	variable: 86 index: 122
80	Operation	operator: 89 operands: 87
81	Operation	operator: 91 operands: 88
82	Operation	operator: 89 operands: 90
83	Operation	operator: 91 operands: 92
84	ExprTuple	93
85	ExprTuple	94
86	Variable
87	NamedExprs	state: 95 part: 122
88	ExprTuple	96, 97
89	Literal
90	NamedExprs	state: 98 part: 120
91	Literal
92	ExprTuple	125, 99
93	Operation	operator: 116 operands: 100
94	Operation	operator: 116 operands: 101
95	IndexedVar	variable: 118 index: 120
96	Operation	operator: 116 operands: 102
97	IndexedVar	variable: 110 index: 120
98	Operation	operator: 104 operands: 105
99	IndexedVar	variable: 110 index: 113
100	ExprTuple	107, 121
101	ExprTuple	122, 121
102	ExprTuple	108, 125
103	ExprTuple	120
104	Literal
105	ExprTuple	109
106	ExprTuple	113
107	Variable
108	IndexedVar	variable: 110 index: 114
109	ExprRange	lambda_map: 112 start_index: 125 end_index: 113
110	Variable
111	ExprTuple	114
112	Lambda	parameter: 122 body: 115
113	Variable
114	Operation	operator: 116 operands: 117
115	IndexedVar	variable: 118 index: 122
116	Literal
117	ExprTuple	120, 121
118	Variable
119	ExprTuple	122
120	Variable
121	Operation	operator: 123 operand: 125
122	Variable
123	Literal
124	ExprTuple	125
125	Literal