Expression of type Forall ¶

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, ExprRange, IndexedVar, N, Variable, VertExprArray, m, n
from proveit.core_expr_types import A_1_to_m, n_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import CartExp, Forall
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}")
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [n_1_to_m], instance_expr = 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, IndexedVar(n, sub_expr2))), one, m)]).with_wrapping(), domain = NaturalPos).with_wrapping(), domain = NaturalPos)

# 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())

\forall_{m \in \mathbb{N}^+}~\left[\begin{array}{l}\forall_{n_{1}, n_{2}, \ldots, n_{m} \in \mathbb{N}^+}~\\
\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}\right]\end{array}\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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',)

	core type	sub-expressions
0	Operation	operator: 24 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 134 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 24 operand: 8
5	Operation	operator: 65 operands: 7
6	ExprTuple	8
7	ExprTuple	134, 30
8	Lambda	parameters: 9 body: 10
9	ExprTuple	11
10	Conditional	value: 12 condition: 13
11	ExprRange	lambda_map: 14 start_index: 146 end_index: 134
12	Operation	operator: 24 operand: 17
13	Operation	operator: 52 operands: 16
14	Lambda	parameter: 143 body: 100
15	ExprTuple	17
16	ExprTuple	18
17	Lambda	parameters: 126 body: 19
18	ExprRange	lambda_map: 20 start_index: 146 end_index: 134
19	Conditional	value: 21 condition: 22
20	Lambda	parameter: 143 body: 23
21	Operation	operator: 24 operand: 28
22	Operation	operator: 52 operands: 26
23	Operation	operator: 65 operands: 27
24	Literal
25	ExprTuple	28
26	ExprTuple	29
27	ExprTuple	100, 30
28	Lambda	parameters: 31 body: 32
29	ExprRange	lambda_map: 33 start_index: 146 end_index: 134
30	Literal
31	ExprTuple	34
32	Conditional	value: 35 condition: 36
33	Lambda	parameter: 143 body: 37
34	ExprRange	lambda_map: 38 start_index: 87 end_index: 134
35	Operation	operator: 39 operands: 40
36	Operation	operator: 52 operands: 41
37	Operation	operator: 65 operands: 42
38	Lambda	parameter: 143 body: 88
39	Literal
40	ExprTuple	43, 44
41	ExprTuple	45, 46
42	ExprTuple	136, 47
43	Operation	operator: 49 operand: 56
44	Operation	operator: 49 operand: 57
45	ExprRange	lambda_map: 51 start_index: 87 end_index: 134
46	Operation	operator: 52 operands: 53
47	Operation	operator: 54 operands: 55
48	ExprTuple	56
49	Literal
50	ExprTuple	57
51	Lambda	parameter: 143 body: 58
52	Literal
53	ExprTuple	59, 60
54	Literal
55	ExprTuple	61, 62
56	ExprTuple	63
57	ExprTuple	64
58	Operation	operator: 65 operands: 66
59	Operation	operator: 80 operands: 67
60	ExprRange	lambda_map: 68 start_index: 146 end_index: 134
61	Literal
62	Operation	operator: 69 operands: 70
63	ExprRange	lambda_map: 71 start_index: 146 end_index: 134
64	ExprRange	lambda_map: 72 start_index: 146 end_index: 134
65	Literal
66	ExprTuple	88, 73
67	ExprTuple	74, 87
68	Lambda	parameter: 143 body: 75
69	Literal
70	ExprTuple	76, 100
71	Lambda	parameter: 141 body: 77
72	Lambda	parameter: 128 body: 78
73	Literal
74	IndexedVar	variable: 131 index: 87
75	Operation	operator: 80 operands: 81
76	Literal
77	ExprRange	lambda_map: 82 start_index: 146 end_index: 83
78	ExprRange	lambda_map: 84 start_index: 85 end_index: 86
79	ExprTuple	87
80	Literal
81	ExprTuple	88, 89
82	Lambda	parameter: 143 body: 90
83	IndexedVar	variable: 107 index: 141
84	Lambda	parameter: 141 body: 91
85	Operation	operator: 137 operands: 92
86	IndexedVar	variable: 131 index: 128
87	Literal
88	IndexedVar	variable: 131 index: 143
89	Operation	operator: 137 operands: 94
90	Operation	operator: 96 operands: 95
91	Operation	operator: 96 operands: 97
92	ExprTuple	98, 146
93	ExprTuple	128
94	ExprTuple	99, 100
95	NamedExprs	element: 101 targets: 102
96	Literal
97	NamedExprs	element: 103 targets: 104
98	IndexedVar	variable: 131 index: 114
99	IndexedVar	variable: 131 index: 115
100	IndexedVar	variable: 107 index: 143
101	Operation	operator: 110 operands: 108
102	Operation	operator: 112 operands: 109
103	Operation	operator: 110 operands: 111
104	Operation	operator: 112 operands: 113
105	ExprTuple	114
106	ExprTuple	115
107	Variable
108	NamedExprs	state: 116 part: 143
109	ExprTuple	117, 118
110	Literal
111	NamedExprs	state: 119 part: 141
112	Literal
113	ExprTuple	146, 120
114	Operation	operator: 137 operands: 121
115	Operation	operator: 137 operands: 122
116	IndexedVar	variable: 139 index: 141
117	Operation	operator: 137 operands: 123
118	IndexedVar	variable: 131 index: 141
119	Operation	operator: 125 operands: 126
120	IndexedVar	variable: 131 index: 134
121	ExprTuple	128, 142
122	ExprTuple	143, 142
123	ExprTuple	129, 146
124	ExprTuple	141
125	Literal
126	ExprTuple	130
127	ExprTuple	134
128	Variable
129	IndexedVar	variable: 131 index: 135
130	ExprRange	lambda_map: 133 start_index: 146 end_index: 134
131	Variable
132	ExprTuple	135
133	Lambda	parameter: 143 body: 136
134	Variable
135	Operation	operator: 137 operands: 138
136	IndexedVar	variable: 139 index: 143
137	Literal
138	ExprTuple	141, 142
139	Variable
140	ExprTuple	143
141	Variable
142	Operation	operator: 144 operand: 146
143	Variable
144	Literal
145	ExprTuple	146
146	Literal

Expression of type Forall¶

from the theory of proveit.physics.quantum.circuits¶

Expression of type Forall ¶

from the theory of proveit.physics.quantum.circuits ¶