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
from proveit.linear_algebra import TensorProd
from proveit.logic import Forall
from proveit.numbers import Add, Interval, Natural, one, subtract
from proveit.physics.quantum.circuits import Input, MultiQubitElem, N_0_to_m, 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 = 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 = 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)]))), domain = Natural, condition = each_Nk_is_partial_sum).with_wrapping()

# 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())

\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{
\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)\right)\end{array}

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