Expression of type Forall

from the theory of proveit.physics.quantum.QPE

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 ExprRange, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import Unitary
from proveit.logic import Forall
from proveit.numbers import Add, Exp, Interval, NaturalPos, one, two
from proveit.physics.quantum import I
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import QPE, QPE1
from proveit.physics.quantum.circuits import Gate, MultiQubitElem, Qcircuit, QcircuitEquiv

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(t, s)
sub_expr3 = Add(t, one)
sub_expr4 = Interval(one, sub_expr2)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE(U, t), part = sub_expr1), targets = sub_expr4)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(U, t), part = sub_expr1), targets = sub_expr4)
expr = Forall(instance_param_or_params = [s, t], instance_expr = Forall(instance_param_or_params = [U], instance_expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, sub_expr5, one, t), ExprRange(sub_expr1, sub_expr5, sub_expr3, sub_expr2)])), Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, sub_expr6, one, t), ExprRange(sub_expr1, sub_expr6, sub_expr3, sub_expr2)], [ExprRange(sub_expr1, MultiQubitElem(element = Gate(operation = InverseFourierTransform(t), part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, s)]))), domain = Unitary(Exp(two, s))), domain = NaturalPos)

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())

\forall_{s, t \in \mathbb{N}^+}~\left[\forall_{U \in \textrm{U}\left(2^{s}\right)}~\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multigate{1}{\textrm{QPE}\left(U, t\right)} & { /^{t} } \qw \\
& \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multigate{1}{\textrm{QPE}_1\left(U, t\right)} & \gate{{\mathrm {FT}}^{\dag}_{t}} & { /^{t} } \qw \\
& \ghost{\textrm{QPE}_1\left(U, t\right)} & { /^{s} } \qw & { /^{s} } \qw
} \end{array}\right)\right)\right]

stored_expr.style_options()

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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	89, 90
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12, 13
11	Lambda	parameter: 88 body: 15
12	Operation	operator: 23 operands: 16
13	Operation	operator: 23 operands: 17
14	ExprTuple	88
15	Conditional	value: 18 condition: 19
16	ExprTuple	89, 20
17	ExprTuple	90, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Literal
21	Literal
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	88, 27
25	Operation	operator: 29 operand: 33
26	Operation	operator: 29 operands: 30
27	Operation	operator: 31 operand: 36
28	ExprTuple	33
29	Literal
30	ExprTuple	34, 35
31	Literal
32	ExprTuple	36
33	ExprTuple	37, 38
34	ExprTuple	39, 40
35	ExprTuple	41, 42
36	Operation	operator: 43 operands: 44
37	ExprRange	lambda_map: 45 start_index: 80 end_index: 90
38	ExprRange	lambda_map: 45 start_index: 47 end_index: 77
39	ExprRange	lambda_map: 46 start_index: 80 end_index: 90
40	ExprRange	lambda_map: 46 start_index: 47 end_index: 77
41	ExprRange	lambda_map: 48 start_index: 80 end_index: 90
42	ExprRange	lambda_map: 49 start_index: 80 end_index: 89
43	Literal
44	ExprTuple	50, 89
45	Lambda	parameter: 79 body: 51
46	Lambda	parameter: 79 body: 52
47	Operation	operator: 84 operands: 53
48	Lambda	parameter: 79 body: 54
49	Lambda	parameter: 79 body: 56
50	Literal
51	Operation	operator: 59 operands: 57
52	Operation	operator: 59 operands: 58
53	ExprTuple	90, 80
54	Operation	operator: 59 operands: 60
55	ExprTuple	79
56	Operation	operator: 71 operands: 61
57	NamedExprs	element: 62 targets: 64
58	NamedExprs	element: 63 targets: 64
59	Literal
60	NamedExprs	element: 65 targets: 66
61	NamedExprs	operation: 67
62	Operation	operator: 71 operands: 68
63	Operation	operator: 71 operands: 69
64	Operation	operator: 73 operands: 70
65	Operation	operator: 71 operands: 72
66	Operation	operator: 73 operands: 74
67	Literal
68	NamedExprs	operation: 75 part: 79
69	NamedExprs	operation: 76 part: 79
70	ExprTuple	80, 77
71	Literal
72	NamedExprs	operation: 78 part: 79
73	Literal
74	ExprTuple	80, 90
75	Operation	operator: 81 operands: 83
76	Operation	operator: 82 operands: 83
77	Operation	operator: 84 operands: 85
78	Operation	operator: 86 operand: 90
79	Variable
80	Literal
81	Literal
82	Literal
83	ExprTuple	88, 90
84	Literal
85	ExprTuple	90, 89
86	Literal
87	ExprTuple	90
88	Variable
89	Variable
90	Variable