Expression of type Lambda

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 Conditional, ExprRange, Lambda, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import Unitary
from proveit.logic import And, Forall, InSet
from proveit.numbers import Add, Exp, Interval, NaturalPos, one, two
from proveit.physics.quantum.QPE import QPE1, QPE1_U_t_circuit
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 = MultiQubitElem(element = Gate(operation = QPE1(U, t), part = sub_expr1), targets = Interval(one, sub_expr2))
expr = Lambda([s, t], Conditional(Forall(instance_param_or_params = [U], instance_expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, sub_expr3, one, t), ExprRange(sub_expr1, sub_expr3, Add(t, one), sub_expr2)])), QPE1_U_t_circuit), domain = Unitary(Exp(two, s))), And(InSet(s, NaturalPos), InSet(t, 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())

\left(s, t\right) \mapsto \left\{\forall_{U \in \textrm{U}\left(2^{s}\right)}~\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multigate{1}{\textrm{QPE}_1\left(U, t\right)} & { /^{t} } \qw \\
& \ghost{\textrm{QPE}_1\left(U, t\right)} & { /^{s} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \control{} \qw \qwx[1] & \qw & \gate{\cdots} \qwx[1] & \qw & \qw \\
& \qw \qwx[1] & \control{} \qw \qwx[1] & \gate{\cdots} \qwx[1] & \qw & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} & \qw \\
& \qw \qwx[1] & \qw \qwx[1] & \gate{\cdots} \qwx[1] & \control{} \qw \qwx[1] & \qw \\
& \gate{U^{2^{t - 1}}} & \gate{U^{2^{t - 2}}} & \gate{\cdots} & \gate{U^{2^{0}}} & { /^{s} } \qw
} \end{array}\right)\right) \textrm{ if } s \in \mathbb{N}^+ ,  t \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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