Expression of type Implies

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 MatrixMult, ScalarMult
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import Add, Exp, Interval, IntervalCO, Mult, NaturalPos, Real, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import I, NumKet, Z, ket_plus, var_ket_u
from proveit.physics.quantum.QPE import QPE, phase, two_pow_t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [s, t, U, var_ket_u, phase]
sub_expr3 = Add(t, one)
sub_expr4 = Add(t, s)
sub_expr5 = InSet(phase, Real)
sub_expr6 = InSet(t, NaturalPos)
sub_expr7 = Interval(sub_expr3, sub_expr4)
sub_expr8 = Mult(two_pow_t, phase)
sub_expr9 = MultiQubitElem(element = Gate(operation = QPE(U, t), part = sub_expr1), targets = Interval(one, sub_expr4))
sub_expr10 = InSet(sub_expr8, Interval(zero, subtract(two_pow_t, one)))
sub_expr11 = Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u))
sub_expr12 = Equals(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = var_ket_u, part = sub_expr1), targets = sub_expr7), one, s)], [ExprRange(sub_expr1, sub_expr9, one, t), ExprRange(sub_expr1, sub_expr9, sub_expr3, sub_expr4)], [ExprRange(sub_expr1, Measure(basis = Z), one, t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, s)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(sub_expr8, t), part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = var_ket_u, part = sub_expr1), targets = sub_expr7), one, s)]))), one)
sub_expr13 = Conditional(sub_expr12, And(sub_expr5, sub_expr10, sub_expr11, InSet(phase, IntervalCO(zero, one))))
sub_expr14 = Conditional(sub_expr12, And(sub_expr5, sub_expr10, sub_expr11))
expr = Implies(Forall(instance_param_or_params = sub_expr2, instance_expr = Equals(sub_expr13, sub_expr14), condition = sub_expr6), Equals(Lambda(sub_expr2, Conditional(sub_expr13, sub_expr6)), Lambda(sub_expr2, Conditional(sub_expr14, sub_expr6))).with_wrapping_at(2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\forall_{s, t, U, \lvert u \rangle, \varphi~|~t \in \mathbb{N}^+}~\left(\left\{\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = 1 \textrm{ if } \varphi \in \mathbb{R} ,  \left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 1\} ,  \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right) ,  \varphi \in \left[0,1\right)\right.. = \left\{\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = 1 \textrm{ if } \varphi \in \mathbb{R} ,  \left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 1\} ,  \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)\right..\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\left(s, t, U, \lvert u \rangle, \varphi\right) \mapsto \left\{\left\{\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = 1 \textrm{ if } \varphi \in \mathbb{R} ,  \left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 1\} ,  \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right) ,  \varphi \in \left[0,1\right)\right.. \textrm{ if } t \in \mathbb{N}^+\right..\right] =  \\ \left[\left(s, t, U, \lvert u \rangle, \varphi\right) \mapsto \left\{\left\{\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = 1 \textrm{ if } \varphi \in \mathbb{R} ,  \left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 1\} ,  \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)\right.. \textrm{ if } t \in \mathbb{N}^+\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 40 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 13 body: 11
9	Lambda	parameters: 13 body: 12
10	Lambda	parameters: 13 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 19 condition: 16
13	ExprTuple	145, 153, 142, 131, 149
14	Conditional	value: 20 condition: 16
15	Operation	operator: 40 operands: 17
16	Operation	operator: 38 operands: 18
17	ExprTuple	19, 20
18	ExprTuple	153, 21
19	Conditional	value: 23 condition: 22
20	Conditional	value: 23 condition: 24
21	Literal
22	Operation	operator: 27 operands: 25
23	Operation	operator: 40 operands: 26
24	Operation	operator: 27 operands: 28
25	ExprTuple	31, 32, 33, 29
26	ExprTuple	30, 144
27	Literal
28	ExprTuple	31, 32, 33
29	Operation	operator: 38 operands: 34
30	Operation	operator: 35 operand: 43
31	Operation	operator: 38 operands: 37
32	Operation	operator: 38 operands: 39
33	Operation	operator: 40 operands: 41
34	ExprTuple	149, 42
35	Literal
36	ExprTuple	43
37	ExprTuple	149, 44
38	Literal
39	ExprTuple	143, 45
40	Literal
41	ExprTuple	46, 47
42	Operation	operator: 48 operands: 49
43	Operation	operator: 50 operands: 51
44	Literal
45	Operation	operator: 126 operands: 52
46	Operation	operator: 53 operands: 54
47	Operation	operator: 55 operands: 56
48	Literal
49	ExprTuple	61, 144
50	Literal
51	ExprTuple	57, 58, 59, 60
52	ExprTuple	61, 62
53	Literal
54	ExprTuple	142, 131
55	Literal
56	ExprTuple	63, 131
57	ExprTuple	64, 65
58	ExprTuple	66, 67
59	ExprTuple	68, 69
60	ExprTuple	70, 71
61	Literal
62	Operation	operator: 140 operands: 72
63	Operation	operator: 150 operands: 73
64	ExprRange	lambda_map: 74 start_index: 144 end_index: 153
65	ExprRange	lambda_map: 75 start_index: 144 end_index: 145
66	ExprRange	lambda_map: 76 start_index: 144 end_index: 153
67	ExprRange	lambda_map: 76 start_index: 133 end_index: 134
68	ExprRange	lambda_map: 77 start_index: 144 end_index: 153
69	ExprRange	lambda_map: 78 start_index: 144 end_index: 145
70	ExprRange	lambda_map: 79 start_index: 144 end_index: 153
71	ExprRange	lambda_map: 80 start_index: 144 end_index: 145
72	ExprTuple	148, 81
73	ExprTuple	82, 83
74	Lambda	parameter: 132 body: 84
75	Lambda	parameter: 132 body: 85
76	Lambda	parameter: 132 body: 86
77	Lambda	parameter: 132 body: 87
78	Lambda	parameter: 132 body: 88
79	Lambda	parameter: 132 body: 89
80	Lambda	parameter: 132 body: 91
81	Operation	operator: 92 operand: 144
82	Literal
83	Operation	operator: 146 operands: 94
84	Operation	operator: 118 operands: 95
85	Operation	operator: 102 operands: 96
86	Operation	operator: 102 operands: 97
87	Operation	operator: 98 operands: 99
88	Operation	operator: 119 operands: 100
89	Operation	operator: 102 operands: 101
90	ExprTuple	132
91	Operation	operator: 102 operands: 103
92	Literal
93	ExprTuple	144
94	ExprTuple	152, 104, 105, 149
95	NamedExprs	state: 106
96	NamedExprs	element: 107 targets: 115
97	NamedExprs	element: 108 targets: 109
98	Literal
99	NamedExprs	basis: 110
100	NamedExprs	operation: 111
101	NamedExprs	element: 112 targets: 113
102	Literal
103	NamedExprs	element: 114 targets: 115
104	Literal
105	Literal
106	Operation	operator: 116 operand: 128
107	Operation	operator: 118 operands: 125
108	Operation	operator: 119 operands: 120
109	Operation	operator: 126 operands: 121
110	Literal
111	Literal
112	Operation	operator: 124 operands: 122
113	Operation	operator: 126 operands: 123
114	Operation	operator: 124 operands: 125
115	Operation	operator: 126 operands: 127
116	Literal
117	ExprTuple	128
118	Literal
119	Literal
120	NamedExprs	operation: 129 part: 132
121	ExprTuple	144, 134
122	NamedExprs	state: 130 part: 132
123	ExprTuple	144, 153
124	Literal
125	NamedExprs	state: 131 part: 132
126	Literal
127	ExprTuple	133, 134
128	Literal
129	Operation	operator: 135 operands: 136
130	Operation	operator: 137 operands: 138
131	Variable
132	Variable
133	Operation	operator: 140 operands: 139
134	Operation	operator: 140 operands: 141
135	Literal
136	ExprTuple	142, 153
137	Literal
138	ExprTuple	143, 153
139	ExprTuple	153, 144
140	Literal
141	ExprTuple	153, 145
142	Variable
143	Operation	operator: 146 operands: 147
144	Literal
145	Variable
146	Literal
147	ExprTuple	148, 149
148	Operation	operator: 150 operands: 151
149	Variable
150	Literal
151	ExprTuple	152, 153
152	Literal
153	Variable