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 Lambda, m, n
from proveit.logic import Bijections, Forall, Implies, InSet, NotEquals, TRUE
from proveit.physics.quantum.QPE import _Omega, _m_domain, _phase_est_circuit

# build up the expression from sub-expressions
expr = Implies(Forall(instance_param_or_params = [m, n], instance_expr = TRUE, domain = _m_domain, condition = NotEquals(m, n)), InSet(Lambda(m, _phase_est_circuit), Bijections(_m_domain, _Omega)))

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[\forall_{m, n \in \{0~\ldotp \ldotp~2^{t} - 1\}~|~m \neq n}~\top\right] \Rightarrow \left(\left[m \mapsto \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \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 m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)\right] \in \left[\{0~\ldotp \ldotp~2^{t} - 1\} \xrightarrow[\text{onto}]{\text{1-to-1}} \Omega\right]\right)

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.	()	()	('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: 39 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameters: 42 body: 11
9	Lambda	parameter: 117 body: 13
10	Operation	operator: 14 operands: 15
11	Conditional	value: 16 condition: 17
12	ExprTuple	117
13	Operation	operator: 18 operands: 19
14	Literal
15	ExprTuple	50, 20
16	Literal
17	Operation	operator: 21 operands: 22
18	Literal
19	ExprTuple	23, 24, 25, 26
20	Literal
21	Literal
22	ExprTuple	27, 28, 29
23	ExprTuple	30, 31
24	ExprTuple	32, 33
25	ExprTuple	34, 35
26	ExprTuple	36, 37
27	Operation	operator: 39 operands: 38
28	Operation	operator: 39 operands: 40
29	Operation	operator: 41 operands: 42
30	ExprRange	lambda_map: 43 start_index: 118 end_index: 119
31	ExprRange	lambda_map: 44 start_index: 118 end_index: 120
32	ExprRange	lambda_map: 45 start_index: 118 end_index: 119
33	ExprRange	lambda_map: 45 start_index: 102 end_index: 103
34	ExprRange	lambda_map: 46 start_index: 118 end_index: 119
35	ExprRange	lambda_map: 47 start_index: 118 end_index: 120
36	ExprRange	lambda_map: 48 start_index: 118 end_index: 119
37	ExprRange	lambda_map: 49 start_index: 118 end_index: 120
38	ExprTuple	117, 50
39	Literal
40	ExprTuple	51, 50
41	Literal
42	ExprTuple	117, 51
43	Lambda	parameter: 101 body: 52
44	Lambda	parameter: 101 body: 53
45	Lambda	parameter: 101 body: 54
46	Lambda	parameter: 101 body: 55
47	Lambda	parameter: 101 body: 56
48	Lambda	parameter: 101 body: 57
49	Lambda	parameter: 101 body: 59
50	Operation	operator: 93 operands: 60
51	Variable
52	Operation	operator: 85 operands: 61
53	Operation	operator: 68 operands: 62
54	Operation	operator: 68 operands: 63
55	Operation	operator: 64 operands: 65
56	Operation	operator: 86 operands: 66
57	Operation	operator: 68 operands: 67
58	ExprTuple	101
59	Operation	operator: 68 operands: 69
60	ExprTuple	70, 71
61	NamedExprs	state: 72
62	NamedExprs	element: 73 targets: 81
63	NamedExprs	element: 74 targets: 75
64	Literal
65	NamedExprs	basis: 76
66	NamedExprs	operation: 77
67	NamedExprs	element: 78 targets: 79
68	Literal
69	NamedExprs	element: 80 targets: 81
70	Literal
71	Operation	operator: 113 operands: 82
72	Operation	operator: 83 operand: 97
73	Operation	operator: 85 operands: 92
74	Operation	operator: 86 operands: 87
75	Operation	operator: 93 operands: 88
76	Literal
77	Literal
78	Operation	operator: 91 operands: 89
79	Operation	operator: 93 operands: 90
80	Operation	operator: 91 operands: 92
81	Operation	operator: 93 operands: 94
82	ExprTuple	95, 96
83	Literal
84	ExprTuple	97
85	Literal
86	Literal
87	NamedExprs	operation: 98 part: 101
88	ExprTuple	118, 103
89	NamedExprs	state: 99 part: 101
90	ExprTuple	118, 119
91	Literal
92	NamedExprs	state: 100 part: 101
93	Literal
94	ExprTuple	102, 103
95	Operation	operator: 104 operands: 105
96	Operation	operator: 106 operand: 118
97	Literal
98	Operation	operator: 108 operands: 109
99	Operation	operator: 110 operands: 111
100	Literal
101	Variable
102	Operation	operator: 113 operands: 112
103	Operation	operator: 113 operands: 114
104	Literal
105	ExprTuple	115, 119
106	Literal
107	ExprTuple	118
108	Literal
109	ExprTuple	116, 119
110	Literal
111	ExprTuple	117, 119
112	ExprTuple	119, 118
113	Literal
114	ExprTuple	119, 120
115	Literal
116	Literal
117	Variable
118	Literal
119	Literal
120	Literal