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 U
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Mult, e, i, one, pi, two
from proveit.physics.quantum import Qmult, var_ket_u, varphi
from proveit.physics.quantum.circuits import phase_kickback_circuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [varphi], instance_expr = Equals(Prob(phase_kickback_circuit), one), condition = Equals(Qmult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, varphi)), var_ket_u)))

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_{\varphi~|~\left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)}~\left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \control{} \qw \qwx[1] & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \gate{U} & \qout{\lvert u \rangle}
} \end{array}\right) = 1\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 104 body: 5
4	ExprTuple	104
5	Conditional	value: 6 condition: 7
6	Operation	operator: 9 operands: 8
7	Operation	operator: 9 operands: 10
8	ExprTuple	11, 98
9	Literal
10	ExprTuple	12, 13
11	Operation	operator: 14 operand: 19
12	Operation	operator: 16 operands: 17
13	Operation	operator: 81 operands: 18
14	Literal
15	ExprTuple	19
16	Literal
17	ExprTuple	72, 76
18	ExprTuple	87, 76
19	Operation	operator: 20 operands: 21
20	Literal
21	ExprTuple	22, 23, 24
22	ExprTuple	25, 26
23	ExprTuple	27, 28
24	ExprTuple	29, 30
25	Operation	operator: 62 operands: 31
26	ExprRange	lambda_map: 32 start_index: 98 end_index: 37
27	Operation	operator: 53 operands: 33
28	ExprRange	lambda_map: 34 start_index: 98 end_index: 37
29	Operation	operator: 68 operands: 35
30	ExprRange	lambda_map: 36 start_index: 98 end_index: 37
31	NamedExprs	state: 38
32	Lambda	parameter: 77 body: 39
33	NamedExprs	element: 40 targets: 41
34	Lambda	parameter: 77 body: 42
35	NamedExprs	state: 43
36	Lambda	parameter: 77 body: 45
37	Operation	operator: 83 operands: 46
38	Operation	operator: 94 operand: 55
39	Operation	operator: 53 operands: 48
40	Literal
41	Operation	operator: 49 operand: 101
42	Operation	operator: 53 operands: 51
43	Operation	operator: 81 operands: 52
44	ExprTuple	77
45	Operation	operator: 53 operands: 54
46	ExprTuple	89, 98
47	ExprTuple	55
48	NamedExprs	element: 56 targets: 61
49	Literal
50	ExprTuple	101
51	NamedExprs	element: 57 targets: 61
52	ExprTuple	58, 59
53	Literal
54	NamedExprs	element: 60 targets: 61
55	Literal
56	Operation	operator: 62 operands: 69
57	Operation	operator: 63 operands: 64
58	Operation	operator: 90 operands: 65
59	Operation	operator: 66 operands: 67
60	Operation	operator: 68 operands: 69
61	Operation	operator: 70 operands: 71
62	Literal
63	Literal
64	NamedExprs	operation: 72 part: 77
65	ExprTuple	98, 73
66	Literal
67	ExprTuple	74, 75
68	Literal
69	NamedExprs	state: 76 part: 77
70	Literal
71	ExprTuple	101, 78
72	Variable
73	Operation	operator: 92 operands: 79
74	Operation	operator: 94 operand: 86
75	Operation	operator: 81 operands: 82
76	Variable
77	Variable
78	Operation	operator: 83 operands: 84
79	ExprTuple	101, 85
80	ExprTuple	86
81	Literal
82	ExprTuple	87, 88
83	Literal
84	ExprTuple	89, 101
85	Operation	operator: 90 operands: 91
86	Literal
87	Operation	operator: 92 operands: 93
88	Operation	operator: 94 operand: 98
89	Variable
90	Literal
91	ExprTuple	98, 101
92	Literal
93	ExprTuple	96, 97
94	Literal
95	ExprTuple	98
96	Literal
97	Operation	operator: 99 operands: 100
98	Literal
99	Literal
100	ExprTuple	101, 102, 103, 104
101	Literal
102	Literal
103	Literal
104	Variable