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 k, l, m
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import Equals, Forall
from proveit.numbers import NaturalPos
from proveit.physics.quantum import l_ket_domain, m_ket_domain, var_ket_psi, var_ket_u, var_ket_v
from proveit.physics.quantum.circuits import circuit__u_Akl_v, circuit__u_Akl_v__psi_m
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [k, l, m], instance_expr = Forall(instance_param_or_params = [var_ket_u, var_ket_v], instance_expr = Forall(instance_param_or_params = [var_ket_psi], instance_expr = Forall(instance_param_or_params = [A11_to_Akl], instance_expr = Equals(Prob(circuit__u_Akl_v), Prob(circuit__u_Akl_v__psi_m)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain), domain = l_ket_domain), 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_{k, l, m \in \mathbb{N}^+}~\left[\forall_{\lvert u \rangle, \lvert v \rangle \in \mathbb{C}^{2^{l}}}~\left[\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\left[\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, l}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, l}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, l}}~\\
\left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle} \\
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle}
} \end{array}\right) \end{array} \end{array}\right)\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: 34 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	76, 132, 133
4	Conditional	value: 5 condition: 6
5	Operation	operator: 34 operand: 9
6	Operation	operator: 22 operands: 8
7	ExprTuple	9
8	ExprTuple	10, 11, 12
9	Lambda	parameters: 13 body: 14
10	Operation	operator: 36 operands: 15
11	Operation	operator: 36 operands: 16
12	Operation	operator: 36 operands: 17
13	ExprTuple	120, 123
14	Conditional	value: 18 condition: 19
15	ExprTuple	76, 20
16	ExprTuple	132, 20
17	ExprTuple	133, 20
18	Operation	operator: 34 operand: 24
19	Operation	operator: 22 operands: 23
20	Literal
21	ExprTuple	24
22	Literal
23	ExprTuple	25, 26
24	Lambda	parameter: 124 body: 28
25	Operation	operator: 36 operands: 29
26	Operation	operator: 36 operands: 30
27	ExprTuple	124
28	Conditional	value: 31 condition: 32
29	ExprTuple	120, 33
30	ExprTuple	123, 33
31	Operation	operator: 34 operand: 39
32	Operation	operator: 36 operands: 37
33	Operation	operator: 44 operands: 38
34	Literal
35	ExprTuple	39
36	Literal
37	ExprTuple	124, 40
38	ExprTuple	50, 41
39	Lambda	parameters: 42 body: 43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 55 operands: 46
42	ExprTuple	47
43	Operation	operator: 48 operands: 49
44	Literal
45	ExprTuple	50, 51
46	ExprTuple	60, 132
47	ExprRange	lambda_map: 52 start_index: 131 end_index: 76
48	Literal
49	ExprTuple	53, 54
50	Literal
51	Operation	operator: 55 operands: 56
52	Lambda	parameter: 121 body: 88
53	Operation	operator: 58 operand: 61
54	Operation	operator: 58 operand: 62
55	Literal
56	ExprTuple	60, 133
57	ExprTuple	61
58	Literal
59	ExprTuple	62
60	Literal
61	Operation	operator: 64 operands: 63
62	Operation	operator: 64 operands: 65
63	ExprTuple	66, 67, 68
64	Literal
65	ExprTuple	69, 70, 71
66	ExprTuple	73
67	ExprRange	lambda_map: 72 start_index: 131 end_index: 76
68	ExprTuple	77
69	ExprTuple	73, 74
70	ExprRange	lambda_map: 75 start_index: 131 end_index: 76
71	ExprTuple	77, 78
72	Lambda	parameter: 121 body: 79
73	ExprRange	lambda_map: 80 start_index: 131 end_index: 132
74	ExprRange	lambda_map: 81 start_index: 131 end_index: 133
75	Lambda	parameter: 121 body: 83
76	Variable
77	ExprRange	lambda_map: 84 start_index: 131 end_index: 132
78	ExprRange	lambda_map: 85 start_index: 131 end_index: 133
79	ExprTuple	88
80	Lambda	parameter: 125 body: 86
81	Lambda	parameter: 125 body: 87
82	ExprTuple	121
83	ExprTuple	88, 89
84	Lambda	parameter: 125 body: 90
85	Lambda	parameter: 125 body: 91
86	Operation	operator: 97 operands: 92
87	Operation	operator: 97 operands: 93
88	ExprRange	lambda_map: 94 start_index: 131 end_index: 132
89	ExprRange	lambda_map: 95 start_index: 131 end_index: 133
90	Operation	operator: 97 operands: 96
91	Operation	operator: 97 operands: 98
92	NamedExprs	element: 99 targets: 105
93	NamedExprs	element: 100 targets: 107
94	Lambda	parameter: 125 body: 101
95	Lambda	parameter: 125 body: 103
96	NamedExprs	element: 104 targets: 105
97	Literal
98	NamedExprs	element: 106 targets: 107
99	Operation	operator: 109 operands: 108
100	Operation	operator: 109 operands: 117
101	IndexedVar	variable: 110 indices: 111
102	ExprTuple	125
103	Operation	operator: 112 operands: 113
104	Operation	operator: 116 operands: 114
105	Operation	operator: 118 operands: 115
106	Operation	operator: 116 operands: 117
107	Operation	operator: 118 operands: 119
108	NamedExprs	state: 120 part: 125
109	Literal
110	Variable
111	ExprTuple	121, 125
112	Literal
113	NamedExprs	operation: 122
114	NamedExprs	state: 123 part: 125
115	ExprTuple	131, 132
116	Literal
117	NamedExprs	state: 124 part: 125
118	Literal
119	ExprTuple	126, 127
120	Variable
121	Variable
122	Literal
123	Variable
124	Variable
125	Variable
126	Operation	operator: 129 operands: 128
127	Operation	operator: 129 operands: 130
128	ExprTuple	132, 131
129	Literal
130	ExprTuple	132, 133
131	Literal
132	Variable
133	Variable