Theorem trivial_expansion_above of type Forall

from the theory of proveit.physics.quantum.circuits

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.physics.quantum.circuits import trivial_expansion_above

# check that the built expression is the same as the stored expression
assert trivial_expansion_above.expr == stored_expr
assert trivial_expansion_above.expr._style_id == stored_expr._style_id
print("Passed sanity check: trivial_expansion_above matches stored_expr")

Passed sanity check: trivial_expansion_above 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{
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle} \\
\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) \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	78, 143, 142
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	130, 134
14	Conditional	value: 18 condition: 19
15	ExprTuple	78, 20
16	ExprTuple	143, 20
17	ExprTuple	142, 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: 133 body: 28
25	Operation	operator: 36 operands: 29
26	Operation	operator: 36 operands: 30
27	ExprTuple	133
28	Conditional	value: 31 condition: 32
29	ExprTuple	130, 33
30	ExprTuple	134, 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	133, 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, 143
47	ExprRange	lambda_map: 52 start_index: 141 end_index: 78
48	Literal
49	ExprTuple	53, 54
50	Literal
51	Operation	operator: 55 operands: 56
52	Lambda	parameter: 132 body: 95
53	Operation	operator: 58 operand: 61
54	Operation	operator: 58 operand: 62
55	Literal
56	ExprTuple	60, 142
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	72
67	ExprRange	lambda_map: 73 start_index: 141 end_index: 78
68	ExprTuple	74
69	ExprTuple	75, 76
70	ExprRange	lambda_map: 77 start_index: 141 end_index: 78
71	ExprTuple	79, 80
72	ExprRange	lambda_map: 81 start_index: 141 end_index: 143
73	Lambda	parameter: 132 body: 82
74	ExprRange	lambda_map: 83 start_index: 141 end_index: 143
75	ExprRange	lambda_map: 84 start_index: 141 end_index: 142
76	ExprRange	lambda_map: 85 start_index: 141 end_index: 143
77	Lambda	parameter: 132 body: 87
78	Variable
79	ExprRange	lambda_map: 88 start_index: 141 end_index: 142
80	ExprRange	lambda_map: 89 start_index: 141 end_index: 143
81	Lambda	parameter: 135 body: 90
82	ExprTuple	95
83	Lambda	parameter: 135 body: 91
84	Lambda	parameter: 135 body: 92
85	Lambda	parameter: 135 body: 93
86	ExprTuple	132
87	ExprTuple	94, 95
88	Lambda	parameter: 135 body: 96
89	Lambda	parameter: 135 body: 97
90	Operation	operator: 105 operands: 98
91	Operation	operator: 105 operands: 99
92	Operation	operator: 105 operands: 100
93	Operation	operator: 105 operands: 101
94	ExprRange	lambda_map: 102 start_index: 141 end_index: 142
95	ExprRange	lambda_map: 103 start_index: 141 end_index: 143
96	Operation	operator: 105 operands: 104
97	Operation	operator: 105 operands: 106
98	NamedExprs	element: 109 targets: 107
99	NamedExprs	element: 115 targets: 107
100	NamedExprs	element: 108 targets: 114
101	NamedExprs	element: 109 targets: 116
102	Lambda	parameter: 135 body: 110
103	Lambda	parameter: 135 body: 112
104	NamedExprs	element: 113 targets: 114
105	Literal
106	NamedExprs	element: 115 targets: 116
107	Operation	operator: 128 operands: 117
108	Operation	operator: 118 operands: 124
109	Operation	operator: 118 operands: 119
110	Operation	operator: 120 operands: 121
111	ExprTuple	135
112	IndexedVar	variable: 122 indices: 123
113	Operation	operator: 126 operands: 124
114	Operation	operator: 128 operands: 125
115	Operation	operator: 126 operands: 127
116	Operation	operator: 128 operands: 129
117	ExprTuple	141, 143
118	Literal
119	NamedExprs	state: 130 part: 135
120	Literal
121	NamedExprs	operation: 131
122	Variable
123	ExprTuple	132, 135
124	NamedExprs	state: 133 part: 135
125	ExprTuple	141, 142
126	Literal
127	NamedExprs	state: 134 part: 135
128	Literal
129	ExprTuple	136, 137
130	Variable
131	Literal
132	Variable
133	Variable
134	Variable
135	Variable
136	Operation	operator: 139 operands: 138
137	Operation	operator: 139 operands: 140
138	ExprTuple	142, 141
139	Literal
140	ExprTuple	142, 143
141	Literal
142	Variable
143	Variable