Expression of type Lambda

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 Conditional, Lambda, U, m
from proveit.linear_algebra import Unitary
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Exp, NaturalPos, one, two
from proveit.physics.quantum import m_ket_domain, var_ket_psi
from proveit.physics.quantum.circuits import unitary_gate_op
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Lambda(m, Conditional(Forall(instance_param_or_params = [U, var_ket_psi], instance_expr = Equals(Prob(unitary_gate_op), one), domains = [Unitary(Exp(two, m)), m_ket_domain]), InSet(m, 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())

m \mapsto \left\{\forall_{U \in \textrm{U}\left(2^{m}\right), \lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi \rangle} & \gate{U} & \qout{U \thinspace \lvert \psi \rangle}
} \end{array}\right) = 1\right) \textrm{ if } m \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 71 body: 2
1	ExprTuple	71
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 23 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	71, 9
8	Lambda	parameters: 73 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operands: 14
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17, 70
15	Literal
16	ExprTuple	18, 19
17	Operation	operator: 20 operand: 25
18	Operation	operator: 23 operands: 22
19	Operation	operator: 23 operands: 24
20	Literal
21	ExprTuple	25
22	ExprTuple	74, 26
23	Literal
24	ExprTuple	75, 27
25	Operation	operator: 28 operands: 29
26	Operation	operator: 30 operand: 38
27	Operation	operator: 32 operands: 33
28	Literal
29	ExprTuple	34, 35, 36
30	Literal
31	ExprTuple	38
32	Literal
33	ExprTuple	37, 38
34	ExprTuple	39
35	ExprTuple	40
36	ExprTuple	41
37	Literal
38	Operation	operator: 42 operands: 43
39	ExprRange	lambda_map: 44 start_index: 70 end_index: 71
40	ExprRange	lambda_map: 45 start_index: 70 end_index: 71
41	ExprRange	lambda_map: 46 start_index: 70 end_index: 71
42	Literal
43	ExprTuple	47, 71
44	Lambda	parameter: 69 body: 48
45	Lambda	parameter: 69 body: 49
46	Lambda	parameter: 69 body: 51
47	Literal
48	Operation	operator: 54 operands: 52
49	Operation	operator: 54 operands: 53
50	ExprTuple	69
51	Operation	operator: 54 operands: 55
52	NamedExprs	element: 56 targets: 59
53	NamedExprs	element: 57 targets: 59
54	Literal
55	NamedExprs	element: 58 targets: 59
56	Operation	operator: 60 operands: 61
57	Operation	operator: 62 operands: 63
58	Operation	operator: 64 operands: 65
59	Operation	operator: 66 operands: 67
60	Literal
61	NamedExprs	state: 75 part: 69
62	Literal
63	NamedExprs	operation: 74 part: 69
64	Literal
65	NamedExprs	state: 68 part: 69
66	Literal
67	ExprTuple	70, 71
68	Operation	operator: 72 operands: 73
69	Variable
70	Literal
71	Variable
72	Literal
73	ExprTuple	74, 75
74	Variable
75	Variable