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, m
from proveit.linear_algebra import Unitary
from proveit.logic import Equals, Forall
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 = Forall(instance_param_or_params = [m], instance_expr = 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]), 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_{m \in \mathbb{N}^+}~\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)\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 73 body: 4
3	ExprTuple	73
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 25 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	73, 11
10	Lambda	parameters: 75 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 72
17	Literal
18	ExprTuple	20, 21
19	Operation	operator: 22 operand: 27
20	Operation	operator: 25 operands: 24
21	Operation	operator: 25 operands: 26
22	Literal
23	ExprTuple	27
24	ExprTuple	76, 28
25	Literal
26	ExprTuple	77, 29
27	Operation	operator: 30 operands: 31
28	Operation	operator: 32 operand: 40
29	Operation	operator: 34 operands: 35
30	Literal
31	ExprTuple	36, 37, 38
32	Literal
33	ExprTuple	40
34	Literal
35	ExprTuple	39, 40
36	ExprTuple	41
37	ExprTuple	42
38	ExprTuple	43
39	Literal
40	Operation	operator: 44 operands: 45
41	ExprRange	lambda_map: 46 start_index: 72 end_index: 73
42	ExprRange	lambda_map: 47 start_index: 72 end_index: 73
43	ExprRange	lambda_map: 48 start_index: 72 end_index: 73
44	Literal
45	ExprTuple	49, 73
46	Lambda	parameter: 71 body: 50
47	Lambda	parameter: 71 body: 51
48	Lambda	parameter: 71 body: 53
49	Literal
50	Operation	operator: 56 operands: 54
51	Operation	operator: 56 operands: 55
52	ExprTuple	71
53	Operation	operator: 56 operands: 57
54	NamedExprs	element: 58 targets: 61
55	NamedExprs	element: 59 targets: 61
56	Literal
57	NamedExprs	element: 60 targets: 61
58	Operation	operator: 62 operands: 63
59	Operation	operator: 64 operands: 65
60	Operation	operator: 66 operands: 67
61	Operation	operator: 68 operands: 69
62	Literal
63	NamedExprs	state: 77 part: 71
64	Literal
65	NamedExprs	operation: 76 part: 71
66	Literal
67	NamedExprs	state: 70 part: 71
68	Literal
69	ExprTuple	72, 73
70	Operation	operator: 74 operands: 75
71	Variable
72	Literal
73	Variable
74	Literal
75	ExprTuple	76, 77
76	Variable
77	Variable