Expression of type ExprTuple

from the theory of proveit.physics.quantum.QPE

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 ExprTuple, U, eps, n, s, t
from proveit.linear_algebra import MatrixMult, ScalarMult, Unitary
from proveit.logic import Equals, InSet
from proveit.numbers import Add, Ceil, Exp, IntervalCO, IntervalOC, Log, Mult, NaturalPos, Real, e, frac, greater_eq, i, one, pi, two, zero
from proveit.physics.quantum import normalized_var_ket_u, var_ket_u
from proveit.physics.quantum.QPE import phase, s_ket_domain, two_pow_s

# build up the expression from sub-expressions
expr = ExprTuple(InSet(eps, IntervalOC(zero, one)), InSet(s, NaturalPos), InSet(n, NaturalPos), greater_eq(n, two), InSet(U, Unitary(two_pow_s)), InSet(var_ket_u, s_ket_domain), InSet(phase, Real), InSet(phase, IntervalCO(zero, one)), normalized_var_ket_u, Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u)), InSet(t, NaturalPos), greater_eq(t, Add(n, Ceil(Log(two, Add(two, frac(one, Mult(two, eps))))))))

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())

\left(\epsilon \in \left(0,1\right], s \in \mathbb{N}^+, n \in \mathbb{N}^+, n \geq 2, U \in \textrm{U}\left(2^{s}\right), \lvert u \rangle \in \mathbb{C}^{2^{s}}, \varphi \in \mathbb{R}, \varphi \in \left[0,1\right), \left \|\lvert u \rangle\right \| = 1, \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right), t \in \mathbb{N}^+, t \geq \left(n + \left\lceil \textrm{log}_2\left(2 + \frac{1}{2 \cdot \epsilon}\right)\right\rceil\right)\right)

stored_expr.style_options()

name	description	default	current value	related methods
wrap_positions	position(s) at which wrapping is to occur; 'n' is after the nth comma.	()	()	('with_wrapping_at',)
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	left	left	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
1	Operation	operator: 24 operands: 13
2	Operation	operator: 24 operands: 14
3	Operation	operator: 24 operands: 15
4	Operation	operator: 26 operands: 16
5	Operation	operator: 24 operands: 17
6	Operation	operator: 24 operands: 18
7	Operation	operator: 24 operands: 19
8	Operation	operator: 24 operands: 20
9	Operation	operator: 22 operands: 21
10	Operation	operator: 22 operands: 23
11	Operation	operator: 24 operands: 25
12	Operation	operator: 26 operands: 27
13	ExprTuple	87, 28
14	ExprTuple	66, 36
15	ExprTuple	59, 36
16	ExprTuple	86, 59
17	ExprTuple	56, 29
18	ExprTuple	58, 30
19	ExprTuple	75, 31
20	ExprTuple	75, 32
21	ExprTuple	33, 82
22	Literal
23	ExprTuple	34, 35
24	Literal
25	ExprTuple	38, 36
26	Literal
27	ExprTuple	37, 38
28	Operation	operator: 39 operands: 45
29	Operation	operator: 40 operand: 54
30	Operation	operator: 42 operands: 43
31	Literal
32	Operation	operator: 44 operands: 45
33	Operation	operator: 46 operand: 58
34	Operation	operator: 48 operands: 49
35	Operation	operator: 50 operands: 51
36	Literal
37	Operation	operator: 77 operands: 52
38	Variable
39	Literal
40	Literal
41	ExprTuple	54
42	Literal
43	ExprTuple	53, 54
44	Literal
45	ExprTuple	55, 82
46	Literal
47	ExprTuple	58
48	Literal
49	ExprTuple	56, 58
50	Literal
51	ExprTuple	57, 58
52	ExprTuple	59, 60
53	Literal
54	Operation	operator: 62 operands: 61
55	Literal
56	Variable
57	Operation	operator: 62 operands: 63
58	Variable
59	Variable
60	Operation	operator: 64 operand: 69
61	ExprTuple	86, 66
62	Literal
63	ExprTuple	67, 68
64	Literal
65	ExprTuple	69
66	Variable
67	Literal
68	Operation	operator: 84 operands: 70
69	Operation	operator: 71 operands: 72
70	ExprTuple	86, 73, 74, 75
71	Literal
72	ExprTuple	86, 76
73	Literal
74	Literal
75	Variable
76	Operation	operator: 77 operands: 78
77	Literal
78	ExprTuple	86, 79
79	Operation	operator: 80 operands: 81
80	Literal
81	ExprTuple	82, 83
82	Literal
83	Operation	operator: 84 operands: 85
84	Literal
85	ExprTuple	86, 87
86	Literal
87	Variable