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, s, t
from proveit.linear_algebra import MatrixMult, ScalarMult, Unitary
from proveit.logic import And, Equals, InSet
from proveit.numbers import Exp, Interval, Mult, NaturalPos, Real, e, i, one, pi, subtract, 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, two_pow_t

# build up the expression from sub-expressions
expr = ExprTuple(InSet(s, NaturalPos), InSet(t, NaturalPos), InSet(U, Unitary(two_pow_s)), InSet(var_ket_u, s_ket_domain), normalized_var_ket_u, And(InSet(phase, Real), InSet(Mult(two_pow_t, phase), Interval(zero, subtract(two_pow_t, one))), Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u))))

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(s \in \mathbb{N}^+, t \in \mathbb{N}^+, U \in \textrm{U}\left(2^{s}\right), \lvert u \rangle \in \mathbb{C}^{2^{s}}, \left \|\lvert u \rangle\right \| = 1, \left(\varphi \in \mathbb{R}\right) \land \left(\left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 1\}\right) \land \left(\left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)\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
1	Operation	operator: 28 operands: 7
2	Operation	operator: 28 operands: 8
3	Operation	operator: 28 operands: 9
4	Operation	operator: 28 operands: 10
5	Operation	operator: 30 operands: 11
6	Operation	operator: 12 operands: 13
7	ExprTuple	47, 14
8	ExprTuple	66, 14
9	ExprTuple	50, 15
10	ExprTuple	52, 16
11	ExprTuple	17, 67
12	Literal
13	ExprTuple	18, 19, 20
14	Literal
15	Operation	operator: 21 operand: 33
16	Operation	operator: 23 operands: 24
17	Operation	operator: 25 operand: 52
18	Operation	operator: 28 operands: 27
19	Operation	operator: 28 operands: 29
20	Operation	operator: 30 operands: 31
21	Literal
22	ExprTuple	33
23	Literal
24	ExprTuple	32, 33
25	Literal
26	ExprTuple	52
27	ExprTuple	71, 34
28	Literal
29	ExprTuple	35, 36
30	Literal
31	ExprTuple	37, 38
32	Literal
33	Operation	operator: 60 operands: 39
34	Literal
35	Operation	operator: 64 operands: 40
36	Operation	operator: 41 operands: 42
37	Operation	operator: 43 operands: 44
38	Operation	operator: 45 operands: 46
39	ExprTuple	68, 47
40	ExprTuple	56, 71
41	Literal
42	ExprTuple	48, 49
43	Literal
44	ExprTuple	50, 52
45	Literal
46	ExprTuple	51, 52
47	Variable
48	Literal
49	Operation	operator: 53 operands: 54
50	Variable
51	Operation	operator: 60 operands: 55
52	Variable
53	Literal
54	ExprTuple	56, 57
55	ExprTuple	58, 59
56	Operation	operator: 60 operands: 61
57	Operation	operator: 62 operand: 67
58	Literal
59	Operation	operator: 64 operands: 65
60	Literal
61	ExprTuple	68, 66
62	Literal
63	ExprTuple	67
64	Literal
65	ExprTuple	68, 69, 70, 71
66	Variable
67	Literal
68	Literal
69	Literal
70	Literal
71	Variable