Expression of type Conditional

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 Conditional, Variable, t
from proveit.linear_algebra import MatrixExp, MatrixMult, ScalarMult
from proveit.logic import Equals, InSet
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, i, one, pi, two, zero
from proveit.physics.quantum.QPE import _U, _ket_u, _phase

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Exp(two, Neg(sub_expr1))
expr = Conditional(Equals(MatrixMult(MatrixExp(_U, sub_expr2), _ket_u), ScalarMult(Exp(e, Mult(two, pi, i, Mult(sub_expr2, _phase))), _ket_u)), InSet(sub_expr1, Interval(Add(Neg(t), one), zero)))

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\{\left(U^{2^{-{_{-}a}}} \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(2^{-{_{-}a}} \cdot \varphi\right)} \cdot \lvert u \rangle\right) \textrm{ if } {_{-}a} \in \{-t + 1~\ldotp \ldotp~0\}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operands: 4
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7, 8
5	Literal
6	ExprTuple	47, 9
7	Operation	operator: 10 operands: 11
8	Operation	operator: 12 operands: 13
9	Operation	operator: 14 operands: 15
10	Literal
11	ExprTuple	16, 18
12	Literal
13	ExprTuple	17, 18
14	Literal
15	ExprTuple	19, 20
16	Operation	operator: 21 operands: 22
17	Operation	operator: 41 operands: 23
18	Literal
19	Operation	operator: 24 operands: 25
20	Literal
21	Literal
22	ExprTuple	26, 39
23	ExprTuple	27, 28
24	Literal
25	ExprTuple	29, 30
26	Literal
27	Literal
28	Operation	operator: 37 operands: 31
29	Operation	operator: 45 operand: 36
30	Literal
31	ExprTuple	43, 33, 34, 35
32	ExprTuple	36
33	Literal
34	Literal
35	Operation	operator: 37 operands: 38
36	Variable
37	Literal
38	ExprTuple	39, 40
39	Operation	operator: 41 operands: 42
40	Literal
41	Literal
42	ExprTuple	43, 44
43	Literal
44	Operation	operator: 45 operand: 47
45	Literal
46	ExprTuple	47
47	Variable