Expression of type And

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 k, t
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import And, Equals, InSet
from proveit.numbers import Exp, Interval, Mult, NaturalPos, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _phase, _psi_t_ket, two_pow_t

# build up the expression from sub-expressions
expr = And(InSet(t, NaturalPos), Equals(_psi_t_ket, ScalarMult(frac(one, Exp(two, frac(t, two))), VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, t)), domain = Interval(zero, subtract(two_pow_t, one))))))

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(t \in \mathbb{N}^+\right) \land \left(\lvert \psi_{t} \rangle = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right)\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 31 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	60, 8
6	Literal
7	ExprTuple	9, 10
8	Literal
9	Operation	operator: 11 operand: 60
10	Operation	operator: 29 operands: 13
11	Literal
12	ExprTuple	60
13	ExprTuple	14, 15
14	Operation	operator: 27 operands: 16
15	Operation	operator: 17 operand: 20
16	ExprTuple	61, 19
17	Literal
18	ExprTuple	20
19	Operation	operator: 55 operands: 21
20	Lambda	parameter: 52 body: 23
21	ExprTuple	59, 24
22	ExprTuple	52
23	Conditional	value: 25 condition: 26
24	Operation	operator: 27 operands: 28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operands: 32
27	Literal
28	ExprTuple	60, 59
29	Literal
30	ExprTuple	33, 34
31	Literal
32	ExprTuple	52, 35
33	Operation	operator: 55 operands: 36
34	Operation	operator: 37 operands: 38
35	Operation	operator: 39 operands: 40
36	ExprTuple	41, 42
37	Literal
38	ExprTuple	52, 60
39	Literal
40	ExprTuple	43, 44
41	Literal
42	Operation	operator: 45 operands: 46
43	Literal
44	Operation	operator: 47 operands: 48
45	Literal
46	ExprTuple	59, 49, 50, 51, 52
47	Literal
48	ExprTuple	53, 54
49	Literal
50	Literal
51	Literal
52	Variable
53	Operation	operator: 55 operands: 56
54	Operation	operator: 57 operand: 61
55	Literal
56	ExprTuple	59, 60
57	Literal
58	ExprTuple	61
59	Literal
60	Variable
61	Literal