Expression of type Mult

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, m
from proveit.numbers import Exp, Mult, Neg, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = frac(one, Exp(two, frac(_t, two)))
expr = Mult(sub_expr1, Mult(sub_expr1, Sum(index_or_indices = [k], summand = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))), domain = _m_domain)))

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())

\frac{1}{2^{\frac{t}{2}}} \cdot \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 \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\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: 47 operands: 1
1	ExprTuple	4, 2
2	Operation	operator: 47 operands: 3
3	ExprTuple	4, 5
4	Operation	operator: 40 operands: 6
5	Operation	operator: 7 operand: 10
6	ExprTuple	46, 9
7	Literal
8	ExprTuple	10
9	Operation	operator: 49 operands: 11
10	Lambda	parameter: 53 body: 13
11	ExprTuple	55, 14
12	ExprTuple	53
13	Conditional	value: 15 condition: 16
14	Operation	operator: 40 operands: 17
15	Operation	operator: 47 operands: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	56, 55
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	53, 23
21	Operation	operator: 49 operands: 24
22	Operation	operator: 49 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	29, 28
25	ExprTuple	29, 30
26	Literal
27	ExprTuple	31, 32
28	Operation	operator: 47 operands: 33
29	Literal
30	Operation	operator: 42 operand: 38
31	Literal
32	Operation	operator: 35 operands: 36
33	ExprTuple	55, 51, 52, 37, 53
34	ExprTuple	38
35	Literal
36	ExprTuple	45, 39
37	Literal
38	Operation	operator: 40 operands: 41
39	Operation	operator: 42 operand: 46
40	Literal
41	ExprTuple	44, 45
42	Literal
43	ExprTuple	46
44	Operation	operator: 47 operands: 48
45	Operation	operator: 49 operands: 50
46	Literal
47	Literal
48	ExprTuple	55, 51, 52, 53, 54
49	Literal
50	ExprTuple	55, 56
51	Literal
52	Literal
53	Variable
54	Variable
55	Literal
56	Literal