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, 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 \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)

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