Expression of type Equals

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.logic import Equals
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _t, _two_pow_t

# build up the expression from sub-expressions
expr = Equals(Qmult(NumBra(m, _t), InverseFourierTransform(_t), NumKet(k, _t)), Mult(frac(one, Exp(two, frac(_t, two))), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))))

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}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t}\right) = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\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',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# 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: 5 operands: 6
4	Operation	operator: 35 operands: 7
5	Literal
6	ExprTuple	8, 9, 10
7	ExprTuple	11, 12
8	Operation	operator: 13 operands: 14
9	Operation	operator: 15 operand: 44
10	Operation	operator: 17 operands: 18
11	Operation	operator: 31 operands: 19
12	Operation	operator: 37 operands: 20
13	Literal
14	ExprTuple	42, 44
15	Literal
16	ExprTuple	44
17	Literal
18	ExprTuple	41, 44
19	ExprTuple	21, 22
20	ExprTuple	23, 24
21	Literal
22	Operation	operator: 37 operands: 25
23	Literal
24	Operation	operator: 26 operand: 29
25	ExprTuple	43, 28
26	Literal
27	ExprTuple	29
28	Operation	operator: 31 operands: 30
29	Operation	operator: 31 operands: 32
30	ExprTuple	44, 43
31	Literal
32	ExprTuple	33, 34
33	Operation	operator: 35 operands: 36
34	Operation	operator: 37 operands: 38
35	Literal
36	ExprTuple	43, 39, 40, 41, 42
37	Literal
38	ExprTuple	43, 44
39	Literal
40	Literal
41	Variable
42	Variable
43	Literal
44	Literal