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, Mod, Mult, Neg, e, frac, i, pi, two
from proveit.physics.quantum.QPE import _two_pow_t

# build up the expression from sub-expressions
expr = Equals(Exp(e, Neg(Mult(frac(Mult(two, pi, i, Mod(m, _two_pow_t)), _two_pow_t), k))), Exp(e, Neg(Mult(frac(Mult(two, pi, i, m), _two_pow_t), k))))

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())

\mathsf{e}^{-\left(\frac{2 \cdot \pi \cdot \mathsf{i} \cdot \left(m ~\textup{mod}~ 2^{t}\right)}{2^{t}} \cdot k\right)} = \mathsf{e}^{-\left(\frac{2 \cdot \pi \cdot \mathsf{i} \cdot m}{2^{t}} \cdot k\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: 35 operands: 5
4	Operation	operator: 35 operands: 6
5	ExprTuple	8, 7
6	ExprTuple	8, 9
7	Operation	operator: 11 operand: 13
8	Literal
9	Operation	operator: 11 operand: 14
10	ExprTuple	13
11	Literal
12	ExprTuple	14
13	Operation	operator: 26 operands: 15
14	Operation	operator: 26 operands: 16
15	ExprTuple	17, 19
16	ExprTuple	18, 19
17	Operation	operator: 21 operands: 20
18	Operation	operator: 21 operands: 22
19	Variable
20	ExprTuple	23, 34
21	Literal
22	ExprTuple	24, 34
23	Operation	operator: 26 operands: 25
24	Operation	operator: 26 operands: 27
25	ExprTuple	37, 29, 30, 28
26	Literal
27	ExprTuple	37, 29, 30, 33
28	Operation	operator: 31 operands: 32
29	Literal
30	Literal
31	Literal
32	ExprTuple	33, 34
33	Variable
34	Operation	operator: 35 operands: 36
35	Literal
36	ExprTuple	37, 38
37	Literal
38	Literal