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
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Neg, e, frac, i, pi, two
from proveit.physics.quantum.QPE import _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
expr = Equals(Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), sub_expr1), Mult(Exp(e, Neg(Mult(two, pi, i, k, _phase))), sub_expr1))

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(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot \left(2^{t} \cdot \varphi\right)}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right) = \left(\mathsf{e}^{-\left(2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot \varphi\right)} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \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: 32 operands: 5
4	Operation	operator: 32 operands: 6
5	ExprTuple	7, 9
6	ExprTuple	8, 9
7	Operation	operator: 36 operands: 10
8	Operation	operator: 36 operands: 11
9	Operation	operator: 36 operands: 12
10	ExprTuple	15, 13
11	ExprTuple	15, 14
12	ExprTuple	15, 16
13	Operation	operator: 18 operand: 21
14	Operation	operator: 18 operand: 22
15	Literal
16	Operation	operator: 32 operands: 20
17	ExprTuple	21
18	Literal
19	ExprTuple	22
20	ExprTuple	38, 28, 29, 35, 30
21	Operation	operator: 23 operands: 24
22	Operation	operator: 32 operands: 25
23	Literal
24	ExprTuple	26, 34
25	ExprTuple	38, 28, 29, 30, 35
26	Operation	operator: 32 operands: 27
27	ExprTuple	38, 28, 29, 30, 31
28	Literal
29	Literal
30	Variable
31	Operation	operator: 32 operands: 33
32	Literal
33	ExprTuple	34, 35
34	Operation	operator: 36 operands: 37
35	Literal
36	Literal
37	ExprTuple	38, 39
38	Literal
39	Literal