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([Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), sub_expr1], [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}}}, \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)}, \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	ExprTuple	5, 7
4	ExprTuple	6, 7
5	Operation	operator: 34 operands: 8
6	Operation	operator: 34 operands: 9
7	Operation	operator: 34 operands: 10
8	ExprTuple	13, 11
9	ExprTuple	13, 12
10	ExprTuple	13, 14
11	Operation	operator: 16 operand: 19
12	Operation	operator: 16 operand: 20
13	Literal
14	Operation	operator: 30 operands: 18
15	ExprTuple	19
16	Literal
17	ExprTuple	20
18	ExprTuple	36, 26, 27, 33, 28
19	Operation	operator: 21 operands: 22
20	Operation	operator: 30 operands: 23
21	Literal
22	ExprTuple	24, 32
23	ExprTuple	36, 26, 27, 28, 33
24	Operation	operator: 30 operands: 25
25	ExprTuple	36, 26, 27, 28, 29
26	Literal
27	Literal
28	Variable
29	Operation	operator: 30 operands: 31
30	Literal
31	ExprTuple	32, 33
32	Operation	operator: 34 operands: 35
33	Literal
34	Literal
35	ExprTuple	36, 37
36	Literal
37	Literal