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 t
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Mult, Neg, e, i, one, pi, subtract, two
from proveit.physics.quantum import ket1
from proveit.physics.quantum.QPE import _phase

# build up the expression from sub-expressions
expr = Equals(ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(Neg(t), one))), _phase)), ket1), ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, subtract(t, one)), _phase)), ket1))

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}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 10
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 26 operands: 11
9	Operation	operator: 26 operands: 12
10	Operation	operator: 13 operand: 39
11	ExprTuple	15, 14
12	ExprTuple	15, 16
13	Literal
14	Operation	operator: 18 operands: 17
15	Literal
16	Operation	operator: 18 operands: 19
17	ExprTuple	29, 21, 22, 20, 24
18	Literal
19	ExprTuple	29, 21, 22, 23, 24
20	Operation	operator: 26 operands: 25
21	Literal
22	Literal
23	Operation	operator: 26 operands: 27
24	Literal
25	ExprTuple	29, 28
26	Literal
27	ExprTuple	29, 30
28	Operation	operator: 40 operand: 33
29	Literal
30	Operation	operator: 35 operands: 32
31	ExprTuple	33
32	ExprTuple	42, 34
33	Operation	operator: 35 operands: 36
34	Operation	operator: 40 operand: 39
35	Literal
36	ExprTuple	38, 39
37	ExprTuple	39
38	Operation	operator: 40 operand: 42
39	Literal
40	Literal
41	ExprTuple	42
42	Variable