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.QPE import _phase, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr2 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
expr = Equals(Mult(sub_expr2, sub_expr1, sub_expr3), Mult(sub_expr2, Mult(sub_expr1, sub_expr3))).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left(\frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right) =  \\ \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right) \end{array} \end{array}

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.	()	(2)	('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, 13, 14
6	ExprTuple	7, 8
7	Operation	operator: 28 operands: 9
8	Operation	operator: 32 operands: 10
9	ExprTuple	11, 12
10	ExprTuple	13, 14
11	Literal
12	Operation	operator: 34 operands: 15
13	Operation	operator: 34 operands: 16
14	Operation	operator: 34 operands: 17
15	ExprTuple	40, 18
16	ExprTuple	20, 19
17	ExprTuple	20, 21
18	Operation	operator: 28 operands: 22
19	Operation	operator: 32 operands: 23
20	Literal
21	Operation	operator: 24 operand: 27
22	ExprTuple	41, 40
23	ExprTuple	40, 36, 37, 26, 38
24	Literal
25	ExprTuple	27
26	Literal
27	Operation	operator: 28 operands: 29
28	Literal
29	ExprTuple	30, 31
30	Operation	operator: 32 operands: 33
31	Operation	operator: 34 operands: 35
32	Literal
33	ExprTuple	40, 36, 37, 38, 39
34	Literal
35	ExprTuple	40, 41
36	Literal
37	Literal
38	Variable
39	Variable
40	Literal
41	Literal