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 Conditional, Lambda, k
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = InSet(k, _m_domain)
expr = Equals(Lambda(k, Conditional(Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), sub_expr1)), Lambda(k, Conditional(one, sub_expr1))).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[k \mapsto \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} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{1 \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\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	Lambda	parameter: 40 body: 5
4	Lambda	parameter: 40 body: 7
5	Conditional	value: 8 condition: 9
6	ExprTuple	40
7	Conditional	value: 36 condition: 9
8	Operation	operator: 42 operands: 10
9	Operation	operator: 11 operands: 12
10	ExprTuple	13, 14
11	Literal
12	ExprTuple	40, 15
13	Operation	operator: 46 operands: 16
14	Operation	operator: 46 operands: 17
15	Operation	operator: 18 operands: 19
16	ExprTuple	21, 20
17	ExprTuple	21, 22
18	Literal
19	ExprTuple	23, 24
20	Operation	operator: 33 operand: 29
21	Literal
22	Operation	operator: 42 operands: 26
23	Literal
24	Operation	operator: 27 operands: 28
25	ExprTuple	29
26	ExprTuple	48, 38, 39, 45, 40
27	Literal
28	ExprTuple	44, 30
29	Operation	operator: 31 operands: 32
30	Operation	operator: 33 operand: 36
31	Literal
32	ExprTuple	35, 44
33	Literal
34	ExprTuple	36
35	Operation	operator: 42 operands: 37
36	Literal
37	ExprTuple	48, 38, 39, 40, 41
38	Literal
39	Literal
40	Variable
41	Operation	operator: 42 operands: 43
42	Literal
43	ExprTuple	44, 45
44	Operation	operator: 46 operands: 47
45	Literal
46	Literal
47	ExprTuple	48, 49
48	Literal
49	Literal