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

# build up the expression from sub-expressions
sub_expr1 = InSet(k, _m_domain)
expr = Equals(Lambda(k, Conditional(Exp(e, Neg(frac(Mult(two, pi, i, k, Mod(m, _two_pow_t)), _two_pow_t))), sub_expr1)), Lambda(k, Conditional(Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))), 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(m ~\textup{mod}~ 2^{t}\right)}{2^{t}}} \textrm{ if } k \in \{0~\ldotp \ldotp~2^{t} - 1\}\right..\right] =  \\ \left[k \mapsto \left\{\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \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: 43 body: 5
4	Lambda	parameter: 43 body: 7
5	Conditional	value: 8 condition: 10
6	ExprTuple	43
7	Conditional	value: 9 condition: 10
8	Operation	operator: 49 operands: 11
9	Operation	operator: 49 operands: 12
10	Operation	operator: 13 operands: 14
11	ExprTuple	16, 15
12	ExprTuple	16, 17
13	Literal
14	ExprTuple	43, 18
15	Operation	operator: 38 operand: 23
16	Literal
17	Operation	operator: 38 operand: 24
18	Operation	operator: 21 operands: 22
19	ExprTuple	23
20	ExprTuple	24
21	Literal
22	ExprTuple	25, 26
23	Operation	operator: 28 operands: 27
24	Operation	operator: 28 operands: 29
25	Literal
26	Operation	operator: 30 operands: 31
27	ExprTuple	32, 48
28	Literal
29	ExprTuple	33, 48
30	Literal
31	ExprTuple	48, 34
32	Operation	operator: 36 operands: 35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operand: 44
35	ExprTuple	51, 41, 42, 43, 40
36	Literal
37	ExprTuple	51, 41, 42, 43, 47
38	Literal
39	ExprTuple	44
40	Operation	operator: 45 operands: 46
41	Literal
42	Literal
43	Variable
44	Literal
45	Literal
46	ExprTuple	47, 48
47	Variable
48	Operation	operator: 49 operands: 50
49	Literal
50	ExprTuple	51, 52
51	Literal
52	Literal