Expression of type Lambda

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, Integer, Mult, Sum, e, frac, i, one, pi, subtract, two
from proveit.physics.quantum.QPE import _alpha_m_mod_two_pow_t, _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
expr = Lambda(m, Conditional(Equals(_alpha_m_mod_two_pow_t, Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Exp(Exp(e, Mult(two, pi, i, subtract(_phase, frac(m, _two_pow_t)))), k), domain = _m_domain))), InSet(m, Integer)))

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())

m \mapsto \left\{\alpha_{m ~\textup{mod}~ 2^{t}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} (\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\varphi - \frac{m}{2^{t}}\right)})^{k}\right)\right) \textrm{ if } m \in \mathbb{Z}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 57 body: 2
1	ExprTuple	57
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operands: 6
4	Operation	operator: 27 operands: 7
5	Literal
6	ExprTuple	8, 9
7	ExprTuple	57, 10
8	Operation	operator: 11 operand: 14
9	Operation	operator: 39 operands: 13
10	Literal
11	Literal
12	ExprTuple	14
13	ExprTuple	15, 16
14	Operation	operator: 17 operands: 56
15	Operation	operator: 55 operands: 18
16	Operation	operator: 19 operand: 21
17	Literal
18	ExprTuple	51, 58
19	Literal
20	ExprTuple	21
21	Lambda	parameter: 30 body: 23
22	ExprTuple	30
23	Conditional	value: 24 condition: 25
24	Operation	operator: 59 operands: 26
25	Operation	operator: 27 operands: 28
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	30, 31
29	Operation	operator: 59 operands: 32
30	Variable
31	Operation	operator: 33 operands: 34
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	37, 38
35	Literal
36	Operation	operator: 39 operands: 40
37	Literal
38	Operation	operator: 46 operands: 41
39	Literal
40	ExprTuple	61, 42, 43, 44
41	ExprTuple	58, 45
42	Literal
43	Literal
44	Operation	operator: 46 operands: 47
45	Operation	operator: 52 operand: 51
46	Literal
47	ExprTuple	49, 50
48	ExprTuple	51
49	Literal
50	Operation	operator: 52 operand: 54
51	Literal
52	Literal
53	ExprTuple	54
54	Operation	operator: 55 operands: 56
55	Literal
56	ExprTuple	57, 58
57	Variable
58	Operation	operator: 59 operands: 60
59	Literal
60	ExprTuple	61, 62
61	Literal
62	Literal