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, Neg, Sum, e, frac, i, one, pi, 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 = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, 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} \left(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right)\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: 63 body: 2
1	ExprTuple	63
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operands: 6
4	Operation	operator: 28 operands: 7
5	Literal
6	ExprTuple	8, 9
7	ExprTuple	63, 10
8	Operation	operator: 11 operand: 14
9	Operation	operator: 56 operands: 13
10	Literal
11	Literal
12	ExprTuple	14
13	ExprTuple	15, 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 49 operands: 19
16	Operation	operator: 20 operand: 22
17	Literal
18	ExprTuple	63, 54
19	ExprTuple	55, 54
20	Literal
21	ExprTuple	22
22	Lambda	parameter: 62 body: 24
23	ExprTuple	62
24	Conditional	value: 25 condition: 26
25	Operation	operator: 56 operands: 27
26	Operation	operator: 28 operands: 29
27	ExprTuple	30, 31
28	Literal
29	ExprTuple	62, 32
30	Operation	operator: 58 operands: 33
31	Operation	operator: 58 operands: 34
32	Operation	operator: 35 operands: 36
33	ExprTuple	38, 37
34	ExprTuple	38, 39
35	Literal
36	ExprTuple	40, 41
37	Operation	operator: 51 operand: 46
38	Literal
39	Operation	operator: 56 operands: 43
40	Literal
41	Operation	operator: 44 operands: 45
42	ExprTuple	46
43	ExprTuple	64, 60, 61, 47, 62
44	Literal
45	ExprTuple	54, 48
46	Operation	operator: 49 operands: 50
47	Literal
48	Operation	operator: 51 operand: 55
49	Literal
50	ExprTuple	53, 54
51	Literal
52	ExprTuple	55
53	Operation	operator: 56 operands: 57
54	Operation	operator: 58 operands: 59
55	Literal
56	Literal
57	ExprTuple	64, 60, 61, 62, 63
58	Literal
59	ExprTuple	64, 65
60	Literal
61	Literal
62	Variable
63	Variable
64	Literal
65	Literal