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, Mult, Neg, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _alpha_m, _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
expr = Lambda(m, Conditional(Equals(_alpha_m, 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, _m_domain)))

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} = \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 \{0~\ldotp \ldotp~2^{t} - 1\}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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