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.linear_algebra import ScalarMult
from proveit.logic import Equals, InSet
from proveit.numbers import Exp, Mult, e, i, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = NumKet(k, _t)
sub_expr4 = Exp(e, Mult(two, pi, i, _phase, k))
expr = Lambda(k, Conditional(Equals(Qmult(sub_expr1, sub_expr2, ScalarMult(sub_expr4, sub_expr3)), Mult(sub_expr4, Qmult(sub_expr1, sub_expr2, sub_expr3))), InSet(k, _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())

k \mapsto \left\{\left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t}\right)\right) \textrm{ if } k \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: 54 body: 2
1	ExprTuple	54
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operands: 6
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9, 10
7	Literal
8	ExprTuple	54, 11
9	Operation	operator: 22 operands: 12
10	Operation	operator: 48 operands: 13
11	Operation	operator: 14 operands: 15
12	ExprTuple	27, 28, 16
13	ExprTuple	26, 17
14	Literal
15	ExprTuple	18, 19
16	Operation	operator: 20 operands: 21
17	Operation	operator: 22 operands: 23
18	Literal
19	Operation	operator: 24 operands: 25
20	Literal
21	ExprTuple	26, 29
22	Literal
23	ExprTuple	27, 28, 29
24	Literal
25	ExprTuple	30, 31
26	Operation	operator: 39 operands: 32
27	Operation	operator: 33 operands: 34
28	Operation	operator: 35 operand: 46
29	Operation	operator: 37 operands: 38
30	Operation	operator: 39 operands: 40
31	Operation	operator: 41 operand: 47
32	ExprTuple	43, 44
33	Literal
34	ExprTuple	45, 46
35	Literal
36	ExprTuple	46
37	Literal
38	ExprTuple	54, 46
39	Literal
40	ExprTuple	50, 46
41	Literal
42	ExprTuple	47
43	Literal
44	Operation	operator: 48 operands: 49
45	Variable
46	Literal
47	Literal
48	Literal
49	ExprTuple	50, 51, 52, 53, 54
50	Literal
51	Literal
52	Literal
53	Literal
54	Variable