Expression of type Lambda

from the theory of proveit.physics.quantum.QFT

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, l, n
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Exp, Interval, Mult, NaturalPos, Neg, e, frac, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform

# build up the expression from sub-expressions
sub_expr1 = Exp(two, n)
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [k, l], instance_expr = Equals(Qmult(NumBra(l, n), InverseFourierTransform(n), NumKet(k, n)), Mult(frac(one, Exp(two, frac(n, two))), Exp(e, frac(Neg(Mult(two, pi, i, k, l)), sub_expr1)))), domain = Interval(zero, subtract(sub_expr1, one))), InSet(n, NaturalPos)))

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())

n \mapsto \left\{\forall_{k, l \in \{0~\ldotp \ldotp~2^{n} - 1\}}~\left(\left({_{n}}\langle l \rvert \thinspace {\mathrm {FT}}^{\dag}_{n} \thinspace \lvert k \rangle_{n}\right) = \left(\frac{1}{2^{\frac{n}{2}}} \cdot \mathsf{e}^{\frac{-\left(2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot l\right)}{2^{n}}}\right)\right) \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 64 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 25 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	64, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	71, 72
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operands: 14
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	19, 20
17	Operation	operator: 21 operands: 22
18	Operation	operator: 66 operands: 23
19	Operation	operator: 25 operands: 24
20	Operation	operator: 25 operands: 26
21	Literal
22	ExprTuple	27, 28, 29
23	ExprTuple	30, 31
24	ExprTuple	71, 32
25	Literal
26	ExprTuple	72, 32
27	Operation	operator: 33 operands: 34
28	Operation	operator: 35 operand: 64
29	Operation	operator: 37 operands: 38
30	Operation	operator: 56 operands: 39
31	Operation	operator: 59 operands: 40
32	Operation	operator: 41 operands: 42
33	Literal
34	ExprTuple	72, 64
35	Literal
36	ExprTuple	64
37	Literal
38	ExprTuple	71, 64
39	ExprTuple	65, 43
40	ExprTuple	44, 45
41	Literal
42	ExprTuple	46, 47
43	Operation	operator: 59 operands: 48
44	Literal
45	Operation	operator: 56 operands: 49
46	Literal
47	Operation	operator: 50 operands: 51
48	ExprTuple	68, 52
49	ExprTuple	53, 54
50	Literal
51	ExprTuple	54, 55
52	Operation	operator: 56 operands: 57
53	Operation	operator: 61 operand: 63
54	Operation	operator: 59 operands: 60
55	Operation	operator: 61 operand: 65
56	Literal
57	ExprTuple	64, 68
58	ExprTuple	63
59	Literal
60	ExprTuple	68, 64
61	Literal
62	ExprTuple	65
63	Operation	operator: 66 operands: 67
64	Variable
65	Literal
66	Literal
67	ExprTuple	68, 69, 70, 71, 72
68	Literal
69	Literal
70	Literal
71	Variable
72	Variable