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

# build up the expression from sub-expressions
sub_expr1 = Exp(two, n)
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [j, k], instance_expr = Equals(Qmult(NumBra(k, n), FourierTransform(n), NumKet(j, n)), Qmult(frac(one, Exp(two, frac(n, two))), Exp(e, frac(Mult(two, pi, i, j, k), 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_{j, k \in \{0~\ldotp \ldotp~2^{n} - 1\}}~\left(\left({_{n}}\langle k \rvert \thinspace {\mathrm {FT}}_{n} \thinspace \lvert j \rangle_{n}\right) = \left(\frac{1}{2^{\frac{n}{2}}} \thinspace \mathsf{e}^{\frac{2 \cdot \pi \cdot \mathsf{i} \cdot j \cdot k}{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: 69 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	69, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	66, 67
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: 22 operands: 21
18	Operation	operator: 22 operands: 23
19	Operation	operator: 25 operands: 24
20	Operation	operator: 25 operands: 26
21	ExprTuple	27, 28, 29
22	Literal
23	ExprTuple	30, 31
24	ExprTuple	66, 32
25	Literal
26	ExprTuple	67, 32
27	Operation	operator: 33 operands: 34
28	Operation	operator: 35 operand: 69
29	Operation	operator: 37 operands: 38
30	Operation	operator: 56 operands: 39
31	Operation	operator: 60 operands: 40
32	Operation	operator: 41 operands: 42
33	Literal
34	ExprTuple	67, 69
35	Literal
36	ExprTuple	69
37	Literal
38	ExprTuple	66, 69
39	ExprTuple	70, 43
40	ExprTuple	44, 45
41	Literal
42	ExprTuple	46, 47
43	Operation	operator: 60 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: 58 operands: 59
54	Operation	operator: 60 operands: 61
55	Operation	operator: 62 operand: 70
56	Literal
57	ExprTuple	69, 68
58	Literal
59	ExprTuple	68, 64, 65, 66, 67
60	Literal
61	ExprTuple	68, 69
62	Literal
63	ExprTuple	70
64	Literal
65	Literal
66	Variable
67	Variable
68	Literal
69	Variable
70	Literal