Expression of type Forall

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 k, l, n
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Interval, Mult, 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 = 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)))

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())

\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)

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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