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, 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 = Forall(instance_param_or_params = [n], instance_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))), domain = 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())

\forall_{n \in \mathbb{N}^+}~\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)\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: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 66 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 27 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	66, 10
9	Lambda	parameters: 11 body: 12
10	Literal
11	ExprTuple	73, 74
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Operation	operator: 23 operands: 24
20	Operation	operator: 68 operands: 25
21	Operation	operator: 27 operands: 26
22	Operation	operator: 27 operands: 28
23	Literal
24	ExprTuple	29, 30, 31
25	ExprTuple	32, 33
26	ExprTuple	73, 34
27	Literal
28	ExprTuple	74, 34
29	Operation	operator: 35 operands: 36
30	Operation	operator: 37 operand: 66
31	Operation	operator: 39 operands: 40
32	Operation	operator: 58 operands: 41
33	Operation	operator: 61 operands: 42
34	Operation	operator: 43 operands: 44
35	Literal
36	ExprTuple	74, 66
37	Literal
38	ExprTuple	66
39	Literal
40	ExprTuple	73, 66
41	ExprTuple	67, 45
42	ExprTuple	46, 47
43	Literal
44	ExprTuple	48, 49
45	Operation	operator: 61 operands: 50
46	Literal
47	Operation	operator: 58 operands: 51
48	Literal
49	Operation	operator: 52 operands: 53
50	ExprTuple	70, 54
51	ExprTuple	55, 56
52	Literal
53	ExprTuple	56, 57
54	Operation	operator: 58 operands: 59
55	Operation	operator: 63 operand: 65
56	Operation	operator: 61 operands: 62
57	Operation	operator: 63 operand: 67
58	Literal
59	ExprTuple	66, 70
60	ExprTuple	65
61	Literal
62	ExprTuple	70, 66
63	Literal
64	ExprTuple	67
65	Operation	operator: 68 operands: 69
66	Variable
67	Literal
68	Literal
69	ExprTuple	70, 71, 72, 73, 74
70	Literal
71	Literal
72	Literal
73	Variable
74	Variable