Expression of type Forall

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 k, m
from proveit.logic import Equals, Forall
from proveit.numbers import Exp, Mult, Neg, e, frac, i, one, 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, _t, _two_pow_t

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [k], instance_expr = Equals(Qmult(NumBra(m, _t), InverseFourierTransform(_t), NumKet(k, _t)), Mult(frac(one, Exp(two, frac(_t, two))), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))))), domain = _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())

\forall_{k \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \lvert k \rangle_{t}\right) = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\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	parameter: 59 body: 5
4	ExprTuple	59
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	59, 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 53 operands: 17
14	Operation	operator: 18 operands: 19
15	Literal
16	ExprTuple	20, 21, 22
17	ExprTuple	23, 24
18	Literal
19	ExprTuple	25, 26
20	Operation	operator: 27 operands: 28
21	Operation	operator: 29 operand: 62
22	Operation	operator: 31 operands: 32
23	Operation	operator: 49 operands: 33
24	Operation	operator: 55 operands: 34
25	Literal
26	Operation	operator: 35 operands: 36
27	Literal
28	ExprTuple	60, 62
29	Literal
30	ExprTuple	62
31	Literal
32	ExprTuple	59, 62
33	ExprTuple	47, 37
34	ExprTuple	38, 39
35	Literal
36	ExprTuple	52, 40
37	Operation	operator: 55 operands: 41
38	Literal
39	Operation	operator: 43 operand: 46
40	Operation	operator: 43 operand: 47
41	ExprTuple	61, 45
42	ExprTuple	46
43	Literal
44	ExprTuple	47
45	Operation	operator: 49 operands: 48
46	Operation	operator: 49 operands: 50
47	Literal
48	ExprTuple	62, 61
49	Literal
50	ExprTuple	51, 52
51	Operation	operator: 53 operands: 54
52	Operation	operator: 55 operands: 56
53	Literal
54	ExprTuple	61, 57, 58, 59, 60
55	Literal
56	ExprTuple	61, 62
57	Literal
58	Literal
59	Variable
60	Variable
61	Literal
62	Literal