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, Integer, Mult, Sum, e, frac, i, one, pi, subtract, two
from proveit.physics.quantum.QPE import _alpha_m_mod_two_pow_t, _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m], instance_expr = Equals(_alpha_m_mod_two_pow_t, Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Exp(Exp(e, Mult(two, pi, i, subtract(_phase, frac(m, _two_pow_t)))), k), domain = _m_domain))), domain = Integer)

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_{m \in \mathbb{Z}}~\left(\alpha_{m ~\textup{mod}~ 2^{t}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} (\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\varphi - \frac{m}{2^{t}}\right)})^{k}\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameter: 60 body: 5
4	ExprTuple	60
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 30 operands: 10
8	Literal
9	ExprTuple	11, 12
10	ExprTuple	60, 13
11	Operation	operator: 14 operand: 17
12	Operation	operator: 42 operands: 16
13	Literal
14	Literal
15	ExprTuple	17
16	ExprTuple	18, 19
17	Operation	operator: 20 operands: 59
18	Operation	operator: 58 operands: 21
19	Operation	operator: 22 operand: 24
20	Literal
21	ExprTuple	54, 61
22	Literal
23	ExprTuple	24
24	Lambda	parameter: 33 body: 26
25	ExprTuple	33
26	Conditional	value: 27 condition: 28
27	Operation	operator: 62 operands: 29
28	Operation	operator: 30 operands: 31
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	33, 34
32	Operation	operator: 62 operands: 35
33	Variable
34	Operation	operator: 36 operands: 37
35	ExprTuple	38, 39
36	Literal
37	ExprTuple	40, 41
38	Literal
39	Operation	operator: 42 operands: 43
40	Literal
41	Operation	operator: 49 operands: 44
42	Literal
43	ExprTuple	64, 45, 46, 47
44	ExprTuple	61, 48
45	Literal
46	Literal
47	Operation	operator: 49 operands: 50
48	Operation	operator: 55 operand: 54
49	Literal
50	ExprTuple	52, 53
51	ExprTuple	54
52	Literal
53	Operation	operator: 55 operand: 57
54	Literal
55	Literal
56	ExprTuple	57
57	Operation	operator: 58 operands: 59
58	Literal
59	ExprTuple	60, 61
60	Variable
61	Operation	operator: 62 operands: 63
62	Literal
63	ExprTuple	64, 65
64	Literal
65	Literal