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, Neg, Sum, e, frac, i, one, pi, 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 = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, 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} \left(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right)\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: 66 body: 5
4	ExprTuple	66
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 31 operands: 10
8	Literal
9	ExprTuple	11, 12
10	ExprTuple	66, 13
11	Operation	operator: 14 operand: 17
12	Operation	operator: 59 operands: 16
13	Literal
14	Literal
15	ExprTuple	17
16	ExprTuple	18, 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 52 operands: 22
19	Operation	operator: 23 operand: 25
20	Literal
21	ExprTuple	66, 57
22	ExprTuple	58, 57
23	Literal
24	ExprTuple	25
25	Lambda	parameter: 65 body: 27
26	ExprTuple	65
27	Conditional	value: 28 condition: 29
28	Operation	operator: 59 operands: 30
29	Operation	operator: 31 operands: 32
30	ExprTuple	33, 34
31	Literal
32	ExprTuple	65, 35
33	Operation	operator: 61 operands: 36
34	Operation	operator: 61 operands: 37
35	Operation	operator: 38 operands: 39
36	ExprTuple	41, 40
37	ExprTuple	41, 42
38	Literal
39	ExprTuple	43, 44
40	Operation	operator: 54 operand: 49
41	Literal
42	Operation	operator: 59 operands: 46
43	Literal
44	Operation	operator: 47 operands: 48
45	ExprTuple	49
46	ExprTuple	67, 63, 64, 50, 65
47	Literal
48	ExprTuple	57, 51
49	Operation	operator: 52 operands: 53
50	Literal
51	Operation	operator: 54 operand: 58
52	Literal
53	ExprTuple	56, 57
54	Literal
55	ExprTuple	58
56	Operation	operator: 59 operands: 60
57	Operation	operator: 61 operands: 62
58	Literal
59	Literal
60	ExprTuple	67, 63, 64, 65, 66
61	Literal
62	ExprTuple	67, 68
63	Literal
64	Literal
65	Variable
66	Variable
67	Literal
68	Literal