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, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _alpha_m, _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, 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 = _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_{m \in \{0~\ldotp \ldotp~2^{t} - 1\}}~\left(\alpha_{m} = \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: 61 body: 4
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operands: 8
6	Operation	operator: 26 operands: 9
7	Literal
8	ExprTuple	10, 11
9	ExprTuple	61, 30
10	Operation	operator: 12 operand: 61
11	Operation	operator: 54 operands: 14
12	Literal
13	ExprTuple	61
14	ExprTuple	15, 16
15	Operation	operator: 47 operands: 17
16	Operation	operator: 18 operand: 20
17	ExprTuple	53, 52
18	Literal
19	ExprTuple	20
20	Lambda	parameter: 60 body: 22
21	ExprTuple	60
22	Conditional	value: 23 condition: 24
23	Operation	operator: 54 operands: 25
24	Operation	operator: 26 operands: 27
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	60, 30
28	Operation	operator: 56 operands: 31
29	Operation	operator: 56 operands: 32
30	Operation	operator: 33 operands: 34
31	ExprTuple	36, 35
32	ExprTuple	36, 37
33	Literal
34	ExprTuple	38, 39
35	Operation	operator: 49 operand: 44
36	Literal
37	Operation	operator: 54 operands: 41
38	Literal
39	Operation	operator: 42 operands: 43
40	ExprTuple	44
41	ExprTuple	62, 58, 59, 45, 60
42	Literal
43	ExprTuple	52, 46
44	Operation	operator: 47 operands: 48
45	Literal
46	Operation	operator: 49 operand: 53
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	53
51	Operation	operator: 54 operands: 55
52	Operation	operator: 56 operands: 57
53	Literal
54	Literal
55	ExprTuple	62, 58, 59, 60, 61
56	Literal
57	ExprTuple	62, 63
58	Literal
59	Literal
60	Variable
61	Variable
62	Literal
63	Literal