Expression of type Implies ¶

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, l
from proveit.logic import Forall, Implies
from proveit.numbers import Abs, Exp, LessEq, Mult, Sum, four, frac, one, subtract, two
from proveit.physics.quantum.QPE import ModAdd, SubIndexed, _alpha, _b_floor, _delta_b_floor, _diff_l_scaled_delta_floor, _pos_domain, _rel_indexed_alpha, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [l]
expr = Implies(Forall(instance_param_or_params = [k], instance_expr = LessEq(Exp(Abs(SubIndexed(_alpha, [ModAdd(_b_floor, k)])), two), frac(one, Mult(four, Exp(subtract(k, Mult(_two_pow_t, _delta_b_floor)), two)))), domain = _pos_domain), LessEq(Sum(index_or_indices = sub_expr1, summand = Exp(Abs(_rel_indexed_alpha), two), domain = _pos_domain), Sum(index_or_indices = sub_expr1, summand = frac(one, Mult(four, Exp(_diff_l_scaled_delta_floor, two))), domain = _pos_domain)))

# 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())

\left[\forall_{k \in \{e + 1~\ldotp \ldotp~2^{t - 1}\}}~\left(\left|\alpha_{b_{\textit{f}} \oplus k}\right|^{2} \leq \frac{1}{4 \cdot \left(k - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right)\right] \Rightarrow \left(\left(\sum_{l = e + 1}^{2^{t - 1}} \left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2}\right) \leq \left(\sum_{l = e + 1}^{2^{t - 1}} \frac{1}{4 \cdot \left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right)\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

	core type	sub-expressions
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 20 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 77 body: 12
9	Operation	operator: 14 operand: 18
10	Operation	operator: 14 operand: 19
11	ExprTuple	77
12	Conditional	value: 16 condition: 17
13	ExprTuple	18
14	Literal
15	ExprTuple	19
16	Operation	operator: 20 operands: 21
17	Operation	operator: 36 operands: 22
18	Lambda	parameter: 78 body: 23
19	Lambda	parameter: 78 body: 25
20	Literal
21	ExprTuple	26, 27
22	ExprTuple	77, 42
23	Conditional	value: 28 condition: 30
24	ExprTuple	78
25	Conditional	value: 29 condition: 30
26	Operation	operator: 90 operands: 31
27	Operation	operator: 34 operands: 32
28	Operation	operator: 90 operands: 33
29	Operation	operator: 34 operands: 35
30	Operation	operator: 36 operands: 37
31	ExprTuple	38, 94
32	ExprTuple	85, 39
33	ExprTuple	40, 94
34	Literal
35	ExprTuple	85, 41
36	Literal
37	ExprTuple	78, 42
38	Operation	operator: 45 operand: 50
39	Operation	operator: 86 operands: 44
40	Operation	operator: 45 operand: 52
41	Operation	operator: 86 operands: 47
42	Operation	operator: 48 operands: 49
43	ExprTuple	50
44	ExprTuple	53, 51
45	Literal
46	ExprTuple	52
47	ExprTuple	53, 54
48	Literal
49	ExprTuple	55, 56
50	Operation	operator: 59 operand: 64
51	Operation	operator: 90 operands: 58
52	Operation	operator: 59 operand: 66
53	Literal
54	Operation	operator: 90 operands: 61
55	Operation	operator: 75 operands: 62
56	Operation	operator: 90 operands: 63
57	ExprTuple	64
58	ExprTuple	65, 94
59	Literal
60	ExprTuple	66
61	ExprTuple	67, 94
62	ExprTuple	68, 85
63	ExprTuple	94, 69
64	Operation	operator: 72 operands: 70
65	Operation	operator: 75 operands: 71
66	Operation	operator: 72 operands: 73
67	Operation	operator: 75 operands: 74
68	Variable
69	Operation	operator: 75 operands: 76
70	ExprTuple	96, 77
71	ExprTuple	77, 79
72	Literal
73	ExprTuple	96, 78
74	ExprTuple	78, 79
75	Literal
76	ExprTuple	95, 80
77	Variable
78	Variable
79	Operation	operator: 82 operand: 84
80	Operation	operator: 82 operand: 85
81	ExprTuple	84
82	Literal
83	ExprTuple	85
84	Operation	operator: 86 operands: 87
85	Literal
86	Literal
87	ExprTuple	88, 89
88	Operation	operator: 90 operands: 91
89	Operation	operator: 92 operand: 96
90	Literal
91	ExprTuple	94, 95
92	Literal
93	ExprTuple	96
94	Literal
95	Literal
96	Literal

Expression of type Implies¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Implies ¶

from the theory of proveit.physics.quantum.QPE ¶