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 ExprRange, IndexedVar, Variable, a, c, l
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Exp, Mult, Sum, frac, one, two, zero
from proveit.physics.quantum.QPE import _diff_l_scaled_delta_floor, _neg_domain

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = [l]
sub_expr3 = IndexedVar(a, one)
sub_expr4 = ExprRange(sub_expr1, IndexedVar(c, sub_expr1), one, zero)
sub_expr5 = frac(one, Exp(_diff_l_scaled_delta_floor, two))
expr = Implies(Forall(instance_param_or_params = sub_expr2, instance_expr = InSet(sub_expr5, Complex), domain = _neg_domain), Forall(instance_param_or_params = [sub_expr3, sub_expr4], instance_expr = Equals(Mult(sub_expr3, Sum(index_or_indices = sub_expr2, summand = sub_expr5, domain = _neg_domain), sub_expr4), Sum(index_or_indices = sub_expr2, summand = Mult(sub_expr3, sub_expr5, sub_expr4), domain = _neg_domain)).with_wrapping_at(2), domain = Complex).with_wrapping()).with_wrapping_at(2)

# 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())

\begin{array}{c} \begin{array}{l} \left[\forall_{l \in \{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\}}~\left(\frac{1}{\left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}} \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\begin{array}{l}\forall_{a_{1}, c_{1}, c_{2}, \ldots, c_{0} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot \left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \frac{1}{\left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{0}\right) =  \\ \left(\sum_{l = -2^{t - 1} + 1}^{-\left(e + 1\right)} \left(a_{1} \cdot \frac{1}{\left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{0}\right)\right) \end{array} \end{array}\right)\end{array}\right] \end{array} \end{array}

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