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 Exp, LessEq, Mult, Sum, frac, one, subtract, two
from proveit.physics.quantum.QPE import _delta_b_floor, _diff_l_scaled_delta_floor, _pos_domain, _two_pow_t

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

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

stored_expr.style_options()

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')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
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: 61 body: 12
9	Operation	operator: 14 operand: 18
10	Operation	operator: 14 operand: 19
11	ExprTuple	61
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: 63 body: 23
19	Lambda	parameter: 63 body: 25
20	Literal
21	ExprTuple	26, 27
22	ExprTuple	61, 42
23	Conditional	value: 28 condition: 30
24	ExprTuple	63
25	Conditional	value: 29 condition: 30
26	Operation	operator: 34 operands: 31
27	Operation	operator: 34 operands: 32
28	Operation	operator: 34 operands: 33
29	Operation	operator: 34 operands: 35
30	Operation	operator: 36 operands: 37
31	ExprTuple	77, 38
32	ExprTuple	77, 39
33	ExprTuple	77, 40
34	Literal
35	ExprTuple	77, 41
36	Literal
37	ExprTuple	63, 42
38	Operation	operator: 78 operands: 43
39	Operation	operator: 78 operands: 44
40	Operation	operator: 78 operands: 45
41	Operation	operator: 78 operands: 46
42	Operation	operator: 47 operands: 48
43	ExprTuple	49, 82
44	ExprTuple	50, 82
45	ExprTuple	51, 82
46	ExprTuple	52, 82
47	Literal
48	ExprTuple	53, 54
49	Operation	operator: 67 operands: 55
50	Operation	operator: 67 operands: 56
51	Operation	operator: 67 operands: 57
52	Operation	operator: 67 operands: 58
53	Operation	operator: 67 operands: 59
54	Operation	operator: 78 operands: 60
55	ExprTuple	61, 62
56	ExprTuple	61, 70
57	ExprTuple	63, 62
58	ExprTuple	63, 70
59	ExprTuple	64, 77
60	ExprTuple	82, 65
61	Variable
62	Operation	operator: 73 operand: 69
63	Variable
64	Variable
65	Operation	operator: 67 operands: 68
66	ExprTuple	69
67	Literal
68	ExprTuple	83, 70
69	Operation	operator: 71 operands: 72
70	Operation	operator: 73 operand: 77
71	Literal
72	ExprTuple	75, 76
73	Literal
74	ExprTuple	77
75	Operation	operator: 78 operands: 79
76	Operation	operator: 80 operand: 84
77	Literal
78	Literal
79	ExprTuple	82, 83
80	Literal
81	ExprTuple	84
82	Literal
83	Literal
84	Literal