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 Conditional, Lambda, l
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Exp, Mult, four, frac, one, two
from proveit.physics.quantum.QPE import _diff_l_scaled_delta_floor, _pos_domain

# build up the expression from sub-expressions
sub_expr1 = Exp(_diff_l_scaled_delta_floor, two)
sub_expr2 = InSet(l, _pos_domain)
sub_expr3 = frac(one, Mult(four, sub_expr1))
sub_expr4 = Mult(frac(one, four), frac(one, sub_expr1))
expr = Implies(Forall(instance_param_or_params = [l], instance_expr = Equals(sub_expr3, sub_expr4), domain = _pos_domain), Equals(Lambda(l, Conditional(sub_expr3, sub_expr2)), Lambda(l, Conditional(sub_expr4, sub_expr2))).with_wrapping_at(2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\forall_{l \in \{e + 1~\ldotp \ldotp~2^{t - 1}\}}~\left(\frac{1}{4 \cdot \left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}} = \left(\frac{1}{4} \cdot \frac{1}{\left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}}\right)\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[l \mapsto \left\{\frac{1}{4 \cdot \left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}} \textrm{ if } l \in \{e + 1~\ldotp \ldotp~2^{t - 1}\}\right..\right] =  \\ \left[l \mapsto \left\{\frac{1}{4} \cdot \frac{1}{\left(l - \left(2^{t} \cdot \delta_{b_{\textit{f}}}\right)\right)^{2}} \textrm{ if } l \in \{e + 1~\ldotp \ldotp~2^{t - 1}\}\right..\right] \end{array} \end{array}\right) \end{array} \end{array}

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.	()	(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')

# 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: 17 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 50 body: 11
9	Lambda	parameter: 50 body: 12
10	Lambda	parameter: 50 body: 14
11	Conditional	value: 15 condition: 16
12	Conditional	value: 21 condition: 16
13	ExprTuple	50
14	Conditional	value: 22 condition: 16
15	Operation	operator: 17 operands: 18
16	Operation	operator: 19 operands: 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	50, 23
21	Operation	operator: 35 operands: 24
22	Operation	operator: 56 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	52, 28
25	ExprTuple	29, 30
26	Literal
27	ExprTuple	31, 32
28	Operation	operator: 56 operands: 33
29	Operation	operator: 35 operands: 34
30	Operation	operator: 35 operands: 36
31	Operation	operator: 47 operands: 37
32	Operation	operator: 60 operands: 38
33	ExprTuple	39, 40
34	ExprTuple	52, 39
35	Literal
36	ExprTuple	52, 40
37	ExprTuple	41, 52
38	ExprTuple	64, 42
39	Literal
40	Operation	operator: 60 operands: 43
41	Variable
42	Operation	operator: 47 operands: 44
43	ExprTuple	45, 64
44	ExprTuple	65, 46
45	Operation	operator: 47 operands: 48
46	Operation	operator: 53 operand: 52
47	Literal
48	ExprTuple	50, 51
49	ExprTuple	52
50	Variable
51	Operation	operator: 53 operand: 55
52	Literal
53	Literal
54	ExprTuple	55
55	Operation	operator: 56 operands: 57
56	Literal
57	ExprTuple	58, 59
58	Operation	operator: 60 operands: 61
59	Operation	operator: 62 operand: 66
60	Literal
61	ExprTuple	64, 65
62	Literal
63	ExprTuple	66
64	Literal
65	Literal
66	Literal