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, e, l
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Abs, Add, Exp, Interval, Neg, one, subtract, two
from proveit.physics.quantum.QPE import _neg_domain, _rel_indexed_alpha, _two_pow__t_minus_one

# build up the expression from sub-expressions
sub_expr1 = Exp(Abs(_rel_indexed_alpha), two)
sub_expr2 = Conditional(sub_expr1, InSet(l, _neg_domain))
sub_expr3 = Conditional(sub_expr1, InSet(l, Interval(Add(Neg(_two_pow__t_minus_one), one), subtract(Neg(e), one))))
expr = Implies(Forall(instance_param_or_params = [l], instance_expr = Equals(sub_expr2, sub_expr3)), Equals(Lambda(l, sub_expr2), Lambda(l, sub_expr3)).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}~\left(\left\{\left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\}\right.. = \left\{\left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~-e - 1\}\right..\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[l \mapsto \left\{\left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\}\right..\right] =  \\ \left[l \mapsto \left\{\left|\alpha_{b_{\textit{f}} \oplus l}\right|^{2} \textrm{ if } l \in \{-2^{t - 1} + 1~\ldotp \ldotp~-e - 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: 13 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 51 body: 11
9	Lambda	parameter: 51 body: 15
10	Lambda	parameter: 51 body: 16
11	Operation	operator: 13 operands: 14
12	ExprTuple	51
13	Literal
14	ExprTuple	15, 16
15	Conditional	value: 18 condition: 17
16	Conditional	value: 18 condition: 19
17	Operation	operator: 22 operands: 20
18	Operation	operator: 54 operands: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	51, 24
21	ExprTuple	25, 56
22	Literal
23	ExprTuple	51, 26
24	Operation	operator: 30 operands: 27
25	Operation	operator: 28 operand: 33
26	Operation	operator: 30 operands: 31
27	ExprTuple	34, 32
28	Literal
29	ExprTuple	33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 62 operand: 41
33	Operation	operator: 37 operand: 42
34	Operation	operator: 58 operands: 39
35	Operation	operator: 58 operands: 40
36	ExprTuple	41
37	Literal
38	ExprTuple	42
39	ExprTuple	43, 64
40	ExprTuple	44, 61
41	Operation	operator: 58 operands: 45
42	Operation	operator: 46 operands: 47
43	Operation	operator: 62 operand: 52
44	Operation	operator: 62 operand: 53
45	ExprTuple	53, 64
46	Literal
47	ExprTuple	50, 51
48	ExprTuple	52
49	ExprTuple	53
50	Literal
51	Variable
52	Operation	operator: 54 operands: 55
53	Variable
54	Literal
55	ExprTuple	56, 57
56	Literal
57	Operation	operator: 58 operands: 59
58	Literal
59	ExprTuple	60, 61
60	Literal
61	Operation	operator: 62 operand: 64
62	Literal
63	ExprTuple	64
64	Literal