Expression of type Lambda

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 Lambda, e, l
from proveit.logic import And, Forall, Iff, InSet, Union
from proveit.numbers import ModAbs, greater
from proveit.physics.quantum.QPE import _e_domain, _full_domain, _neg_domain, _pos_domain, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = greater(ModAbs(l, _two_pow_t), e)
expr = Lambda(l, Forall(instance_param_or_params = [e], instance_expr = Iff(And(InSet(l, Union(_neg_domain, _pos_domain)), sub_expr1), And(InSet(l, _full_domain), sub_expr1)).with_wrapping_at(1), domain = _e_domain).with_wrapping())

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())

l \mapsto \left[\begin{array}{l}\forall_{e \in \{1~\ldotp \ldotp~2^{t - 1} - 2\}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left(l \in \left(\{-2^{t - 1} + 1~\ldotp \ldotp~-\left(e + 1\right)\} \cup \{e + 1~\ldotp \ldotp~2^{t - 1}\}\right)\right) \land \left(\left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right)\right) \\  \Leftrightarrow \left(\left(l \in \{-2^{t - 1} + 1~\ldotp \ldotp~2^{t - 1}\}\right) \land \left(\left|l\right|_{\textup{mod}\thinspace 2^{t}} > e\right)\right) \end{array} \end{array}\right)\end{array}\right]

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 42 body: 2
1	ExprTuple	42
2	Operation	operator: 3 operand: 5
3	Literal
4	ExprTuple	5
5	Lambda	parameter: 57 body: 7
6	ExprTuple	57
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operands: 11
9	Operation	operator: 25 operands: 12
10	Literal
11	ExprTuple	13, 14
12	ExprTuple	57, 15
13	Operation	operator: 17 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 45 operands: 19
16	ExprTuple	20, 22
17	Literal
18	ExprTuple	21, 22
19	ExprTuple	68, 23
20	Operation	operator: 25 operands: 24
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operands: 28
23	Operation	operator: 62 operands: 29
24	ExprTuple	42, 30
25	Literal
26	ExprTuple	42, 31
27	Literal
28	ExprTuple	57, 32
29	ExprTuple	56, 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 45 operands: 36
32	Operation	operator: 37 operands: 38
33	Operation	operator: 66 operand: 60
34	Literal
35	ExprTuple	40, 41
36	ExprTuple	48, 56
37	Literal
38	ExprTuple	42, 43
39	ExprTuple	60
40	Operation	operator: 45 operands: 44
41	Operation	operator: 45 operands: 46
42	Variable
43	Operation	operator: 58 operands: 47
44	ExprTuple	48, 49
45	Literal
46	ExprTuple	53, 56
47	ExprTuple	60, 64
48	Operation	operator: 62 operands: 50
49	Operation	operator: 66 operand: 53
50	ExprTuple	52, 68
51	ExprTuple	53
52	Operation	operator: 66 operand: 56
53	Operation	operator: 62 operands: 55
54	ExprTuple	56
55	ExprTuple	57, 68
56	Operation	operator: 58 operands: 59
57	Variable
58	Literal
59	ExprTuple	60, 61
60	Literal
61	Operation	operator: 62 operands: 63
62	Literal
63	ExprTuple	64, 65
64	Literal
65	Operation	operator: 66 operand: 68
66	Literal
67	ExprTuple	68
68	Literal