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 Conditional, ConditionalSet, ExprRange, Lambda, U, Variable, s, t
from proveit.linear_algebra import MatrixExp
from proveit.logic import Equals, NotEquals, Set
from proveit.numbers import Add, Exp, Interval, Neg, one, two
from proveit.physics.quantum import CONTROL, I
from proveit.physics.quantum.circuits import Gate, MultiQubitElem

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr3 = Add(t, one)
sub_expr4 = Add(sub_expr1, t)
expr = Lambda(sub_expr1, [ExprRange(sub_expr2, ConditionalSet(Conditional(MultiQubitElem(element = CONTROL, targets = Set(sub_expr3)), Equals(sub_expr4, sub_expr2)), Conditional(Gate(operation = I).with_implicit_representation(), NotEquals(sub_expr4, sub_expr2))), one, t).with_case_simplification(), ExprRange(sub_expr2, MultiQubitElem(element = Gate(operation = MatrixExp(U, Exp(two, Neg(sub_expr1))), part = sub_expr2), targets = Interval(sub_expr3, Add(t, s))), one, s)])

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())

{_{-}b} \mapsto \left(\left\{\begin{array}{ccc}\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{CONTROL~\mbox{on}~\left\{t + 1\right\}} & \qw 
} \end{array} \textrm{ if } \left({_{-}b} + t\right) = 1 \\ \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} \textrm{ if } \left({_{-}b} + t\right) \neq 1\end{array}\right.., \left\{\begin{array}{ccc}\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{CONTROL~\mbox{on}~\left\{t + 1\right\}} & \qw 
} \end{array} \textrm{ if } \left({_{-}b} + t\right) = 2 \\ \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} \textrm{ if } \left({_{-}b} + t\right) \neq 2\end{array}\right.., \ldots, \left\{\begin{array}{ccc}\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{CONTROL~\mbox{on}~\left\{t + 1\right\}} & \qw 
} \end{array} \textrm{ if } \left({_{-}b} + t\right) = t \\ \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} \textrm{ if } \left({_{-}b} + t\right) \neq t\end{array}\right..,\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{-{_{-}b}}}~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{-{_{-}b}}}~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{-{_{-}b}}}~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \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: 57 body: 1
1	ExprTuple	2, 3
2	ExprRange	lambda_map: 4 start_index: 54 end_index: 53
3	ExprRange	lambda_map: 5 start_index: 54 end_index: 45
4	Lambda	parameter: 36 body: 6
5	Lambda	parameter: 36 body: 8
6	Operation	operator: 9 operands: 10
7	ExprTuple	36
8	Operation	operator: 23 operands: 11
9	Literal
10	ExprTuple	12, 13
11	NamedExprs	element: 14 targets: 15
12	Conditional	value: 16 condition: 17
13	Conditional	value: 18 condition: 19
14	Operation	operator: 26 operands: 20
15	Operation	operator: 21 operands: 22
16	Operation	operator: 23 operands: 24
17	Operation	operator: 25 operands: 29
18	Operation	operator: 26 operands: 27
19	Operation	operator: 28 operands: 29
20	NamedExprs	operation: 30 part: 36
21	Literal
22	ExprTuple	46, 31
23	Literal
24	NamedExprs	element: 32 targets: 33
25	Literal
26	Literal
27	NamedExprs	operation: 34
28	Literal
29	ExprTuple	35, 36
30	Operation	operator: 37 operands: 38
31	Operation	operator: 49 operands: 39
32	Literal
33	Operation	operator: 40 operand: 46
34	Literal
35	Operation	operator: 49 operands: 42
36	Variable
37	Literal
38	ExprTuple	43, 44
39	ExprTuple	53, 45
40	Literal
41	ExprTuple	46
42	ExprTuple	57, 53
43	Variable
44	Operation	operator: 47 operands: 48
45	Variable
46	Operation	operator: 49 operands: 50
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	53, 54
51	Literal
52	Operation	operator: 55 operand: 57
53	Variable
54	Literal
55	Literal
56	ExprTuple	57
57	Variable