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Expression of type Conditional

from the theory of proveit.physics.quantum.circuits

In [1]:
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, j, k, l, m
from proveit.core_expr_types import A_1_to_j, B_1_to_k, C_1_to_l, D_1_to_m
from proveit.logic import And, Forall, Implies, InSet
from proveit.numbers import Natural
from proveit.physics.quantum.circuits import QcircuitEquiv, circuit_AjBkCl, circuit_AjDmCl, circuit_Bk, circuit_Dm
In [2]:
# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A_1_to_j, B_1_to_k, C_1_to_l, D_1_to_m], instance_expr = Implies(QcircuitEquiv(circuit_Bk, circuit_Dm), QcircuitEquiv(circuit_AjBkCl, circuit_AjDmCl).with_wrapping_at(1)).with_wrapping_at(2)).with_wrapping(), And(InSet(j, Natural), InSet(k, Natural), InSet(l, Natural), InSet(m, Natural)))
expr:
In [3]:
# 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
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
\left\{\begin{array}{l}\forall_{A_{1}, A_{2}, \ldots, A_{j}, B_{1}, B_{2}, \ldots, B_{k}, C_{1}, C_{2}, \ldots, C_{l}, D_{1}, D_{2}, \ldots, D_{m}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\B_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\B_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\B_{k} \\ \downarrow\end{array}} & \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\D_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\D_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\D_{m} \\ \downarrow\end{array}} & \qw
} \end{array}\right)\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\A_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\A_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\A_{j} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\B_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\B_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\B_{k} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\C_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\C_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\C_{l} \\ \downarrow\end{array}} & \qw
} \end{array}\right) \\  \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\A_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\A_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\A_{j} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\D_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\D_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\D_{m} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\C_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\C_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\C_{l} \\ \downarrow\end{array}} & \qw
} \end{array}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array} \textrm{ if } j \in \mathbb{N} ,  k \in \mathbb{N} ,  l \in \mathbb{N} ,  m \in \mathbb{N}\right..
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
condition_delimiter'comma' or 'and'commacomma('with_comma_delimiter', 'with_conjunction_delimiter')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Conditionalvalue: 1
condition: 2
1Operationoperator: 3
operand: 7
2Operationoperator: 5
operands: 6
3Literal
4ExprTuple7
5Literal
6ExprTuple8, 9, 10, 11
7Lambdaparameters: 12
body: 13
8Operationoperator: 17
operands: 14
9Operationoperator: 17
operands: 15
10Operationoperator: 17
operands: 16
11Operationoperator: 17
operands: 18
12ExprTuple37, 36, 39, 38
13Operationoperator: 19
operands: 20
14ExprTuple43, 21
15ExprTuple41, 21
16ExprTuple48, 21
17Literal
18ExprTuple45, 21
19Literal
20ExprTuple22, 23
21Literal
22Operationoperator: 25
operands: 24
23Operationoperator: 25
operands: 26
24ExprTuple27, 28
25Literal
26ExprTuple29, 30
27Operationoperator: 34
operands: 31
28Operationoperator: 34
operands: 32
29Operationoperator: 34
operands: 33
30Operationoperator: 34
operands: 35
31ExprTuple36
32ExprTuple38
33ExprTuple37, 36, 39
34Literal
35ExprTuple37, 38, 39
36ExprRangelambda_map: 40
start_index: 47
end_index: 41
37ExprRangelambda_map: 42
start_index: 47
end_index: 43
38ExprRangelambda_map: 44
start_index: 47
end_index: 45
39ExprRangelambda_map: 46
start_index: 47
end_index: 48
40Lambdaparameter: 58
body: 49
41Variable
42Lambdaparameter: 58
body: 50
43Variable
44Lambdaparameter: 58
body: 51
45Variable
46Lambdaparameter: 58
body: 52
47Literal
48Variable
49IndexedVarvariable: 53
index: 58
50IndexedVarvariable: 54
index: 58
51IndexedVarvariable: 55
index: 58
52IndexedVarvariable: 56
index: 58
53Variable
54Variable
55Variable
56Variable
57ExprTuple58
58Variable