Expression of type Lambda

from the theory of proveit.physics.quantum.circuits

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, 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

# build up the expression from sub-expressions
expr = Lambda([j, k, l, m], 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:

# 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())

\left(j, k, l, m\right) \mapsto \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..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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