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, a, b, k, m
from proveit.core_expr_types import U_1_to_m
from proveit.logic import And, Forall, Implies, InSet
from proveit.numbers import NaturalPos
from proveit.physics.quantum.circuits import QcircuitEquiv, circuit_aU, circuit_aUb, circuit_b

# build up the expression from sub-expressions
expr = Lambda([k, m], Conditional(Forall(instance_param_or_params = [a, b, U_1_to_m], instance_expr = Implies(circuit_aUb, QcircuitEquiv(circuit_aU, circuit_b))), And(InSet(k, NaturalPos), InSet(m, NaturalPos))))

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(k, m\right) \mapsto \left\{\forall_{a, b, U_{1}, U_{2}, \ldots, U_{m}}~\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{a} & \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{m} \\ \downarrow\end{array}} & \qout{b}
} \end{array}\right) \Rightarrow \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{a} & \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{m} \\ \downarrow\end{array}} & { /^{k} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{b} & { /^{k} } \qw
} \end{array}\right)\right)\right) \textrm{ if } k \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	66, 39
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
9	Lambda	parameters: 12 body: 13
10	Operation	operator: 15 operands: 14
11	Operation	operator: 15 operands: 16
12	ExprTuple	62, 63, 34
13	Operation	operator: 17 operands: 18
14	ExprTuple	66, 19
15	Literal
16	ExprTuple	39, 19
17	Literal
18	ExprTuple	20, 21
19	Literal
20	Operation	operator: 30 operands: 22
21	Operation	operator: 23 operands: 24
22	ExprTuple	33, 34, 25
23	Literal
24	ExprTuple	26, 27
25	ExprTuple	28
26	Operation	operator: 30 operands: 29
27	Operation	operator: 30 operand: 35
28	ExprRange	lambda_map: 32 start_index: 65 end_index: 66
29	ExprTuple	33, 34
30	Literal
31	ExprTuple	35
32	Lambda	parameter: 64 body: 36
33	ExprTuple	37
34	ExprRange	lambda_map: 38 start_index: 65 end_index: 39
35	ExprTuple	40
36	Operation	operator: 52 operands: 41
37	ExprRange	lambda_map: 42 start_index: 65 end_index: 66
38	Lambda	parameter: 64 body: 43
39	Variable
40	ExprRange	lambda_map: 44 start_index: 65 end_index: 66
41	NamedExprs	element: 45 targets: 56
42	Lambda	parameter: 64 body: 46
43	IndexedVar	variable: 47 index: 64
44	Lambda	parameter: 64 body: 49
45	Operation	operator: 50 operands: 59
46	Operation	operator: 52 operands: 51
47	Variable
48	ExprTuple	64
49	Operation	operator: 52 operands: 53
50	Literal
51	NamedExprs	element: 54 targets: 56
52	Literal
53	NamedExprs	element: 55 targets: 56
54	Operation	operator: 58 operands: 57
55	Operation	operator: 58 operands: 59
56	Operation	operator: 60 operands: 61
57	NamedExprs	state: 62 part: 64
58	Literal
59	NamedExprs	state: 63 part: 64
60	Literal
61	ExprTuple	65, 66
62	Variable
63	Variable
64	Variable
65	Literal
66	Variable