Expression of type Conditional

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, k, l
from proveit.core_expr_types.expr_arrays import A11_to_Akl, A11_to_Akl_varray, B11_to_Bkl, B11_to_Bkl_varray
from proveit.logic import And, Forall, Implies, InSet, NotEquals
from proveit.numbers import NaturalPos
from proveit.physics.quantum.circuits import circuit_Akl, circuit_Bkl

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A11_to_Akl, B11_to_Bkl], instance_expr = Implies(NotEquals(A11_to_Akl_varray, B11_to_Bkl_varray), NotEquals(circuit_Akl, circuit_Bkl)).with_wrapping_at(2)).with_wrapping(), And(InSet(k, NaturalPos), InSet(l, 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\{\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, l}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, l}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, l}, B_{1, 1}, B_{1, 2}, \ldots, B_{1, l}, B_{2, 1}, B_{2, 2}, \ldots, B_{2, l}, \ldots\ldots, B_{k, 1}, B_{k, 2}, \ldots, B_{k, l}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left(\begin{array}{cccc} 
 A_{1, 1} & A_{2, 1} & \cdots & A_{k, 1} \\
A_{1, 2} & A_{2, 2} & \cdots & A_{k, 2} \\
\vdots & \vdots & \ddots & \vdots \\
A_{1, l} & A_{2, l} & \cdots & A_{k, l} \\
\end{array}
\right) \neq \left(\begin{array}{cccc} 
 B_{1, 1} & B_{2, 1} & \cdots & B_{k, 1} \\
B_{1, 2} & B_{2, 2} & \cdots & B_{k, 2} \\
\vdots & \vdots & \ddots & \vdots \\
B_{1, l} & B_{2, l} & \cdots & B_{k, l} \\
\end{array}
\right)\right) \Rightarrow  \\ \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \qw \\
& \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \qw \\
& \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \qw
} \end{array}\right) \neq \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{B_{1, 1}} \qwx[1] & \gate{B_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{B_{k, 1}} \qwx[1] & \qw \\
& \gate{B_{1, 2}} \qwx[1] & \gate{B_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{B_{k, 2}} \qwx[1] & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \qw \\
& \gate{B_{1, l}} & \gate{B_{2, l}} & \gate{\cdots} & \gate{B_{k, l}} & \qw
} \end{array}\right)\right) \end{array} \end{array}\right)\end{array} \textrm{ if } k \in \mathbb{N}^+ ,  l \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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