Expression of type Forall

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.core_expr_types import A_1_to_j, B_1_to_k, C_1_to_l, D_1_to_m
from proveit.logic import Forall, Implies
from proveit.physics.quantum.circuits import QcircuitEquiv, circuit_AjBkCl, circuit_AjDmCl, circuit_Bk, circuit_Dm

# build up the expression from sub-expressions
expr = 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()

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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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