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.expr_arrays import A11_to_Akl, A11_to_Akl_varray, B11_to_Bkl, B11_to_Bkl_varray
from proveit.logic import Equals, Forall, Implies
from proveit.physics.quantum.circuits import circuit_Akl, circuit_Bkl

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [A11_to_Akl, B11_to_Bkl], instance_expr = Implies(Equals(A11_to_Akl_varray, B11_to_Bkl_varray), Equals(circuit_Akl, circuit_Bkl)).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, 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) = \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) = \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}

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	6, 7
5	Operation	operator: 8 operands: 9
6	ExprRange	lambda_map: 10 start_index: 34 end_index: 26
7	ExprRange	lambda_map: 11 start_index: 34 end_index: 26
8	Literal
9	ExprTuple	12, 13
10	Lambda	parameter: 42 body: 30
11	Lambda	parameter: 42 body: 31
12	Operation	operator: 15 operands: 14
13	Operation	operator: 15 operands: 16
14	ExprTuple	19, 21
15	Literal
16	ExprTuple	17, 18
17	Operation	operator: 20 operands: 19
18	Operation	operator: 20 operands: 21
19	ExprTuple	22
20	Literal
21	ExprTuple	23
22	ExprRange	lambda_map: 24 start_index: 34 end_index: 26
23	ExprRange	lambda_map: 25 start_index: 34 end_index: 26
24	Lambda	parameter: 42 body: 27
25	Lambda	parameter: 42 body: 29
26	Variable
27	ExprTuple	30
28	ExprTuple	42
29	ExprTuple	31
30	ExprRange	lambda_map: 32 start_index: 34 end_index: 35
31	ExprRange	lambda_map: 33 start_index: 34 end_index: 35
32	Lambda	parameter: 43 body: 36
33	Lambda	parameter: 43 body: 38
34	Literal
35	Variable
36	IndexedVar	variable: 39 indices: 41
37	ExprTuple	43
38	IndexedVar	variable: 40 indices: 41
39	Variable
40	Variable
41	ExprTuple	42, 43
42	Variable
43	Variable