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 import A, B, S, k
from proveit.logic import Equals, Forall
from proveit.numbers import NaturalPos
from proveit.physics.quantum.circuits import MultiQubitElem, Output

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [k], instance_expr = Forall(instance_param_or_params = [A, B, S], instance_expr = Equals(Equals(MultiQubitElem(element = Output(state = A, part = k), targets = S), MultiQubitElem(element = Output(state = B, part = k), targets = S)), Equals(A, B))), domain = 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())

\forall_{k \in \mathbb{N}^+}~\left[\forall_{A, B, S}~\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{A~\mbox{part}~k~\mbox{on}~S} 
} \end{array} = \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{B~\mbox{part}~k~\mbox{on}~S} 
} \end{array}\right) = \left(A = B\right)\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 34 body: 4
3	ExprTuple	34
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	34, 12
11	Lambda	parameters: 13 body: 14
12	Literal
13	ExprTuple	32, 33, 28
14	Operation	operator: 19 operands: 15
15	ExprTuple	16, 17
16	Operation	operator: 19 operands: 18
17	Operation	operator: 19 operands: 20
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	32, 33
21	Operation	operator: 24 operands: 23
22	Operation	operator: 24 operands: 25
23	NamedExprs	element: 26 targets: 28
24	Literal
25	NamedExprs	element: 27 targets: 28
26	Operation	operator: 30 operands: 29
27	Operation	operator: 30 operands: 31
28	Variable
29	NamedExprs	state: 32 part: 34
30	Literal
31	NamedExprs	state: 33 part: 34
32	Variable
33	Variable
34	Variable