Expression of type Implies

from the theory of proveit.physics.quantum.QPE

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 ExprRange, Variable, m, n
from proveit.logic import And, Boolean, Forall, Implies, InSet, NotEquals
from proveit.numbers import Interval, one
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _t
from proveit.physics.quantum.circuits import MultiQubitElem, Output

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Interval(one, _t)
sub_expr3 = InSet(NotEquals(MultiQubitElem(element = Output(state = NumKet(m, _t), part = sub_expr1), targets = sub_expr2), MultiQubitElem(element = Output(state = NumKet(n, _t), part = sub_expr1), targets = sub_expr2)), Boolean)
expr = Implies(Forall(instance_param_or_params = [sub_expr1], instance_expr = sub_expr3, domain = sub_expr2), And(ExprRange(sub_expr1, sub_expr3, one, _t)))

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[\forall_{{_{-}a} \in \{1~\ldotp \ldotp~t\}}~\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~{_{-}a}~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert n \rangle_{t}~\mbox{part}~{_{-}a}~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}\right) \in \mathbb{B}\right)\right] \Rightarrow \left(\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert n \rangle_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}\right) \in \mathbb{B}\right) \land  \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert n \rangle_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}\right) \in \mathbb{B}\right) \land  \ldots \land  \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert n \rangle_{t}~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array}\right) \in \mathbb{B}\right)\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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