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
from proveit.logic import And, Boolean, Forall, Implies, InSet, NotEquals
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum.QPE import _ket_u, _s, _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 = MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = Interval(Add(_t, one), Add(_t, _s)))
sub_expr3 = InSet(NotEquals(sub_expr2, sub_expr2), Boolean)
expr = Implies(Forall(instance_param_or_params = [sub_expr1], instance_expr = sub_expr3, domain = Interval(one, _s)), And(ExprRange(sub_expr1, sub_expr3, one, _s)))

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~s\}}~\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~{_{-}a}~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~{_{-}a}~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right) \in \mathbb{B}\right)\right] \Rightarrow \left(\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right) \in \mathbb{B}\right) \land  \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array}\right) \in \mathbb{B}\right) \land  \ldots \land  \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \neq \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \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: 35 body: 11
10	ExprRange	lambda_map: 12 start_index: 41 end_index: 43
11	Conditional	value: 15 condition: 13
12	Lambda	parameter: 35 body: 15
13	Operation	operator: 17 operands: 16
14	ExprTuple	35
15	Operation	operator: 17 operands: 18
16	ExprTuple	35, 19
17	Literal
18	ExprTuple	20, 21
19	Operation	operator: 32 operands: 22
20	Operation	operator: 23 operands: 24
21	Literal
22	ExprTuple	41, 43
23	Literal
24	ExprTuple	25, 25
25	Operation	operator: 26 operands: 27
26	Literal
27	NamedExprs	element: 28 targets: 29
28	Operation	operator: 30 operands: 31
29	Operation	operator: 32 operands: 33
30	Literal
31	NamedExprs	state: 34 part: 35
32	Literal
33	ExprTuple	36, 37
34	Literal
35	Variable
36	Operation	operator: 39 operands: 38
37	Operation	operator: 39 operands: 40
38	ExprTuple	42, 41
39	Literal
40	ExprTuple	42, 43
41	Literal
42	Literal
43	Literal