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Expression of type Forall

from the theory of proveit.physics.quantum.algebra

In [1]:
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, IndexedVar, Variable, a, k
from proveit.core_expr_types import a_1_to_k
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals, Forall
from proveit.numbers import BinarySequence, Bit, NaturalPos, one
from proveit.physics.quantum import Ket, NumKet
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [k], instance_expr = Forall(instance_param_or_params = [a_1_to_k], instance_expr = Equals(NumKet(BinarySequence(a_1_to_k), k), TensorProd(ExprRange(sub_expr1, Ket(IndexedVar(a, sub_expr1)), one, k))), domain = Bit), domain = NaturalPos)
Out[2]:
expr:
In [3]:
# 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
In [4]:
# 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_{1}, a_{2}, \ldots, a_{k} \in \mathbb{N}^{\leq 1}}~\left(\lvert \texttt{\#}\texttt{b}(a_{1}, a_{2}, \ldots, a_{k}) \rangle_{k} = \left(\lvert a_{1} \rangle {\otimes}  \lvert a_{2} \rangle {\otimes}  \ldots {\otimes}  \lvert a_{k} \rangle\right)\right)\right]
In [5]:
stored_expr.style_options()
Out[5]:
namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()
Out[6]:
 core typesub-expressionsexpression
0Operationoperator: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 40
body: 4
3ExprTuple40
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 33
operands: 9
7Literal
8ExprTuple10
9ExprTuple40, 11
10Lambdaparameters: 31
body: 12
11Literal
12Conditionalvalue: 13
condition: 14
13Operationoperator: 15
operands: 16
14Operationoperator: 17
operands: 18
15Literal
16ExprTuple19, 20
17Literal
18ExprTuple21
19Operationoperator: 22
operands: 23
20Operationoperator: 24
operands: 25
21ExprRangelambda_map: 26
start_index: 39
end_index: 40
22Literal
23ExprTuple27, 40
24Literal
25ExprTuple28
26Lambdaparameter: 46
body: 29
27Operationoperator: 30
operands: 31
28ExprRangelambda_map: 32
start_index: 39
end_index: 40
29Operationoperator: 33
operands: 34
30Literal
31ExprTuple35
32Lambdaparameter: 46
body: 36
33Literal
34ExprTuple43, 37
35ExprRangelambda_map: 38
start_index: 39
end_index: 40
36Operationoperator: 41
operand: 43
37Literal
38Lambdaparameter: 46
body: 43
39Literal
40Variable
41Literal
42ExprTuple43
43IndexedVarvariable: 44
index: 46
44Variable
45ExprTuple46
46Variable