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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 j, l
from proveit.core_expr_types import A_1_to_l, b_1_to_j
from proveit.logic import Equals, Forall, InClass
from proveit.numbers import Complex, Mult, Natural, NaturalPos
from proveit.physics.quantum import Qmult, QmultCodomain
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [j, l], instance_expr = Forall(instance_param_or_params = [b_1_to_j], instance_expr = Forall(instance_param_or_params = [A_1_to_l], instance_expr = Equals(Qmult(b_1_to_j, A_1_to_l), Qmult(Mult(b_1_to_j), A_1_to_l)).with_wrapping_at(2), condition = InClass(Qmult(A_1_to_l), QmultCodomain)), domain = Complex), domains = [NaturalPos, Natural])
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_{j \in \mathbb{N}^+, l \in \mathbb{N}}~\left[\forall_{b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\left[\forall_{A_{1}, A_{2}, \ldots, A_{l}~|~\left(A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{l}\right) \underset{{\scriptscriptstyle c}}{\in} \mathcal{Q^*}}~\left(\begin{array}{c} \begin{array}{l} \left(b_{1} \thinspace  b_{2} \thinspace  \ldots \thinspace  b_{j}\thinspace A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{l}\right) =  \\ \left(\left(b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\right)\thinspace A_{1} \thinspace  A_{2} \thinspace  \ldots \thinspace  A_{l}\right) \end{array} \end{array}\right)\right]\right]
In [5]:
stored_expr.style_options()
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()
 core typesub-expressionsexpression
0Operationoperator: 19
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple55, 50
4Conditionalvalue: 5
condition: 6
5Operationoperator: 19
operand: 9
6Operationoperator: 21
operands: 8
7ExprTuple9
8ExprTuple10, 11
9Lambdaparameters: 48
body: 12
10Operationoperator: 34
operands: 13
11Operationoperator: 34
operands: 14
12Conditionalvalue: 15
condition: 16
13ExprTuple55, 17
14ExprTuple50, 18
15Operationoperator: 19
operand: 23
16Operationoperator: 21
operands: 22
17Literal
18Literal
19Literal
20ExprTuple23
21Literal
22ExprTuple24
23Lambdaparameters: 44
body: 25
24ExprRangelambda_map: 26
start_index: 54
end_index: 55
25Conditionalvalue: 27
condition: 28
26Lambdaparameter: 60
body: 29
27Operationoperator: 30
operands: 31
28Operationoperator: 32
operands: 33
29Operationoperator: 34
operands: 35
30Literal
31ExprTuple36, 37
32Literal
33ExprTuple38, 39
34Literal
35ExprTuple57, 40
36Operationoperator: 43
operands: 41
37Operationoperator: 43
operands: 42
38Operationoperator: 43
operands: 44
39Literal
40Literal
41ExprTuple51, 46
42ExprTuple45, 46
43Literal
44ExprTuple46
45Operationoperator: 47
operands: 48
46ExprRangelambda_map: 49
start_index: 54
end_index: 50
47Literal
48ExprTuple51
49Lambdaparameter: 60
body: 52
50Variable
51ExprRangelambda_map: 53
start_index: 54
end_index: 55
52IndexedVarvariable: 56
index: 60
53Lambdaparameter: 60
body: 57
54Literal
55Variable
56Variable
57IndexedVarvariable: 58
index: 60
58Variable
59ExprTuple60
60Variable