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Expression of type Forall

from the theory of proveit.linear_algebra.tensors

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 A, B, K, U, V, W
from proveit.linear_algebra import TensorProd, VecSpaces
from proveit.logic import And, Exists, Forall, InSet
In [2]:
# build up the expression from sub-expressions
sub_expr1 = VecSpaces(K)
expr = Forall(instance_param_or_params = [K], instance_expr = Forall(instance_param_or_params = [W], instance_expr = Forall(instance_param_or_params = [A, B], instance_expr = Exists(instance_param_or_params = [U, V], instance_expr = And(InSet(A, U), InSet(B, V)), domain = sub_expr1), condition = InSet(TensorProd(A, B), W)), domain = sub_expr1))
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}~\left[\forall_{W \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{A, B~|~\left(A {\otimes} B\right) \in W}~\left[\exists_{U, V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left(\left(A \in U\right) \land \left(B \in V\right)\right)\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: 10
operand: 2
1ExprTuple2
2Lambdaparameter: 49
body: 3
3Operationoperator: 10
operand: 5
4ExprTuple5
5Lambdaparameter: 22
body: 7
6ExprTuple22
7Conditionalvalue: 8
condition: 9
8Operationoperator: 10
operand: 13
9Operationoperator: 40
operands: 12
10Literal
11ExprTuple13
12ExprTuple22, 46
13Lambdaparameters: 26
body: 14
14Conditionalvalue: 15
condition: 16
15Operationoperator: 17
operand: 20
16Operationoperator: 37
operands: 19
17Literal
18ExprTuple20
19ExprTuple21, 22
20Lambdaparameters: 23
body: 24
21Operationoperator: 25
operands: 26
22Variable
23ExprTuple44, 45
24Conditionalvalue: 27
condition: 28
25Literal
26ExprTuple42, 43
27Operationoperator: 30
operands: 29
28Operationoperator: 30
operands: 31
29ExprTuple32, 33
30Literal
31ExprTuple34, 35
32Operationoperator: 37
operands: 36
33Operationoperator: 37
operands: 38
34Operationoperator: 40
operands: 39
35Operationoperator: 40
operands: 41
36ExprTuple42, 44
37Literal
38ExprTuple43, 45
39ExprTuple44, 46
40Literal
41ExprTuple45, 46
42Variable
43Variable
44Variable
45Variable
46Operationoperator: 47
operand: 49
47Literal
48ExprTuple49
49Variable