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Expression of type Forall

from the theory of proveit.logic.booleans.quantification.existence

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 P, Q, alpha
from proveit.core_expr_types import P__x_1_to_n, Q__x_1_to_n, x_1_to_n
from proveit.logic import And, Forall, Implies
from proveit.logic.booleans.quantification import general_exists_Py_st_Qy
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [P, Q], instance_expr = Implies(And(general_exists_Py_st_Qy, Forall(instance_param_or_params = [x_1_to_n], instance_expr = Implies(P__x_1_to_n, alpha), condition = Q__x_1_to_n)), alpha))
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_{P, Q}~\left(\left(\left[\exists_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~P\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right] \land \left[\forall_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~\left(P\left(x_{1}, x_{2}, \ldots, x_{n}\right) \Rightarrow \alpha\right)\right]\right) \Rightarrow \alpha\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: 13
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple31, 26
4Operationoperator: 24
operands: 5
5ExprTuple6, 29
6Operationoperator: 7
operands: 8
7Literal
8ExprTuple9, 10
9Operationoperator: 11
operand: 15
10Operationoperator: 13
operand: 16
11Literal
12ExprTuple15
13Literal
14ExprTuple16
15Lambdaparameters: 23
body: 17
16Lambdaparameters: 32
body: 18
17Conditionalvalue: 19
condition: 20
18Conditionalvalue: 21
condition: 22
19Operationoperator: 31
operands: 23
20Operationoperator: 26
operands: 23
21Operationoperator: 24
operands: 25
22Operationoperator: 26
operands: 32
23ExprTuple27
24Literal
25ExprTuple28, 29
26Variable
27ExprRangelambda_map: 30
start_index: 37
end_index: 38
28Operationoperator: 31
operands: 32
29Variable
30Lambdaparameter: 42
body: 33
31Variable
32ExprTuple34
33IndexedVarvariable: 35
index: 42
34ExprRangelambda_map: 36
start_index: 37
end_index: 38
35Variable
36Lambdaparameter: 42
body: 39
37Literal
38Variable
39IndexedVarvariable: 40
index: 42
40Variable
41ExprTuple42
42Variable