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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
from proveit.logic import Forall, Implies, Not
from proveit.logic.booleans.quantification import general_exists_Px_st_Qx, general_forall__Py_not_T__st_Qy
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [P, Q], instance_expr = Implies(general_exists_Px_st_Qx, Not(general_forall__Py_not_T__st_Qy)).with_wrapping_at(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_{P, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\exists_{x_{1}, x_{2}, \ldots, x_{n}~|~Q\left(x_{1}, x_{2}, \ldots, x_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ (\lnot \left[\forall_{y_{1}, y_{2}, \ldots, y_{n}~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(P\left(y_{1}, y_{2}, \ldots, y_{n}\right) \neq \top\right)\right]) \end{array} \end{array}\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: 16
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple34, 29
4Operationoperator: 5
operands: 6
5Literal
6ExprTuple7, 8
7Operationoperator: 9
operand: 13
8Operationoperator: 11
operand: 14
9Literal
10ExprTuple13
11Literal
12ExprTuple14
13Lambdaparameters: 21
body: 15
14Operationoperator: 16
operand: 20
15Conditionalvalue: 18
condition: 19
16Literal
17ExprTuple20
18Operationoperator: 34
operands: 21
19Operationoperator: 29
operands: 21
20Lambdaparameters: 35
body: 22
21ExprTuple23
22Conditionalvalue: 24
condition: 25
23ExprRangelambda_map: 26
start_index: 38
end_index: 39
24Operationoperator: 27
operands: 28
25Operationoperator: 29
operands: 35
26Lambdaparameter: 43
body: 30
27Literal
28ExprTuple31, 32
29Variable
30IndexedVarvariable: 33
index: 43
31Operationoperator: 34
operands: 35
32Literal
33Variable
34Variable
35ExprTuple36
36ExprRangelambda_map: 37
start_index: 38
end_index: 39
37Lambdaparameter: 43
body: 40
38Literal
39Variable
40IndexedVarvariable: 41
index: 43
41Variable
42ExprTuple43
43Variable