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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, n
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
from proveit.numbers import NaturalPos
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [P, Q, alpha], 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)), domain = NaturalPos)
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_{n \in \mathbb{N}^+}~\left[\forall_{P, Q, \alpha}~\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)\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: 22
operand: 2
1ExprTuple2
2Lambdaparameter: 47
body: 4
3ExprTuple47
4Conditionalvalue: 5
condition: 6
5Operationoperator: 22
operand: 10
6Operationoperator: 8
operands: 9
7ExprTuple10
8Literal
9ExprTuple47, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple40, 35, 38
13Operationoperator: 33
operands: 14
14ExprTuple15, 38
15Operationoperator: 16
operands: 17
16Literal
17ExprTuple18, 19
18Operationoperator: 20
operand: 24
19Operationoperator: 22
operand: 25
20Literal
21ExprTuple24
22Literal
23ExprTuple25
24Lambdaparameters: 32
body: 26
25Lambdaparameters: 41
body: 27
26Conditionalvalue: 28
condition: 29
27Conditionalvalue: 30
condition: 31
28Operationoperator: 40
operands: 32
29Operationoperator: 35
operands: 32
30Operationoperator: 33
operands: 34
31Operationoperator: 35
operands: 41
32ExprTuple36
33Literal
34ExprTuple37, 38
35Variable
36ExprRangelambda_map: 39
start_index: 46
end_index: 47
37Operationoperator: 40
operands: 41
38Variable
39Lambdaparameter: 51
body: 42
40Variable
41ExprTuple43
42IndexedVarvariable: 44
index: 51
43ExprRangelambda_map: 45
start_index: 46
end_index: 47
44Variable
45Lambdaparameter: 51
body: 48
46Literal
47Variable
48IndexedVarvariable: 49
index: 51
49Variable
50ExprTuple51
51Variable