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Expression of type Forall

from the theory of proveit.logic.equality

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, n
from proveit.core_expr_types import P__x_1_to_n, P__y_1_to_n, x_1_to_n, x_eq_y__1_to_n, y_1_to_n
from proveit.logic import Forall
from proveit.numbers import Natural
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [P, x_1_to_n, y_1_to_n], instance_expr = P__x_1_to_n, conditions = [P__y_1_to_n, x_eq_y__1_to_n]), domain = 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_{n \in \mathbb{N}}~\left[\forall_{P, x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2}, \ldots, y_{n}~|~P\left(y_{1}, y_{2}, \ldots, y_{n}\right),\left(x_{1} = y_{1}\right), \left(x_{2} = y_{2}\right), \ldots, \left(x_{n} = y_{n}\right)}~P\left(x_{1}, x_{2}, \ldots, x_{n}\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: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 31
body: 4
3ExprTuple31
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple31, 12
11Lambdaparameters: 13
body: 14
12Literal
13ExprTuple24, 20, 27
14Conditionalvalue: 15
condition: 16
15Operationoperator: 24
operands: 17
16Operationoperator: 18
operands: 19
17ExprTuple20
18Literal
19ExprTuple21, 22
20ExprRangelambda_map: 23
start_index: 30
end_index: 31
21Operationoperator: 24
operands: 25
22ExprRangelambda_map: 26
start_index: 30
end_index: 31
23Lambdaparameter: 39
body: 34
24Variable
25ExprTuple27
26Lambdaparameter: 39
body: 28
27ExprRangelambda_map: 29
start_index: 30
end_index: 31
28Operationoperator: 32
operands: 33
29Lambdaparameter: 39
body: 35
30Literal
31Variable
32Literal
33ExprTuple34, 35
34IndexedVarvariable: 36
index: 39
35IndexedVarvariable: 37
index: 39
36Variable
37Variable
38ExprTuple39
39Variable