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Expression of type Forall

from the theory of proveit.logic.sets.cardinality

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 S, x
from proveit.logic import Card, Equals, Forall, InSet, NotInSet, Set, Union
from proveit.numbers import Add, Natural, one
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Card(S)
expr = Forall(instance_param_or_params = [x, S], instance_expr = Equals(Card(Union(S, Set(x))), Add(sub_expr1, one)), conditions = [InSet(sub_expr1, Natural), NotInSet(x, S)])
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_{x, S~|~|S| \in \mathbb{N}, x \notin S}~\left(|S \cup \left\{x\right\}| = \left(|S| + 1\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: 1
operand: 3
1Literal
2ExprTuple3
3Lambdaparameters: 21
body: 4
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operands: 8
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11, 12
9Literal
10ExprTuple13, 14
11Operationoperator: 28
operand: 22
12Operationoperator: 16
operands: 17
13Operationoperator: 18
operands: 19
14Operationoperator: 20
operands: 21
15ExprTuple22
16Literal
17ExprTuple24, 23
18Literal
19ExprTuple24, 25
20Literal
21ExprTuple34, 31
22Operationoperator: 26
operands: 27
23Literal
24Operationoperator: 28
operand: 31
25Literal
26Literal
27ExprTuple31, 30
28Literal
29ExprTuple31
30Operationoperator: 32
operand: 34
31Variable
32Literal
33ExprTuple34
34Variable