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Expression of type Forall

from the theory of proveit.logic.sets.enumeration

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 c, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, d_1_to_k
from proveit.logic import Equals, Forall, Set
from proveit.numbers import Natural
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j, k], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c, d_1_to_k], instance_expr = Equals(Set(a_1_to_i, b_1_to_j, c, d_1_to_k), Set(a_1_to_i, c, b_1_to_j, d_1_to_k))), 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_{i, j, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, c, d_{1}, d_{2}, \ldots, d_{k}}~\left(\left\{a_{1}, a_{2}, \ldots, a_{i},b_{1}, b_{2}, \ldots, b_{j}, c,d_{1}, d_{2}, \ldots, d_{k}\right\} = \left\{a_{1}, a_{2}, \ldots, a_{i}, c,b_{1}, b_{2}, \ldots, b_{j},d_{1}, d_{2}, \ldots, d_{k}\right\}\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
2Lambdaparameters: 3
body: 4
3ExprTuple33, 35, 38
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple12, 13, 14
11Lambdaparameters: 25
body: 15
12Operationoperator: 18
operands: 16
13Operationoperator: 18
operands: 17
14Operationoperator: 18
operands: 19
15Operationoperator: 20
operands: 21
16ExprTuple33, 22
17ExprTuple35, 22
18Literal
19ExprTuple38, 22
20Literal
21ExprTuple23, 24
22Literal
23Operationoperator: 26
operands: 25
24Operationoperator: 26
operands: 27
25ExprTuple28, 30, 29, 31
26Literal
27ExprTuple28, 29, 30, 31
28ExprRangelambda_map: 32
start_index: 37
end_index: 33
29Variable
30ExprRangelambda_map: 34
start_index: 37
end_index: 35
31ExprRangelambda_map: 36
start_index: 37
end_index: 38
32Lambdaparameter: 46
body: 39
33Variable
34Lambdaparameter: 46
body: 40
35Variable
36Lambdaparameter: 46
body: 41
37Literal
38Variable
39IndexedVarvariable: 42
index: 46
40IndexedVarvariable: 43
index: 46
41IndexedVarvariable: 44
index: 46
42Variable
43Variable
44Variable
45ExprTuple46
46Variable