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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 i, j, k, x
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_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, x, b_1_to_j, c_1_to_k], instance_expr = Equals(Set(a_1_to_i, x, b_1_to_j, x, c_1_to_k), Set(a_1_to_i, b_1_to_j, x, c_1_to_k)).with_wrapping_at(2)), 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}, x, b_{1}, b_{2}, \ldots, b_{j}, c_{1}, c_{2}, \ldots, c_{k}}~\left(\begin{array}{c} \begin{array}{l} \left\{a_{1}, a_{2}, \ldots, a_{i}, x,b_{1}, b_{2}, \ldots, b_{j}, x,c_{1}, c_{2}, \ldots, c_{k}\right\} =  \\ \left\{a_{1}, a_{2}, \ldots, a_{i},b_{1}, b_{2}, \ldots, b_{j}, x,c_{1}, c_{2}, \ldots, c_{k}\right\} \end{array} \end{array}\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
3ExprTuple34, 36, 39
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 11
6Operationoperator: 9
operands: 10
7Literal
8ExprTuple11
9Literal
10ExprTuple12, 13, 14
11Lambdaparameters: 15
body: 16
12Operationoperator: 19
operands: 17
13Operationoperator: 19
operands: 18
14Operationoperator: 19
operands: 20
15ExprTuple29, 31, 30, 32
16Operationoperator: 21
operands: 22
17ExprTuple34, 23
18ExprTuple36, 23
19Literal
20ExprTuple39, 23
21Literal
22ExprTuple24, 25
23Literal
24Operationoperator: 27
operands: 26
25Operationoperator: 27
operands: 28
26ExprTuple29, 31, 30, 31, 32
27Literal
28ExprTuple29, 30, 31, 32
29ExprRangelambda_map: 33
start_index: 38
end_index: 34
30ExprRangelambda_map: 35
start_index: 38
end_index: 36
31Variable
32ExprRangelambda_map: 37
start_index: 38
end_index: 39
33Lambdaparameter: 47
body: 40
34Variable
35Lambdaparameter: 47
body: 41
36Variable
37Lambdaparameter: 47
body: 42
38Literal
39Variable
40IndexedVarvariable: 43
index: 47
41IndexedVarvariable: 44
index: 47
42IndexedVarvariable: 45
index: 47
43Variable
44Variable
45Variable
46ExprTuple47
47Variable