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Expression of type Forall

from the theory of proveit.logic.sets.comprehension

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 Q, f, n, x
from proveit.core_expr_types import Q__y_1_to_n, S_1_to_n, f__y_1_to_n, y_1_to_n
from proveit.logic import Equals, Exists, Forall, Implies, InSet
from proveit.logic.sets import general_comprehension_fyn
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 = [S_1_to_n, Q, f, x], instance_expr = Implies(InSet(x, general_comprehension_fyn), Exists(instance_param_or_params = [y_1_to_n], instance_expr = Equals(x, f__y_1_to_n), domains = [S_1_to_n], condition = Q__y_1_to_n)).with_wrapping_at(1)), 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_{S_{1}, S_{2}, \ldots, S_{n}, Q, f, x}~\left(\begin{array}{c} \begin{array}{l} \left(x \in \left\{f\left(y_{1}, y_{2}, \ldots, y_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right\}_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)}\right) \\  \Rightarrow \left[\exists_{\left(y_{1} \in S_{1}\right), \left(y_{2} \in S_{2}\right), \ldots, \left(y_{n} \in S_{n}\right)~|~Q\left(y_{1}, y_{2}, \ldots, y_{n}\right)}~\left(x = f\left(y_{1}, y_{2}, \ldots, y_{n}\right)\right)\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
2Lambdaparameter: 50
body: 4
3ExprTuple50
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 46
operands: 9
7Literal
8ExprTuple10
9ExprTuple50, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple14, 42, 38, 34
13Operationoperator: 15
operands: 16
14ExprRangelambda_map: 17
start_index: 49
end_index: 50
15Literal
16ExprTuple18, 19
17Lambdaparameter: 56
body: 51
18Operationoperator: 46
operands: 20
19Operationoperator: 21
operand: 24
20ExprTuple34, 23
21Literal
22ExprTuple24
23Operationoperator: 25
operand: 28
24Lambdaparameters: 43
body: 27
25Literal
26ExprTuple28
27Conditionalvalue: 29
condition: 33
28Lambdaparameters: 43
body: 30
29Operationoperator: 31
operands: 32
30Conditionalvalue: 35
condition: 33
31Literal
32ExprTuple34, 35
33Operationoperator: 36
operands: 37
34Variable
35Operationoperator: 38
operands: 43
36Literal
37ExprTuple39, 40
38Variable
39ExprRangelambda_map: 41
start_index: 49
end_index: 50
40Operationoperator: 42
operands: 43
41Lambdaparameter: 56
body: 44
42Variable
43ExprTuple45
44Operationoperator: 46
operands: 47
45ExprRangelambda_map: 48
start_index: 49
end_index: 50
46Literal
47ExprTuple52, 51
48Lambdaparameter: 56
body: 52
49Literal
50Variable
51IndexedVarvariable: 53
index: 56
52IndexedVarvariable: 54
index: 56
53Variable
54Variable
55ExprTuple56
56Variable