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Expression of type Lambda

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 Conditional, Lambda, 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 = Lambda(n, Conditional(Forall(instance_param_or_params = [S_1_to_n, Q, f, x], instance_expr = Implies(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), InSet(x, general_comprehension_fyn)).with_wrapping_at(1)), InSet(n, 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())
n \mapsto \left\{\forall_{S_{1}, S_{2}, \ldots, S_{n}, Q, f, x}~\left(\begin{array}{c} \begin{array}{l} \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] \\  \Rightarrow \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) \end{array} \end{array}\right) \textrm{ if } n \in \mathbb{N}^+\right..
In [5]:
stored_expr.style_options()
no style options
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Lambdaparameter: 48
body: 2
1ExprTuple48
2Conditionalvalue: 3
condition: 4
3Operationoperator: 5
operand: 8
4Operationoperator: 44
operands: 7
5Literal
6ExprTuple8
7ExprTuple48, 9
8Lambdaparameters: 10
body: 11
9Literal
10ExprTuple12, 40, 34, 31
11Operationoperator: 13
operands: 14
12ExprRangelambda_map: 15
start_index: 47
end_index: 48
13Literal
14ExprTuple16, 17
15Lambdaparameter: 54
body: 49
16Operationoperator: 18
operand: 21
17Operationoperator: 44
operands: 20
18Literal
19ExprTuple21
20ExprTuple31, 22
21Lambdaparameters: 41
body: 23
22Operationoperator: 24
operand: 27
23Conditionalvalue: 26
condition: 33
24Literal
25ExprTuple27
26Operationoperator: 28
operands: 29
27Lambdaparameters: 41
body: 30
28Literal
29ExprTuple31, 32
30Conditionalvalue: 32
condition: 33
31Variable
32Operationoperator: 34
operands: 41
33Operationoperator: 35
operands: 36
34Variable
35Literal
36ExprTuple37, 38
37ExprRangelambda_map: 39
start_index: 47
end_index: 48
38Operationoperator: 40
operands: 41
39Lambdaparameter: 54
body: 42
40Variable
41ExprTuple43
42Operationoperator: 44
operands: 45
43ExprRangelambda_map: 46
start_index: 47
end_index: 48
44Literal
45ExprTuple50, 49
46Lambdaparameter: 54
body: 50
47Literal
48Variable
49IndexedVarvariable: 51
index: 54
50IndexedVarvariable: 52
index: 54
51Variable
52Variable
53ExprTuple54
54Variable