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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, 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
In [2]:
# build up the expression from sub-expressions
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))
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_{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)
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: 4
body: 5
4ExprTuple6, 34, 30, 26
5Operationoperator: 7
operands: 8
6ExprRangelambda_map: 9
start_index: 41
end_index: 42
7Literal
8ExprTuple10, 11
9Lambdaparameter: 48
body: 43
10Operationoperator: 38
operands: 12
11Operationoperator: 13
operand: 16
12ExprTuple26, 15
13Literal
14ExprTuple16
15Operationoperator: 17
operand: 20
16Lambdaparameters: 35
body: 19
17Literal
18ExprTuple20
19Conditionalvalue: 21
condition: 25
20Lambdaparameters: 35
body: 22
21Operationoperator: 23
operands: 24
22Conditionalvalue: 27
condition: 25
23Literal
24ExprTuple26, 27
25Operationoperator: 28
operands: 29
26Variable
27Operationoperator: 30
operands: 35
28Literal
29ExprTuple31, 32
30Variable
31ExprRangelambda_map: 33
start_index: 41
end_index: 42
32Operationoperator: 34
operands: 35
33Lambdaparameter: 48
body: 36
34Variable
35ExprTuple37
36Operationoperator: 38
operands: 39
37ExprRangelambda_map: 40
start_index: 41
end_index: 42
38Literal
39ExprTuple44, 43
40Lambdaparameter: 48
body: 44
41Literal
42Variable
43IndexedVarvariable: 45
index: 48
44IndexedVarvariable: 46
index: 48
45Variable
46Variable
47ExprTuple48
48Variable