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Expression of type Implies

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 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, Implies, InSet
from proveit.logic.sets import general_comprehension_fyn
In [2]:
# build up the expression from sub-expressions
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())
\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}
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()(1)('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 31
operands: 5
4Operationoperator: 6
operand: 9
5ExprTuple19, 8
6Literal
7ExprTuple9
8Operationoperator: 10
operand: 13
9Lambdaparameters: 28
body: 12
10Literal
11ExprTuple13
12Conditionalvalue: 14
condition: 18
13Lambdaparameters: 28
body: 15
14Operationoperator: 16
operands: 17
15Conditionalvalue: 20
condition: 18
16Literal
17ExprTuple19, 20
18Operationoperator: 21
operands: 22
19Variable
20Operationoperator: 23
operands: 28
21Literal
22ExprTuple24, 25
23Variable
24ExprRangelambda_map: 26
start_index: 34
end_index: 35
25Operationoperator: 27
operands: 28
26Lambdaparameter: 41
body: 29
27Variable
28ExprTuple30
29Operationoperator: 31
operands: 32
30ExprRangelambda_map: 33
start_index: 34
end_index: 35
31Literal
32ExprTuple37, 36
33Lambdaparameter: 41
body: 37
34Literal
35Variable
36IndexedVarvariable: 38
index: 41
37IndexedVarvariable: 39
index: 41
38Variable
39Variable
40ExprTuple41
41Variable