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Expression of type Forall

from the theory of proveit.numbers.summation

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 S, f, fx, g, gx, x
from proveit.logic import Forall, Implies
from proveit.numbers import LessEq, Sum
In [2]:
# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Forall(instance_param_or_params = [S, f, g], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = LessEq(fx, gx), domain = S), LessEq(Sum(index_or_indices = sub_expr1, summand = fx, domain = S), Sum(index_or_indices = sub_expr1, summand = gx, domain = S))))
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, f, g}~\left(\left[\forall_{x \in S}~\left(f\left(x\right) \leq g\left(x\right)\right)\right] \Rightarrow \left(\left[\sum_{x \in S}~f\left(x\right)\right] \leq \left[\sum_{x \in S}~g\left(x\right)\right]\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: 9
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple35, 29, 30
4Operationoperator: 5
operands: 6
5Literal
6ExprTuple7, 8
7Operationoperator: 9
operand: 12
8Operationoperator: 22
operands: 11
9Literal
10ExprTuple12
11ExprTuple13, 14
12Lambdaparameter: 34
body: 15
13Operationoperator: 17
operand: 20
14Operationoperator: 17
operand: 21
15Conditionalvalue: 19
condition: 28
16ExprTuple20
17Literal
18ExprTuple21
19Operationoperator: 22
operands: 23
20Lambdaparameter: 34
body: 24
21Lambdaparameter: 34
body: 25
22Literal
23ExprTuple26, 27
24Conditionalvalue: 26
condition: 28
25Conditionalvalue: 27
condition: 28
26Operationoperator: 29
operand: 34
27Operationoperator: 30
operand: 34
28Operationoperator: 32
operands: 33
29Variable
30Variable
31ExprTuple34
32Literal
33ExprTuple34, 35
34Variable
35Variable