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Expression of type Forall

from the theory of proveit.numbers.multiplication

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, y
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import Forall
from proveit.numbers import Less, Mult, RealPos
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j], instance_expr = Less(Mult(a_1_to_i, x, b_1_to_j), Mult(a_1_to_i, y, b_1_to_j)), domain = RealPos, condition = Less(x, y))
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_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{R}^+~|~x < y}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot x\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\right) < \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot y\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\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: 1
operand: 3
1Literal
2ExprTuple3
3Lambdaparameters: 4
body: 5
4ExprTuple23, 24
5Conditionalvalue: 6
condition: 7
6Operationoperator: 21
operands: 8
7Operationoperator: 9
operands: 10
8ExprTuple11, 12
9Literal
10ExprTuple13, 14, 15
11Operationoperator: 17
operands: 16
12Operationoperator: 17
operands: 18
13ExprRangelambda_map: 19
start_index: 32
end_index: 30
14ExprRangelambda_map: 20
start_index: 32
end_index: 33
15Operationoperator: 21
operands: 22
16ExprTuple23, 27, 24
17Literal
18ExprTuple23, 28, 24
19Lambdaparameter: 43
body: 25
20Lambdaparameter: 43
body: 26
21Literal
22ExprTuple27, 28
23ExprRangelambda_map: 29
start_index: 32
end_index: 30
24ExprRangelambda_map: 31
start_index: 32
end_index: 33
25Operationoperator: 35
operands: 34
26Operationoperator: 35
operands: 36
27Variable
28Variable
29Lambdaparameter: 43
body: 37
30Variable
31Lambdaparameter: 43
body: 38
32Literal
33Variable
34ExprTuple37, 39
35Literal
36ExprTuple38, 39
37IndexedVarvariable: 40
index: 43
38IndexedVarvariable: 41
index: 43
39Literal
40Variable
41Variable
42ExprTuple43
43Variable