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Expression of type Forall

from the theory of proveit.numbers.exponentiation

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 b, m
from proveit.core_expr_types import a_1_to_m
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Exp, Mult, NaturalPos, RealPos
from proveit.numbers.exponentiation import prod_ai_raise_b__1_to_m
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m], instance_expr = Forall(instance_param_or_params = [a_1_to_m, b], instance_expr = Equals(Exp(Mult(a_1_to_m), b), prod_ai_raise_b__1_to_m), domains = [RealPos, Complex]), domain = 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())
\forall_{m \in \mathbb{N}^+}~\left[\forall_{\left(a_{1} \in \mathbb{R}^+\right), \left(a_{2} \in \mathbb{R}^+\right), \ldots, \left(a_{m} \in \mathbb{R}^+\right), b \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{m}\right)^{b} = \left(\left(a_{1}\right)^{b} \cdot  \left(a_{2}\right)^{b} \cdot  \ldots \cdot  \left(a_{m}\right)^{b}\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: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 42
body: 4
3ExprTuple42
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 35
operands: 9
7Literal
8ExprTuple10
9ExprTuple42, 11
10Lambdaparameters: 12
body: 13
11Literal
12ExprTuple37, 46
13Conditionalvalue: 14
condition: 15
14Operationoperator: 16
operands: 17
15Operationoperator: 18
operands: 19
16Literal
17ExprTuple20, 21
18Literal
19ExprTuple22, 23
20Operationoperator: 43
operands: 24
21Operationoperator: 32
operands: 25
22ExprRangelambda_map: 26
start_index: 41
end_index: 42
23Operationoperator: 35
operands: 27
24ExprTuple28, 46
25ExprTuple29
26Lambdaparameter: 49
body: 30
27ExprTuple46, 31
28Operationoperator: 32
operands: 33
29ExprRangelambda_map: 34
start_index: 41
end_index: 42
30Operationoperator: 35
operands: 36
31Literal
32Literal
33ExprTuple37
34Lambdaparameter: 49
body: 38
35Literal
36ExprTuple45, 39
37ExprRangelambda_map: 40
start_index: 41
end_index: 42
38Operationoperator: 43
operands: 44
39Literal
40Lambdaparameter: 49
body: 45
41Literal
42Variable
43Literal
44ExprTuple45, 46
45IndexedVarvariable: 47
index: 49
46Variable
47Variable
48ExprTuple49
49Variable