logo

Expression of type Forall

from the theory of proveit.numbers.numerals.decimals

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 ExprRange, IndexedVar, Variable, a, b, k, m
from proveit.logic import Equals, Forall
from proveit.numbers import Add, DecimalSequence, Digits, Less, Natural, nine, one, zero
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = ExprRange(sub_expr1, IndexedVar(a, sub_expr1), one, m)
expr = Forall(instance_param_or_params = [m, k], instance_expr = Forall(instance_param_or_params = [sub_expr2, b], instance_expr = Equals(Add(DecimalSequence(sub_expr2, b, ExprRange(sub_expr1, nine, one, k)), one), DecimalSequence(sub_expr2, Add(b, one), ExprRange(sub_expr1, zero, one, k))), domain = Digits, condition = Less(b, nine)), domain = Natural)
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, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{m}, b \in \mathbb{N}^{\leq 9}~|~b < 9}~\left(\left(\texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ b ~ 9, 9, ..\left(k - 3\right) \times.., 9) + 1\right) = \texttt{\#}(a_{1}, a_{2}, \ldots, a_{m} ~ \left(b + 1\right) ~ 0, 0, ..\left(k - 3\right) \times.., 0)\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
2Lambdaparameters: 3
body: 4
3ExprTuple52, 55
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 22
operands: 9
7Literal
8ExprTuple10
9ExprTuple11, 12
10Lambdaparameters: 13
body: 14
11Operationoperator: 44
operands: 15
12Operationoperator: 44
operands: 16
13ExprTuple46, 48
14Conditionalvalue: 17
condition: 18
15ExprTuple52, 19
16ExprTuple55, 19
17Operationoperator: 20
operands: 21
18Operationoperator: 22
operands: 23
19Literal
20Literal
21ExprTuple24, 25
22Literal
23ExprTuple26, 27, 28
24Operationoperator: 41
operands: 29
25Operationoperator: 39
operands: 30
26ExprRangelambda_map: 31
start_index: 54
end_index: 52
27Operationoperator: 44
operands: 32
28Operationoperator: 33
operands: 34
29ExprTuple35, 54
30ExprTuple46, 36, 37
31Lambdaparameter: 60
body: 38
32ExprTuple48, 50
33Literal
34ExprTuple48, 57
35Operationoperator: 39
operands: 40
36Operationoperator: 41
operands: 42
37ExprRangelambda_map: 43
start_index: 54
end_index: 55
38Operationoperator: 44
operands: 45
39Literal
40ExprTuple46, 48, 47
41Literal
42ExprTuple48, 54
43Lambdaparameter: 60
body: 49
44Literal
45ExprTuple56, 50
46ExprRangelambda_map: 51
start_index: 54
end_index: 52
47ExprRangelambda_map: 53
start_index: 54
end_index: 55
48Variable
49Literal
50Literal
51Lambdaparameter: 60
body: 56
52Variable
53Lambdaparameter: 60
body: 57
54Literal
55Variable
56IndexedVarvariable: 58
index: 60
57Literal
58Variable
59ExprTuple60
60Variable