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Expression of type Forall

from the theory of proveit.numbers.modular

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 N, a, b, x
from proveit.logic import Card, Equals, Forall, SetOfAll
from proveit.numbers import Integer, Interval, Mod, NaturalPos, greater_eq
In [2]:
# build up the expression from sub-expressions
sub_expr1 = Interval(a, b)
expr = Forall(instance_param_or_params = [N], instance_expr = Forall(instance_param_or_params = [a, b], instance_expr = Equals(Card(SetOfAll(instance_param_or_params = [x], instance_element = Mod(x, N), domain = sub_expr1)), N), domain = Integer, condition = greater_eq(Card(sub_expr1), N)), 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_{N \in \mathbb{N}^+}~\left[\forall_{a, b \in \mathbb{Z}~|~|\{a~\ldotp \ldotp~b\}| \geq N}~\left(|\left\{x ~\textup{mod}~ N\right\}_{x \in \{a~\ldotp \ldotp~b\}}| = N\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: 44
body: 4
3ExprTuple44
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 42
operands: 9
7Literal
8ExprTuple10
9ExprTuple44, 11
10Lambdaparameters: 48
body: 12
11Literal
12Conditionalvalue: 13
condition: 14
13Operationoperator: 15
operands: 16
14Operationoperator: 17
operands: 18
15Literal
16ExprTuple19, 44
17Literal
18ExprTuple20, 21, 22
19Operationoperator: 33
operand: 28
20Operationoperator: 42
operands: 24
21Operationoperator: 42
operands: 25
22Operationoperator: 26
operands: 27
23ExprTuple28
24ExprTuple49, 29
25ExprTuple50, 29
26Literal
27ExprTuple44, 30
28Operationoperator: 31
operand: 35
29Literal
30Operationoperator: 33
operand: 46
31Literal
32ExprTuple35
33Literal
34ExprTuple46
35Lambdaparameter: 45
body: 37
36ExprTuple45
37Conditionalvalue: 38
condition: 39
38Operationoperator: 40
operands: 41
39Operationoperator: 42
operands: 43
40Literal
41ExprTuple45, 44
42Literal
43ExprTuple45, 46
44Variable
45Variable
46Operationoperator: 47
operands: 48
47Literal
48ExprTuple49, 50
49Variable
50Variable