Expression of type Lambda

from the theory of proveit.numbers.modular

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 Conditional, Lambda, N, a, b, x
from proveit.logic import Card, Equals, Forall, InSet, SetOfAll
from proveit.numbers import Integer, Interval, LessEq, Mod, NaturalPos

# build up the expression from sub-expressions
sub_expr1 = Interval(a, b)
sub_expr2 = Card(sub_expr1)
expr = Lambda(N, Conditional(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)), sub_expr2), domain = Integer, condition = LessEq(sub_expr2, N)), InSet(N, NaturalPos)))

expr:

# 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

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

N \mapsto \left\{\forall_{a, b \in \mathbb{Z}~|~|\{a~\ldotp \ldotp~b\}| \leq N}~\left(|\left\{x ~\textup{mod}~ N\right\}_{x \in \{a~\ldotp \ldotp~b\}}| = |\{a~\ldotp \ldotp~b\}|\right) \textrm{ if } N \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 42 body: 2
1	ExprTuple	42
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 40 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	42, 9
8	Lambda	parameters: 46 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operands: 14
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17, 28
15	Literal
16	ExprTuple	18, 19, 20
17	Operation	operator: 31 operand: 26
18	Operation	operator: 40 operands: 22
19	Operation	operator: 40 operands: 23
20	Operation	operator: 24 operands: 25
21	ExprTuple	26
22	ExprTuple	47, 27
23	ExprTuple	48, 27
24	Literal
25	ExprTuple	28, 42
26	Operation	operator: 29 operand: 33
27	Literal
28	Operation	operator: 31 operand: 44
29	Literal
30	ExprTuple	33
31	Literal
32	ExprTuple	44
33	Lambda	parameter: 43 body: 35
34	ExprTuple	43
35	Conditional	value: 36 condition: 37
36	Operation	operator: 38 operands: 39
37	Operation	operator: 40 operands: 41
38	Literal
39	ExprTuple	43, 42
40	Literal
41	ExprTuple	43, 44
42	Variable
43	Variable
44	Operation	operator: 45 operands: 46
45	Literal
46	ExprTuple	47, 48
47	Variable
48	Variable