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, c, x
from proveit.logic import Card, Equals, Forall, InSet, SetOfAll
from proveit.numbers import Add, Integer, Interval, Mod, NaturalPos, greater_eq

# build up the expression from sub-expressions
sub_expr1 = Interval(a, b)
expr = Lambda(N, Conditional(Forall(instance_param_or_params = [a, b, c], instance_expr = Equals(Card(SetOfAll(instance_param_or_params = [x], instance_element = Mod(Add(x, c), N), domain = sub_expr1)), N), domain = Integer, condition = greater_eq(Card(sub_expr1), 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, c \in \mathbb{Z}~|~|\{a~\ldotp \ldotp~b\}| \geq N}~\left(|\left\{\left(x + c\right) ~\textup{mod}~ N\right\}_{x \in \{a~\ldotp \ldotp~b\}}| = N\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: 46 body: 2
1	ExprTuple	46
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 43 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	46, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	54, 55, 53
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operands: 15
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18, 46
16	Literal
17	ExprTuple	19, 20, 21, 22
18	Operation	operator: 34 operand: 29
19	Operation	operator: 43 operands: 24
20	Operation	operator: 43 operands: 25
21	Operation	operator: 43 operands: 26
22	Operation	operator: 27 operands: 28
23	ExprTuple	29
24	ExprTuple	54, 30
25	ExprTuple	55, 30
26	ExprTuple	53, 30
27	Literal
28	ExprTuple	46, 31
29	Operation	operator: 32 operand: 36
30	Literal
31	Operation	operator: 34 operand: 47
32	Literal
33	ExprTuple	36
34	Literal
35	ExprTuple	47
36	Lambda	parameter: 52 body: 38
37	ExprTuple	52
38	Conditional	value: 39 condition: 40
39	Operation	operator: 41 operands: 42
40	Operation	operator: 43 operands: 44
41	Literal
42	ExprTuple	45, 46
43	Literal
44	ExprTuple	52, 47
45	Operation	operator: 48 operands: 49
46	Variable
47	Operation	operator: 50 operands: 51
48	Literal
49	ExprTuple	52, 53
50	Literal
51	ExprTuple	54, 55
52	Variable
53	Variable
54	Variable
55	Variable