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, L, Lambda, S, c, x
from proveit.logic import Card, Equals, Forall, InSet, SetOfAll, SubsetEq
from proveit.numbers import Add, Mod, Real

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Lambda(L, Conditional(Forall(instance_param_or_params = [S], instance_expr = Forall(instance_param_or_params = [c], instance_expr = Equals(Card(SetOfAll(instance_param_or_params = sub_expr1, instance_element = Mod(Add(c, x), L), domain = S)), Card(SetOfAll(instance_param_or_params = sub_expr1, instance_element = Mod(x, L), domain = S))), domain = Real), condition = SubsetEq(S, Real)), InSet(L, Real)))

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())

L \mapsto \left\{\forall_{S~|~S \subseteq \mathbb{R}}~\left[\forall_{c \in \mathbb{R}}~\left(|\left\{\left(c + x\right) ~\textup{mod}~ L\right\}_{x \in S}| = |\left\{x ~\textup{mod}~ L\right\}_{x \in S}|\right)\right] \textrm{ if } L \in \mathbb{R}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 49 body: 2
1	ExprTuple	49
2	Conditional	value: 3 condition: 4
3	Operation	operator: 12 operand: 7
4	Operation	operator: 46 operands: 6
5	ExprTuple	7
6	ExprTuple	49, 26
7	Lambda	parameter: 50 body: 9
8	ExprTuple	50
9	Conditional	value: 10 condition: 11
10	Operation	operator: 12 operand: 16
11	Operation	operator: 14 operands: 15
12	Literal
13	ExprTuple	16
14	Literal
15	ExprTuple	50, 26
16	Lambda	parameter: 53 body: 18
17	ExprTuple	53
18	Conditional	value: 19 condition: 20
19	Operation	operator: 21 operands: 22
20	Operation	operator: 46 operands: 23
21	Literal
22	ExprTuple	24, 25
23	ExprTuple	53, 26
24	Operation	operator: 28 operand: 30
25	Operation	operator: 28 operand: 31
26	Literal
27	ExprTuple	30
28	Literal
29	ExprTuple	31
30	Operation	operator: 33 operand: 35
31	Operation	operator: 33 operand: 36
32	ExprTuple	35
33	Literal
34	ExprTuple	36
35	Lambda	parameter: 54 body: 37
36	Lambda	parameter: 54 body: 39
37	Conditional	value: 40 condition: 42
38	ExprTuple	54
39	Conditional	value: 41 condition: 42
40	Operation	operator: 44 operands: 43
41	Operation	operator: 44 operands: 45
42	Operation	operator: 46 operands: 47
43	ExprTuple	48, 49
44	Literal
45	ExprTuple	54, 49
46	Literal
47	ExprTuple	54, 50
48	Operation	operator: 51 operands: 52
49	Variable
50	Variable
51	Literal
52	ExprTuple	53, 54
53	Variable
54	Variable