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/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import model_theory.basic
/-!
#Elementary Maps Between First-Order Structures
## Main Definitions
* A `first_order.language.ele... |
module Database
import Schema
import Decidable.Equality
public export data DB : String -> Type where
MkDB : (dbFile : String) ->
(dbTables : List (String, Schema)) -> DB dbFile
%name DB db
public export data HasTable : List (String, Schema) -> String -> Schema -> Type where
Here : HasTable ((name, s):... |
Set Implicit Arguments.
Require Import List.
Section NEList.
Variable A:Type.
Definition nelist := (A * list A)%type.
Definition single x : nelist := (x, nil).
Definition append (l1 l2: nelist) : nelist :=
let (x, xs) := l1 in
let (y, ys) := l2 in
(x, xs ++ cons y ys).
Definition hd (l:neli... |
lemma uniformly_continuous_on_closure: fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space" assumes ucont: "uniformly_continuous_on S f" and cont: "continuous_on (closure S) f" shows "uniformly_continuous_on (closure S) f" |
Formal statement is: lemma Lim_null: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net" Informal statement is: A net $f$ converges to $l$ if and only if the net $f - l$ converges to $0$. |
Formal statement is: lemma complete_closed_subset: fixes S :: "'a::metric_space set" assumes "closed S" and "S \<subseteq> t" and "complete t" shows "complete S" Informal statement is: If $S$ is a closed subset of a complete metric space $t$, then $S$ is complete. |
[STATEMENT]
lemma mem_cnb_minus: "x \<in> set l ==> cnb l = crypt_nb x + (cnb l - crypt_nb x)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. x \<in> set l \<Longrightarrow> cnb l = crypt_nb x + (cnb l - crypt_nb x)
[PROOF STEP]
by (induct l, auto) |
import basic_defs_world.level1 -- hide
/- Axiom : A set A is the neighborhood of a point x if there is an open U such that x ∈ U ⊆ A.
is_neighborhood : ∃ U, is_open U ∧ x ∈ U ∧ U ⊆ A
-/
/- Axiom : A point x is an interior point of A if A is a neighborhood of x.
is_interior_point : is_neighborhood x A
-/
/- Axiom : T... |
/- various properties about pfun and roption -/
import basic data.pfun
open set
universes u v w
variables {α : Type u} {β : Type v} {γ : Type w} {n : ℕ}
namespace roption
def compatible (o₁ o₂ : roption α) : Prop := ∀{{x y}}, x ∈ o₁ → y ∈ o₂ → x = y
namespace compatible
variables {o₁ o₂ o₃ : roption α}
infix ... |
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