text stringlengths 0 3.34M |
|---|
(* *)
theory Dagstuhl
imports "$HIPSTER_HOME/IsaHipster"
begin
(* Hard exercise *)
fun qrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
where
"qrev [] acc = acc"
| "qrev (x#xs) acc = qrev xs (x#acc)"
hipster rev qrev
lemma lemma_a [thy_expl]: "qrev (qrev z y) [] = qrev y z"
apply (induct... |
universes u
axiom elimEx (motive : Nat → Nat → Sort u) (x y : Nat)
(diag : (a : Nat) → motive a a)
(upper : (delta a : Nat) → motive a (a + delta.succ))
(lower : (delta a : Nat) → motive (a + delta.succ) a)
: motive y x
theorem ex1 (p q : Nat) : p ≤ q ∨ p > q := by
cases p, q using elimEx with
| lower d ... |
10
* 15
20
- 10 # no error but unary minus here |
[STATEMENT]
lemma strip_guards_whileAnno [simp]:
"strip_guards F (whileAnno b I V c) = whileAnno b I V (strip_guards F c)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. strip_guards F (whileAnno b I V c) = whileAnno b I V (strip_guards F c)
[PROOF STEP]
by (simp add: whileAnno_def while_def) |
[STATEMENT]
lemma word_add_def [code]:
"a + b = word_of_int (uint a + uint b)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. a + b = word_of_int (uint a + uint b)
[PROOF STEP]
by transfer (simp add: take_bit_add) |
import Scripts.set_theory
--===================--
------ TOPOLOGÍA ------
--===================--
-- Un espacio topológico sobre un tipo α viene dado por un predicado 'is_open' sobre los conjuntos de
-- elementos de α junto con las pruebas de que ese predicado cumple los axiomas de la topología
structure topological_s... |
constants (p q : Prop) (Hnq : ¬q) (Hpq : p → q)
theorem t (Hp : p): false := Hnq (Hpq Hp)
check t -- t : p → false
|
theorem ex (h : a = 0) (p : Nat → Prop) : p a → p 0 := by
simp_all
|
data Id (A : Set) : Set where
wrap : A → Id A
data Maybe (A : Set) : Set where
nothing : Maybe A
just : A → Maybe A
maybe : {A : Set} {B : Maybe A → Set} →
((x : A) → B (just x)) → B nothing → (x : Maybe A) → B x
maybe j n (just x) = j x
maybe j n nothing = n
record MaybeT (M : Set → Set) (A : Set... |
import data.list.basic
open list
lemma foldr_ext {α : Type*} {β : Type*} {l : list α} (f f' : α → β → β) (s : β)
(H : ∀ a ∈ l, ∀ b : β, f a b = f' a b) : foldr f s l = foldr f' s l :=
begin induction l with h l IH, {simp}, -- base case}
simp *,
suffices : foldr f s l = foldr f' s l,
rw this,
apply IH,
i... |
import
tactic.induction
tactic.linarith
...compiler
...semantics
open vm_big_step env_big_step
-- convert from environment-preserving to regular big_step
lemma env_vm_big_step {env env' P S R} :
(env, P, S) ⟹ₙᵥ (env', R)
→ (env, P, S) ⟹ᵥₘ R :=
begin
assume hnv,
induction' hnv,
case ERunEmpty {
... |
import Smt
theorem comm (f : Bool → Bool → Bool) (p q : Bool) : f p q == f q p := by
smt
admit
|
(** Instance of identity on propositions *)
Theorem not_False : ~ False.
Proof. unfold not; trivial. Qed.
Definition not_False' : ~ False :=
fun H => H.
Theorem triple_neg : forall P:Prop, ~ ~ ~ P -> ~ P.
Proof. auto. Qed.
Theorem P3PQ : forall P Q:Prop, ~ ~ ~ P -> P -> Q.
Proof. tauto. Qed.
(** instance of t... |
lemma coeffs_reflect_poly [code abstract]: "coeffs (reflect_poly p) = rev (dropWhile ((=) 0) (coeffs p))" |
(** Some Neo-davidsonian Semantics*)
Require Import Bvector.
Require Import Coq.Lists.List.
Require Import Omega.
Load LibTactics.
Ltac AUTO:= cbv delta;intuition;try repeat congruence; jauto;intuition.
Parameter Event Ind LOcation Instrument:Set.
Parameter Agent
Theme: Ind->Event->Prop.
Parameter Caesar Bru... |
/-example : 1 + 2222 = 2223 :=
begin
reflexivity
end
theorem foo : 1 + (2222222222222 : nat) = 2222222222223 :=
begin
reflexivity
end
-/
theorem bar : 100003 + 100003 = 200006 :=
rfl
|
/-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f
-/
import mathzoo.imports.miniF2F
open_locale nat rat real big_operators topological_space
theorem mathd_algebra_234
(d : ℝ)
(h₀... |
--
set_option autoBoundImplicitLocal false
universes u
variable {α : Type u}
variable {β : α → Type v}
theorem ex {p₁ p₂ : Sigma (fun a => β a)} (h₁ : p₁.1 = p₂.1) (h : p₁.2 ≅ p₂.2) : p₁ = p₂ :=
match p₁, p₂, h₁, h with
| ⟨_, _⟩, ⟨_, _⟩, rfl, HEq.refl _ => rfl
|
lemma (in algebra) sets_Collect_finite_All: assumes "\<And>i. i \<in> S \<Longrightarrow> {x\<in>\<Omega>. P i x} \<in> M" "finite S" shows "{x\<in>\<Omega>. \<forall>i\<in>S. P i x} \<in> M" |
informal statement Prove that a subgroup $H$ of $G$ is normal if and only if $[G, H] \leq H$.formal statement theorem exercise_7_1_11 {R : Type*} [ring R] [is_domain R]
{x : R} (hx : x^2 = 1) : x = 1 ∨ x = -1 := |
import data.complex.basic
import number_theory.divisors
theorem AMC_2021_A_25 (N:ℕ)(f:ℕ → ℝ)(h:∀(n:ℕ), f n = (finset.card (nat.divisors n))/(n^(1/3)))(hN:∀(n:ℕ),n≠N → (f N > f n)):
N=2520
:=
begin
sorry
end |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import ring_theory.int.basic
import data.nat.factorization.prime_pow
import algebra.squarefree
/-!
# Lemmas about squarefreeness of natural numbers
A number is squar... |
--
-- The Computer Language Benchmarks Game
-- https://salsa.debian.org/benchmarksgame-team/benchmarksgame/
--
-- Contributed by Don Stewart
-- *reset*
--
-- import System.Environment
import Data.Bits
-- import Text.Printf
--
-- an artificially strict tree.
--
-- normally you would ensure the branches are lazy, but t... |
import topology.instances.real
import data.complex.exponential
import data.real.irrational
open filter real
open_locale topological_space
open_locale big_operators
noncomputable def e : ℕ → ℝ := λ n, ∑ i in finset.range(n+1), 1 / (nat.factorial i)
theorem part_a (n : ℕ) :
∃ p : ℕ, e n = p / (nat.factorial n) :=... |
module Figures
using AverageShiftedHistograms, Distributions, Plots
gr()
# sz = 1000
# o = ash(rand(Gamma(5, 1), sz), rng=0:.05:20, m = 15)
# anim = @animate for i in 1:50
# plot(o, title = "Nobs = $(sz * i)", ylim = (0, .5))
# ash!(o, rand(Gamma(5, 1), sz))
# end
# gif(anim, "animation.gif", fps = 10)
# ... |
import Mathlib
import LeanAide
/-!
## A bit of AI
We use `leanaide` for a bit of AI. For more direct use (such as debugging), can use
```lean
#eval translateViewM "There are infinitely many odd numbers"
```
For more details on using the AI, please see the [README](https://github.com/siddhartha-gadgil/LeanAide#readm... |
theory Ex045
imports Main
begin
theorem "C \<longrightarrow> \<not>A \<or> ((B \<or> C) \<longrightarrow> A)"
proof -
{
assume C
{
assume "\<not>(\<not>A \<or> ((B \<or> C) \<longrightarrow> A))"
{
assume A
{
assume "B \<or> C"
from \<open>A\<close> hav... |
(* Title: Containers/Set_Impl.thy
Author: Andreas Lochbihler, KIT
René Thiemann, UIBK *)
theory Set_Impl imports
Collection_Enum
DList_Set
RBT_Set2
Closure_Set
Containers_Generator
Complex_Main
begin
section \<open>Different implementations of sets\<close>
subsection \<open>... |
import game.world7.level6 -- hide
/-
# Advanced proposition world.
## Level 7: `or_symm`
Proving that $(P\lor Q)\implies(Q\lor P)$ involves an element of danger.
`intro h,` is the obvious start. But now,
even though the goal is an `∨` statement, both `left` and `right` put
you in a situation with an impossible goal... |
import m114 algebra.pi_instances rename_var
lemma abs_inferieur_ssi (x y : ℝ) : |x| ≤ y ↔ -y ≤ x ∧ x ≤ y :=
abs_le
variables {α : Type*} [decidable_linear_order α]
lemma superieur_max_ssi (p q r : α) : r ≥ max p q ↔ r ≥ p ∧ r ≥ q :=
max_le_iff
lemma inferieur_max_gauche (p q : α) : p ≤ max p q :=
le_max_left _ _
l... |
/-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f
-/
import mathzoo.imports.miniF2F
open_locale nat rat real big_operators topological_space
theorem imo_1961_p1
(x y z a b : ℝ)
(... |
lemma homeomorphism_empty [simp]: "homeomorphism {} {} f g" |
theorem easy (P Q : Prop) (HP : P) (HPQ : P → Q) : Q :=
begin
sorry
end
|
set_option pp.analyze false
def p (x y : Nat) := x = y
example (x y : Nat) : p (x + y) (y + x + 0) := by
conv =>
whnf
congr
. skip
. whnf; skip
trace_state
rw [Nat.add_comm]
rfl
example (x y : Nat) : p (x + y) (y + x + 0) := by
conv =>
whnf
rhs
whnf
trace_state
rw [Nat.add_c... |
lemma both_and (P Q : Prop) (p : P) (q : Q) : P ∧ Q :=
begin
split,
{
exact p,
},
{
exact q,
},
end |
def f (x y : Nat) := x + 2*y
namespace Foo
local infix:65 (priority := high) "+" => f
theorem ex1 (x y : Nat) : x + y = f x y := rfl
#check 1 + 2
end Foo
#check 1 + 2
theorem ex2 (x y : Nat) : x + y = HAdd.hAdd x y := rfl
open Foo
theorem ex3 (x y : Nat) : x + y = HAdd.hAdd x y := rfl
#check 1 + 2
section
def ... |
@[default_instance] instance : Pow Int Nat where
pow m n := m ^ n
instance : @Trans Int Int Int (· < ·) (· < ·) (· < ·) where
trans := sorry
example {n : Int} : n ^ 2 < 1 :=
calc
n ^ 2 < 1 ^ 2 := sorry
_ < 1 := sorry
|
[GOAL]
S : Set ℍ
⊢ S ∈ atImInfty ↔ ∃ A, ∀ (z : ℍ), A ≤ im z → z ∈ S
[PROOFSTEP]
simp only [atImInfty_basis.mem_iff, true_and]
[GOAL]
S : Set ℍ
⊢ (∃ i, im ⁻¹' Set.Ici i ⊆ S) ↔ ∃ A, ∀ (z : ℍ), A ≤ im z → z ∈ S
[PROOFSTEP]
rfl
[GOAL]
f g : ℍ → ℂ
hf : IsBoundedAtImInfty f
hg : IsBoundedAtImInfty g
⊢ IsBoundedAtImInfty (f *... |
universe u
def f {α : Type u} [BEq α] (xs : List α) (y : α) : α := do
for x in xs do
if x == y then
return x
return y
structure S :=
(key val : Nat)
instance : BEq S :=
⟨fun a b => a.key == b.key⟩
theorem ex1 : f (α := S) [⟨1, 2⟩, ⟨3, 4⟩, ⟨5, 6⟩] ⟨3, 0⟩ = ⟨3, 4⟩ :=
rfl
theorem ex2 : f (α := S) [⟨1, 2⟩, ⟨3, ... |
lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im] |
theory deMorgan1
imports Main
begin
text\<open> Apply style \<close>
theorem de_morgan_1 : "(\<not> (p \<or> q)) \<longrightarrow> (\<not> p \<and> \<not> q)"
apply (rule impI)
apply (rule conjI)
apply (rule notI)
apply (erule notE)
apply (rule disjI1)
apply assumption
apply (rule notI)
apply (erul... |
/-
Copyright © 2018 François G. Dorais. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
-/
import fin.choice
import .basic .subst .mod .prf
universe u
variables {σ : Type u} {sig : σ → ℕ} {I : Type*} (ax : I → eqn sig)
include sig ax
lemma proof.sound {α : Type*} (a : alg sig... |
[STATEMENT]
theorem terminates_iff_terminates_merge_guards:
"\<Gamma>\<turnstile>c\<down> s = \<Gamma>\<turnstile>merge_guards c\<down> s"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<Gamma>\<turnstile>c \<down> s = \<Gamma>\<turnstile>merge_guards c \<down> s
[PROOF STEP]
by (iprover intro: terminates_to_term... |
(* *********************************************************************)
(* *)
(* The CertiKOS Certified Kit Operating System *)
(* *)
(* The... |
import algebra.group algebra.group_power
variables {G: Type*} (a b : G)
theorem Q_24 [group G] (H1 : ∀ a b : G, (a * b)^3 = a^3 * b^3) (H2 : ∀ x : G, x^3 = 1 → x = 1) :
∀ a b : G, a * b = b * a :=
λ a b : G,
have Lem1 : ∀ x y : G, (y * x)^2 = x^2 * y^2, from
λ x y : G,
have H10 : (a * b) * (a * b) * (a *... |
my_f(x,y) = 2x + y
my_derivative(x,y) = ForwardDiff.derivative(x->my_f(x,y),x) |
theory Scratch
imports Main
begin
lemma a:
assumes "A"
assumes "B"
shows "A \<and> B \<Longrightarrow> True"
using [[simp_trace]]
apply auto
done
end |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import field_theory.splitting_field
import field_theory.perfect_closure
import field_theory.separable
/-!
# Algebraically Closed Field
In this file we define the typeclass f... |
example : 2 + 3 = 5 :=
begin
generalize h : 3 = x,
rw ←h
end
|
[GOAL]
⊢ ∀ (x y : ZMod 4),
OneHom.toFun
{ toFun := ![0, 1, 0, -1], map_one' := (_ : Matrix.vecCons 0 ![1, 0, -1] 1 = Matrix.vecCons 0 ![1, 0, -1] 1) }
(x * y) =
OneHom.toFun
{ toFun := ![0, 1, 0, -1], map_one' := (_ : Matrix.vecCons 0 ![1, 0, -1] 1 = Matrix.vecCons 0 ![1, 0, -1] 1) }... |
%% Copyright (C) 2014, 2016 Colin B. Macdonald
%%
%% This file is part of OctSymPy.
%%
%% OctSymPy is free software; you can redistribute it and/or modify
%% it under the terms of the GNU General Public License as published
%% by the Free Software Foundation; either version 3 of the License,
%% or (at your option) any ... |
/-
This type bundles together a value of a certain type
together with a proof that it satisfies a certain property.
It can be used to write algorithms that return values
that must satisfy certain defining properties.
-/
@[class] structure Prover {α : Type _} (p : α → Prop) :=
value : α
proof : p value
-... |
import unitb.decomposition.component
universe variables u
namespace decomposition
section
open predicate unitb function scheduling
parameter {α : Type}
parameter {t : Type}
parameter [sched t]
parameter {s : t → program α}
parameter {s₀ : α}
parameter (asm : α → α → Prop)
parameter (h₀ : ∀ i, (s i).mch.first s₀)
... |
lemma synthetic_div_0 [simp]: "synthetic_div 0 c = 0" |
import data.nat.basic
def is_even (a : nat) := ∃ b, a = 2 * b
theorem even_plus_even {a b : nat}
(h1 : is_even a) (h2 : is_even b) : is_even (a + b) :=
exists.elim h1 (assume w1, assume hw1 : a = 2 * w1,
exists.elim h2 (assume w2, assume hw2 : b = 2 * w2,
exists.intro (w1 + w2)
(calc
a + b = 2... |
import data.list algebra.ring tactic.omega defs
lemma single_complete: almost_complete [●] :=
begin
apply almost_complete.cmp_rule,
existsi ([]),
intros t h,
exfalso, apply h, apply grow_list.head_grow,
apply grow.single_grow
end
lemma single_grow : ∀ t : bintree, (t ↣ ●) → t = ● :=
begin
intros t ... |
informal statement Let $G=\left\{g_{1}, \ldots, g_{n}\right\}$ be a finite group. Prove that the element $N=g_{1}+g_{2}+\ldots+g_{n}$ is in the center of the group ring $R G$.formal statement theorem exercise_7_3_37 {R : Type*} {p m : ℕ} (hp : p.prime)
(N : ideal $ zmod $ p^m) :
is_nilpotent N ↔ is_nilpotent (id... |
Formal statement is: lemma closure_empty [simp]: "closure {} = {}" Informal statement is: The closure of the empty set is the empty set. |
module Examples
import Data.Vect
import ProcessLib
import Proto
import HoriComp
import VertComp
import Primitives
import SumWire
import Dualise
import SumWire
import CupCap
import Util
%flag C "-O3"
gt5: Hom [Down Int] [Down Int, Down Int]
gt5 = mkPure "gt5: " (\n => if n > 5 then Left n else Right n) -*- splitEithe... |
import Smt
theorem verum : true := by
smt
simp_all
|
(*
Authors: Asta Halkjær From, Agnes Moesgård Eschen & Jørgen Villadsen, DTU Compute
*)
theory LT1 imports System_L1 begin
text \<open>System from Jan Lukasiewicz and Alfred Tarski (1930): Untersuchungen über den Aussagenkalkül\<close>
text \<open>Inspired by Shotaro Tanaka (1965): On Axiom Systems of Propositiona... |
[STATEMENT]
lemma
iNext_iEx_iff_singleton : "(\<circle> t t0 I. P t) = (\<diamond> t {inext t0 I}. P t)" and
iLast_iEx_iff_singleton : "(\<ominus> t t0 I. P t) = (\<diamond> t {iprev t0 I}. P t)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. iNext t0 I P = (\<diamond> t {inext t0 I}. P t) &&& iLast t0 I P ... |
lemma to_fract_eq_0_iff [simp]: "to_fract x = 0 \<longleftrightarrow> x = 0" |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import tactic.apply_fun
import data.equiv.ring
import data.zmod.basic
import linear_algebra.basis
import ring_theory.integral_domain
import ... |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import analysis.normed_space.star.basic
import analysis.normed_space.spectrum
import analysis.normed_space.star.exponential
import analysis.special_functions.exponentia... |
import data.list.basic
open list
universe u
variables {α : Type} (x y z : α) (xs ys zs : list α)
def mk_symm (xs : list α) := xs ++ reverse xs
theorem reverse_mk_symm (xs : list α) :
reverse (mk_symm xs) = mk_symm xs :=
by simp [mk_symm]
|
informal statement Let $X$ be a compact Hausdorff space. Let $\mathcal{A}$ be a collection of closed connected subsets of $X$ that is simply ordered by proper inclusion. Then $Y=\bigcap_{A \in \mathcal{A}} A$ is connected.formal statement theorem exercise_27_4
{X : Type*} [metric_space X] [connected_space X] (hX : ∃ ... |
import SciLean.Algebra
import SciLean.Quot.Monomial
namespace SciLean
namespace Algebra
-- M monomials
-- K ring
inductive Repr (M K X : Type u) : Type u where
| mon (m : M) : Repr M K X
| add (x y : Repr M K X) : Repr M K X
| mul (x y : Repr M K X) : Repr M K X
| lmul (c : K) (x : Repr M K X) : Rep... |
Formal statement is: lemma two_is_prime_nat [simp]: "prime (2::nat)" Informal statement is: The number 2 is prime. |
{- This example goes through now that we allow instantiation of
blocked terms #-}
module Issue439 where
record Σ (A : Set) (B : A → Set) : Set where
constructor _,_
field
p₁ : A
p₂ : B p₁
open Σ
record ⊤ : Set where
data Tree : Set where
leaf : Tree
node : Tree → Tree → Tree
mutual
U : Tree → Set
U... |
def p (x : Nat := 0) : Nat × Nat :=
(x, x)
theorem ex1 : p.1 = 0 :=
rfl
theorem ex2 : (p (x := 1) |>.2) = 1 :=
rfl
def c {α : Type} [Inhabited α] : α × α :=
(default, default)
theorem ex3 {α} [Inhabited α] : c.1 = default (α := α) :=
rfl
theorem ex4 {α} [Inhabited α] : c.2 = default (α := α) :=
rfl
|
[STATEMENT]
lemma orthogonal_complement_antimono_iff[simp]:
fixes A B :: \<open>('a::chilbert_space) set\<close>
assumes \<open>closed_csubspace A\<close> and \<open>closed_csubspace B\<close>
shows \<open>orthogonal_complement A \<subseteq> orthogonal_complement B \<longleftrightarrow> A \<supseteq> B\<close>
... |
State Before: ⊢ ord ∘ aleph = enumOrd {b | ord (card b) = b ∧ ω ≤ b} State After: ⊢ StrictMono (ord ∘ aleph) ∧ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b} Tactic: rw [← eq_enumOrd _ ord_card_unbounded'] State Before: ⊢ StrictMono (ord ∘ aleph) ∧ range (ord ∘ aleph) = {b | ord (card b) = b ∧ ω ≤ b} State After:... |
[STATEMENT]
lemma currentLevelPrefixToLevel_aux:
assumes "l \<ge> i"
shows "currentLevel (prefixToLevel_aux M l i) <= l - i"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. currentLevel (prefixToLevel_aux M l i) \<le> l - i
[PROOF STEP]
using assms
[PROOF STATE]
proof (prove)
using this:
i \<le> l
goal (1 subgo... |
[GOAL]
x y : Path PUnit.unit PUnit.unit
⊢ x = y
[PROOFSTEP]
ext
[GOAL]
x y : FundamentalGroupoid PUnit
⊢ Subsingleton (x ⟶ y)
[PROOFSTEP]
convert_to Subsingleton (Path.Homotopic.Quotient PUnit.unit PUnit.unit)
[GOAL]
x y : FundamentalGroupoid PUnit
⊢ Subsingleton (Path.Homotopic.Quotient PUnit.unit PUnit.unit)
[PROOFST... |
corollary fps_coeff_residues_bigo': fixes f :: "complex \<Rightarrow> complex" and r :: real assumes exp: "f has_fps_expansion F" assumes "open A" "connected A" "cball 0 r \<subseteq> A" "r > 0" assumes "f holomorphic_on A - S" "S \<subseteq> ball 0 r" "finite S" "0 \<notin> S" assumes "eventually (\<lambda>n. g n = -(... |
State Before: m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
l : List α
⊢ mapM' f l = mapM f l State After: no goals Tactic: simp [go, mapM] State Before: m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
l : List α... |
/--
From Sets to Relations
Take-away message. Whereas we represent a set
as a one-place predicate, we will represent a
binary relation as a two-place predicate. Just
as a set is a collection of individual objects
that satisfy a predicate, a relation is a set
of *pairs* of objects, each of which satisfies
the pair mem... |
import tactic
-- True or false? n = 3 ↔ n^2-2n-3=0. If you think it's false
-- then you'll have to modify the statement by putting it in brackets
-- and adding a ¬ in front of it.
lemma part_a : ∀ n : ℤ, n = 3 → n ^ 2 - 2 * n - 3 = 0 :=
begin
norm_num,
end
lemma part_b : ¬ (∀ n : ℤ, n ^ 2 - 2 * n - 3 = 0 → n = 3) ... |
(*
Benedikt Ahrens and Régis Spadotti
Terminal semantics for codata types in intensional Martin-Löf type theory
http://arxiv.org/abs/1401.1053
*)
(*
Content of this file:
- definition of pushforward of comodules along a comonad morphism
- definition of comodule morphism induced by a comonad morphi... |
State Before: p : ℕ
q : ℚ
hq : q = 0
⊢ padicNorm p (-q) = padicNorm p q State After: no goals Tactic: simp [hq] State Before: p : ℕ
q : ℚ
hq : ¬q = 0
⊢ padicNorm p (-q) = padicNorm p q State After: no goals Tactic: simp [padicNorm, hq] |
/-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f
-/
import mathzoo.imports.miniF2F
open_locale nat rat real big_operators topological_space
theorem mathd_algebra_142
(m b : ℝ)
(... |
[STATEMENT]
lemma defined_Integer_simps [simp]:
"defined (MkI\<cdot>i)"
"defined (0::Integer)"
"defined (1::Integer)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. defined (MkI\<cdot>i) &&& defined 0 &&& defined 1
[PROOF STEP]
by (simp_all add: defined_def) |
lemma is_unit_content_iff [simp]: "is_unit (content p) \<longleftrightarrow> content p = 1" |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
Definitions and properties of gcd, lcm, and coprime.
-/
import .div data.nat.gcd
open eq.ops
namespace int
/- gcd -/
definition gcd (a b : ℤ) : ℤ :=... |
import algebra.big_operators.basic
import data.int.gcd
import data.real.basic
import data.finset.basic
import number_theory.divisors
#check int.lcm 42 47
open_locale big_operators
theorem AIME_2021_I_14 (n1:ℕ)(sigma:ℕ → ℕ )(hs:∀(n:pnat), sigma n = ∑ i in nat.divisors n, i)
(h:∀ (a:pnat),2021∈ nat.divisors (sigma (a^n... |
(** * Facts about Static Expressions *)
Require Import hvhdl.Environment.
Require Import hvhdl.StaticExpressions.
Require Import hvhdl.AbstractSyntax.
Require Import hvhdl.SemanticalDomains.
Require Import hvhdl.proofs.EnvironmentFacts.
(** ** Facts about Locally Static Expressions *)
Section LStatic.
End LStatic.... |
function [fit] = compare_new(varargin)
% Determine list of inputs.
inpn = cell(1, length(varargin));
for kn = 1:length(varargin);
inpn{kn} = inputname(kn);
end
v = {varargin{:} inpn};
th = idss(v{2});
th = th('y1', cell(0));
z = v{1};
z = iddata(z(:, 1), z(:, 2:end), 1);
y = pvget(z, 'OutputData');
z1 = z(:, 'y1... |
import data.list.basic
open list
variables {α : Type*} (x y z : α) (xs ys zs : list α)
def mk_symm (xs : list α) := xs ++ reverse xs
theorem reverse_mk_symm (xs : list α) :
reverse (mk_symm xs) = mk_symm xs :=
by simp [mk_symm]
section
local attribute [simp] reverse_mk_symm
example (xs ys : list ℕ) :
reverse ... |
import data.real.basic
import game.functions.bothInjective game.functions.bothSurjective
open function
/-
# Chapter 6 : Functions
## Level 3
Be sure to make use of the results in the previous two levels.
-/
/- Lemma
If $f : X \to Y$ and $g : Y \to Z$ are both bijective functions, then
the function resulting from th... |
theory Exercise2
imports Main
begin
inductive palindrome :: "'a list \<Rightarrow> bool" where
pdrmNil: "palindrome []" |
pdrmSing: "palindrome [x]" |
pdrmRec: "palindrome xs \<Longrightarrow> palindrome (x # xs @ [x])"
theorem "(palindrome xs) \<Longrightarrow> (rev xs = xs)"
apply (induction rule: palindrome.... |
[STATEMENT]
lemma list_of_lazy_sequence_append [simp]:
"list_of_lazy_sequence (append xq yq) = list_of_lazy_sequence xq @ list_of_lazy_sequence yq"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. list_of_lazy_sequence (Lazy_Sequence.append xq yq) = list_of_lazy_sequence xq @ list_of_lazy_sequence yq
[PROOF STEP]
by... |
[STATEMENT]
lemma assert_gpv_simps [simp]:
"assert_gpv True = Done ()"
"assert_gpv False = Fail"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. assert_gpv True = Generative_Probabilistic_Value.Done () &&& assert_gpv False = Fail
[PROOF STEP]
by(simp_all add: assert_gpv_def) |
From Coq Require Import
List
ssreflect
.
From ExtensibleCompiler.Theory Require Import
Algebra
Environment
Eval
Functor
ProgramAlgebra
SubFunctor
Sum1
Types
UniversalProperty
.
Local Open Scope SubFunctor.
Inductive Closure
L
`{F : forall V, ... |
subsection \<open>Implementation of Division on Multivariate Polynomials\<close>
theory MPoly_Divide_Code
imports
MPoly_Divide
Polynomials.MPoly_Type_Class_FMap
Polynomials.MPoly_Type_Univariate
begin
text \<open>
We now set up code equations for some of the operations that we will need, such as div... |
lemma succ_le_succ (a b : mynat) (h : a ≤ b) : succ a ≤ succ b :=
begin
cases h with c hc,
use c,
rwa [hc, succ_add],
end
|
import linear_algebra.finite_dimensional
import missing_mathlib.linear_algebra.dimension
universes u v v' w
open_locale classical
open vector_space cardinal submodule module function
variables {K : Type u} {V : Type v} [field K] [add_comm_group V] [vector_space K V]
{V₂ : Type v'} [add_comm_group V₂] [vector_space K... |
Formal statement is: lemma emeasure_lfp'[consumes 1, case_names cont measurable]: assumes "P M" assumes cont: "sup_continuous F" assumes *: "\<And>M A. P M \<Longrightarrow> (\<And>N. P N \<Longrightarrow> Measurable.pred N A) \<Longrightarrow> Measurable.pred M (F A)" shows "emeasure M {x\<in>space M. lfp F x} = (SUP ... |
write (*,*)
! !"AA"
end |
variables p q : Prop
theorem t1 : p → q → p := λ (hp : p) (hq : q), hp
|
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