claude gets stuck

Stream: lean4

[Jared green]

i tried getting claude to solve this(trying to prove c1), but it got stuck. i also got a confusing type error in rep2:

import Init.Data.List
import Init.PropLemmas
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Join
import Mathlib.Data.List.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Lemmas
import Mathlib.Data.Bool.AllAny
import Mathlib.Data.Bool.Basic
import Mathlib.Logic.Basic
import Batteries.Data.List.Lemmas
import Aesop
open Classical

variable {α : Type}[h : DecidableEq α]
inductive normalizable α (pred : α -> Prop)
  where
  | atom : α -> (normalizable α pred)
  | And : (normalizable α pred) -> (normalizable α pred) -> normalizable α pred
  | Or : (normalizable α pred) -> (normalizable α pred) -> normalizable α pred
  | Not : normalizable α pred -> normalizable α pred
deriving DecidableEq

namespace normalizable

@[reducible]
def toProp (n : normalizable α pred) : Prop :=
  match n with
  | atom a => pred a
  | And a b =>  toProp a ∧ toProp b
  | Or a b => (toProp a) ∨ toProp b
  | Not i => ¬(toProp i)

def wToProp (w : Bool × normalizable α pred) : Prop :=
  if w.fst then toProp w.snd else ¬(toProp w.snd)

def sToProp (s : List (Bool × normalizable α pred)) : Prop :=
  s.all (fun x => wToProp x)

def gToProp (g : List (List (Bool × normalizable α pred))) : Prop :=
  g.any (fun x => sToProp x)

def nToProp (n : List (List (List (Bool × normalizable α pred)))) : Prop :=
  n.all (fun x => gToProp x)

def fToProp (n : List (List (List (normalizable α pred)))) : Prop :=
  n.all (fun x => x.any (fun y => y.all (fun z => toProp z)))

theorem all_and : List.all ( a ++ b) c <-> List.all a c ∧ List.all b c :=
  by
  simp

theorem any_erase :∀ l : List b,∀ a: b -> Prop,
                     ∀ s : b,s ∈ l -> ¬ (a s) -> (List.any l a <-> List.any (List.erase l s) a) :=
  by
  sorry

theorem any_filter : ∀ s : a -> Bool, ∀ l:List a, l.any s <-> (l.filter s).any s :=
  by
  sorry

theorem all_filter (s t : a -> Bool) : ∀ l : List a, l.all s -> (l.filter t).all s :=
  by
  sorry

theorem any_filter_imp (s t : a -> Bool): (∀ x : a, ¬ (s x) -> ¬ (t x)) -> ∀ l : List a,l.any t <-> (l.filter s).any t :=
  by
  sorry

theorem property1 : ∀ n : List (List (List (Bool × normalizable α pred))),
                    ∀ g : List (List (Bool × normalizable α pred)), g ∈ n ->
                    ∀ s : List (Bool × normalizable α pred), s ∈ g ->
                    ∀ i : Bool × normalizable α pred, (nToProp n -> sToProp s -> wToProp i) ->
                    (nToProp n -> (sToProp s <-> sToProp (s.concat i))) :=
  by
  sorry

theorem property2 : ∀ n : List (List (List (Bool × normalizable α pred))),
                    ∀ g : List (List (Bool × normalizable α pred)), g ∈ n ->
                    ∀ s : List (Bool × normalizable α pred), s ∈ g ->
                    ((nToProp n -> ¬(sToProp s)) -> nToProp n -> (gToProp g <-> gToProp (g.erase s))) :=
  by
  sorry

def bcompatible (s : List (Bool × normalizable α pred)) (t : List (Bool × normalizable α pred)) : Bool :=
  s.all (fun x => t.all (fun y =>  x.snd == y.snd -> x.fst == y.fst))

theorem compatibility :∀ a b : List (Bool × normalizable α pred),  (¬ bcompatible a b = true) -> ¬(sToProp a ∧ sToProp b) :=
  by
  sorry

theorem rule1 : ∀ n : List (List (List (Bool × normalizable α pred))),
                ∀ g : List (List (Bool × normalizable α pred)), g ∈ n ->
                ∀ s : List (Bool × normalizable α pred), s ∈ g ->
                ∀ v : Bool × normalizable α pred, ¬(v ∈ s) ->
                (∃ h : List (List (Bool × normalizable α pred)), h ∈ n ∧
                ∀ t : List (Bool × normalizable α pred), t ∈ h ->
                bcompatible t s -> v ∈ t) ->
                nToProp n -> sToProp s -> wToProp v :=
  by
  sorry

theorem rule2 : ∀ n : List (List (List (Bool × normalizable α pred))),
                ∀ g : List (List (Bool × normalizable α pred)), g ∈ n ->
                ∀ s : List (Bool × normalizable α pred),
                (∃ h : List (List (Bool × normalizable α pred)), h ∈ n ∧
                ∀ t : List (Bool × normalizable α pred), t ∈ h ->
                ¬(bcompatible s t)) ->
                nToProp n -> ¬(sToProp s) :=
  by
  sorry

theorem op1 : ∀ n : List (List (List (Bool × normalizable α pred))),
                    ∀ g h : List (List (Bool × normalizable α pred)), g ∈ n ∧ h ∈ n ->
                    nToProp n -> (gToProp g <-> gToProp (g.filter (fun x => h.any (fun y => bcompatible x y )))) :=
  by
  sorry

@[reducible]
def interl (l : List (List a)) [DecidableEq a] : List a :=
  match l with
  | [] => []
  | a :: [] => a
  | (a :: as) => List.inter a (interl as)


theorem interl_all (s : a -> Prop) : ∀ l : List (List a),
          l.any (fun x => x.all (fun y => s y)) -> ∀ b ∈ interl l, s b :=
  by
  sorry

theorem interl_all_filter (s : a -> Prop)(t : List a -> Prop) : (∀ x : List a, ¬ t x -> ¬(x.all fun y => s y))
     -> ∀ l : List (List a), (l.any (fun x => x.all (fun y => s y))
     -> ∀ b ∈ interl (l.filter (fun x => t x)), s b) :=
  by
  sorry

theorem forall_mem_or {b : a -> Prop}{c : a -> Prop}{e : a -> Prop}: (∀ f, (b f ∨ c f) -> e f) <-> (∀ f, b f -> e f) ∧ (∀ f, c f -> e f ) :=
  by
  sorry

theorem interl_filter_filter (d : a -> Prop)(e : List a -> Prop):
        ∀ b : a,∀ c : List (List a),
        b ∈ interl ((c.filter (fun x: List a => e x)).map (fun x => x.filter (fun y : a => (d y) )))
        -> b ∈ interl (c.filter fun x => e x) :=
  by
  sorry

theorem op2 : ∀ n : List (List (List (Bool × normalizable α pred))),
              ∀ g h : List (List (Bool × normalizable α pred)), h ∈ n
              -> g.all (fun x => h.any (fun y => bcompatible x y))
              -> nToProp n
              -> (gToProp g
                <-> gToProp (g.map (fun x =>
                x.append (interl ((h.filter
                (fun y => bcompatible x y)).map
              (fun y => y.filter (fun z => z ∉ x))))))) :=
  by
  sorry

theorem rule3 : ∀ n : List (List (List (Bool × normalizable α pred))), [] ∈ n -> ¬(nToProp n) :=
  by
  sorry

theorem rep1 : ∀ n : List (List (List (Bool × normalizable α pred))),
              ∀ g : List (List (Bool × normalizable α pred)), g ∈ n ->
              ∀ s t : List (Bool × normalizable α pred), s ∈ g ->
              (nToProp n -> (sToProp s <-> sToProp t) ) ->
              (nToProp n -> (gToProp g <-> gToProp (g.replace s t))) :=
  by
  sorry

theorem rep2 : ∀ n : List (List (List (Bool × normalizable α pred))),
             ∀ g : List (List (Bool × normalizable α pred)), g ∈ n ->
             (nToProp n -> (gToProp g <-> gToProp h)) -> (nToProp n <-> nToProp (n.replace g h)) :=
  by
  intro n g h hg hngh
  constructor
  intro hn
  unfold nToProp
  simp
  intro x hx
  unfold nToProp at hn
  simp at hn
  have hxx : x = h ∨ x ∈ n :=  by {
    apply mem_replace_or_mem g h x n
    convert hx
  }
  cases' (hxx) with hxh hxn
  rw [hxh]
  unfold nToProp at hngh
  simp at hngh
  have hgh := hngh hn
  rw [← hgh]
  apply hn
  exact hg
  apply hn
  exact hxn
  intro hn
  unfold nToProp
  simp
  intro x hx
  unfold nToProp at hn
  simp at hn
  cases' Classical.em (x = g) with hxg hxng
  rw [hxg]
  rw [iff_def'] at hngh
  rw [imp_and] at hngh
  obtain ⟨ hnghl,hnghr⟩ := hngh
  apply hnghl
  unfold nToProp
  simp
  intro y hy
  cases' Classical.em (y = g) with hyg hyng
  rw [hyg]
  apply hn
  convert mem_replace_of_eq g h n hg --this should not expect a g = h
  sorry
  apply hn
  rw [← ne_eq] at hyng
  convert mem_replace_of_mem_of_ne_r g h y n hy hyng
  apply hn
  convert mem_replace_of_eq g h n hg
  have hgx := mem_replace_of_mem_of_ne_r g h x n hx hxng
  apply hn
  convert hgx

theorem c1 : ∀ n : List (List (List (Bool × normalizable α pred))),
             ∀ g : List (List (Bool × normalizable α pred)), g ∈ n ->
             ∀ s : List (Bool × normalizable α pred), s ∈ g ->
             ∀ w : Bool × normalizable α pred, ¬(w ∈ s) ->
             (nToProp n -> (sToProp s -> wToProp w)) ->
             ∃ t : List (Bool × normalizable α pred),
             (List.Subset s t) ∧ ¬(w ∈ t) ∧
             (nToProp n -> (sToProp s <-> sToProp t)) ∧
             ∃ h i : List (List (Bool × normalizable α pred)),
             h ∈ n ∧ ¬(h = g) ∧ (nToProp n -> (gToProp h <-> gToProp i)) ∧
             ∀ u : List (Bool × normalizable α pred), u ∈ i ->
             (bcompatible t u) -> w ∈ u :=
  by
  intro n g hg s hs w hw hhw
  by_contra ht
  have h_rule1 := rule1 n g hg s hs w hw
  --i didnt get much more(that was useful) than this out of it

[Jared green]

the theorems rep2 and c1 are true, but i cant prove them at this part of my file.

[Chris Bailey]

You will probably need to shrink this way down (like >10x) and ask a specific question to get help. Many of the people here are very generous with their time, but you have to meet them in the middle.