how do i name side variables in induction?

Stream: lean4

[Jared green]

i was trying to prove this (subnormal) but cannot reference unnamed variables. and i dont know how to make it so they are named.

import Init.Data.List
import Std.Data.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

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)

@[simp]
theorem toProp_not : toProp (Not n₁) ↔ ¬ toProp n₁ := Iff.rfl

@[simp]
theorem toProp_and : toProp (And n₁ n₂) ↔ toProp n₁ ∧ toProp n₂ := Iff.rfl

@[simp]
theorem toProp_or : toProp (Or n₁ n₂) ↔ toProp n₁ ∨ toProp n₂ := Iff.rfl

@[simp]
theorem toProp_atom {a : α} : toProp (atom a : normalizable α pred) ↔ pred a := Iff.rfl


def subnormalize (n : (normalizable α pred)) : List (List (List (normalizable α pred))) :=
  match n with
  | Or a b => [[a,n],[b,n],[Not a,Not b, Not n]] :: (List.append (subnormalize a) (subnormalize b))
  | And a b => [[a,b,n],[Not a,Not n],[Not b,Not n]] :: (List.append (subnormalize a) (subnormalize b))
  | Not i => [[n,Not i],[Not n, i]] :: (subnormalize i)
  | atom _ => [[[n],[Not n]]]

def normalize :  normalizable α pred -> List (List (List (normalizable α pred))) := fun o =>
  [[o]] :: (subnormalize o)

def nStrip (n : normalizable α pred) : Bool × normalizable α pred :=
  match n with
  | Not i => (false,i)
  | i => (true,i)

def booleanize (n : List (List (List (normalizable α pred)))) : List (List (List (Bool × normalizable α pred))) :=
  n.map (fun x => x.map (fun y => y.map (fun z => nStrip z)))

def normalizel (n : normalizable α pred) : List (List (List (Bool × normalizable α pred))) :=
  booleanize (normalize n)

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 nStrip_equiv : ∀ n : normalizable α pred, toProp n <-> wToProp (nStrip n) :=
  by
  intro n
  unfold nStrip
  induction n
  simp
  unfold wToProp
  simp
  simp
  unfold wToProp
  simp
  unfold wToProp
  simp
  simp
  unfold wToProp
  simp

theorem booleanize_eqiv : ∀ n : List (List (List (normalizable α pred))), fToProp n <-> nToProp (booleanize n) :=
  by
  intro n
  unfold nToProp
  simp
  unfold fToProp
  simp
  unfold booleanize
  simp
  unfold gToProp
  simp
  unfold sToProp
  simp
  simp [nStrip_equiv]

theorem andGateTaut : (¬ a ∧ ¬(a ∧ b)) ∨ (¬ b ∧ ¬(a ∧ b)) ∨ (a ∧ b ∧ (a ∧ b)) :=
  by
  cases Classical.em a
  cases Classical.em b
  right
  right
  constructor
  assumption
  constructor
  assumption
  constructor
  assumption
  assumption
  right
  left
  constructor
  assumption
  push_neg
  intro
  assumption
  left
  constructor
  assumption
  rw [and_comm]
  push_neg
  intro
  assumption

theorem orGateTaut : (a ∧ (a ∨ b)) ∨ (b ∧ (a ∨ b)) ∨ ((¬ a) ∧ (¬b) ∧ ¬(a ∨ b)) = true :=
  by
  simp
  cases Classical.em a
  left
  constructor
  assumption
  left
  assumption
  right
  cases Classical.em b
  left
  constructor
  assumption
  right
  assumption
  right
  constructor
  assumption
  constructor
  assumption
  push_neg
  constructor
  assumption
  assumption

theorem notGateTaut : (a ∧ ¬(¬ a)) ∨ (¬ a ∧ ¬ a) = true :=
  by
  simp
  apply Classical.em

theorem all_and : List.all ( a ++ b) c <-> List.all a c ∧ List.all b c :=
  by
  simp
  constructor
  intro ha
  constructor
  intro b
  intro hb
  apply ha
  left
  exact hb
  intro c
  intro hc
  apply ha
  right
  exact hc
  intro ha
  intro b
  intro hb
  cases hb
  apply ha.left
  assumption
  apply ha.right
  assumption

theorem subnormal : ∀ n : normalizable α pred, fToProp (subnormalize n) :=
  by
  intro n
  induction n with a b c d e f g h i --name all side variables here
  unfold subnormalize
  unfold fToProp
  simp
  unfold toProp
  apply Classical.em
  unfold subnormalize
  simp
  unfold fToProp
  rw [List.all_cons]
  simp only [List.any_cons, List.all_cons, List.all_nil, Bool.and_true, List.any_nil,
    Bool.or_false, Bool.and_eq_true, Bool.or_eq_true, decide_eq_true_eq,
    List.mem_append, List.any_eq_true]
  constructor
  rw [toProp_not]
  rw [toProp_and]
  sorry --should be andGateTaut
  rw [all_and]
  constructor
  assumption
  assumption
  unfold fToProp
  unfold subnormalize
  rw [List.all_cons]
  simp only [List.any_cons, List.all_cons, toProp_or, List.all_nil, Bool.and_true, toProp_not,
    List.any_nil, Bool.or_false, List.append_eq, Bool.and_eq_true, Bool.or_eq_true,
    decide_eq_true_eq, List.mem_append, List.any_eq_true]
  constructor
  rw [toProp_not]
  rw [toProp_or]
  sorry  --should be orGateTaut
  rw [all_and]
  constructor
  assumption
  assumption
  unfold fToProp
  unfold subnormalize
  rw [List.all_cons]
  simp only [List.any_cons, List.all_cons, toProp_not, List.all_nil, Bool.and_true, Bool.and_self,
    not_not, List.any_nil, Bool.or_false, Bool.and_eq_true, Bool.or_eq_true, decide_eq_true_eq,
    List.any_eq_true]
  constructor
  sorry --should be notGateTaut
  assumption

[Timo Carlin-Burns]

If you want to name your variables all on one line like that, you can use induction' instead of induction

[Jared green]

theorem subnormal : ∀ n : normalizable α pred, fToProp (subnormalize n) :=
  by
  intro n
  induction' n with a b c d e f g i j k l
  unfold subnormalize
  unfold fToProp
  simp
  unfold toProp
  apply Classical.em
  unfold subnormalize
  simp
  unfold fToProp
  rw [List.all_cons]
  simp only [List.any_cons, List.all_cons, List.all_nil, Bool.and_true, List.any_nil,
    Bool.or_false, Bool.and_eq_true, Bool.or_eq_true, decide_eq_true_eq,
    List.mem_append, List.any_eq_true]
  constructor
  rw [toProp_not]
  rw [toProp_and]
  exact andGateTaut (toProp b)
  rw [all_and]
  constructor
  assumption
  assumption
  unfold fToProp
  unfold subnormalize
  rw [List.all_cons]
  simp only [List.any_cons, List.all_cons, toProp_or, List.all_nil, Bool.and_true, toProp_not,
    List.any_nil, Bool.or_false, List.append_eq, Bool.and_eq_true, Bool.or_eq_true,
    decide_eq_true_eq, List.mem_append, List.any_eq_true]
  constructor
  rw [toProp_not]
  rw [toProp_or]
  exact orGateTaut (toProp f)
  rw [all_and]
  constructor
  assumption
  assumption
  unfold fToProp
  unfold subnormalize
  rw [List.all_cons]
  simp only [List.any_cons, List.all_cons, toProp_not, List.all_nil, Bool.and_true, Bool.and_self,
    not_not, List.any_nil, Bool.or_false, Bool.and_eq_true, Bool.or_eq_true, decide_eq_true_eq,
    List.any_eq_true]
  constructor
  exact
  unfold fToProp at l --apparently this nor 'assumption' work like it did before.

[Ruben Van de Velde]

If you write induction x in Vs code, you should get a blue lightbulb too click on which can create a skeleton that allows naming the variables

[Jared green]

now i have a different problem. shall i start a new discussion for it?