module Spec
  use option.Option
  use int.Int
  use Nat.Nat
  use Functions.Func
  use OptionFuncs.Funcs
  use Sum.Sum
  use Firstorder_symbol_spec.Spec
  use Firstorder_term_spec.Spec
  use Firstorder_formula_spec.Spec
  use Firstorder_formula_list_spec.Spec
  type tableau 'b0 'b1 =
    | Root
    | Node (tableau 'b0 'b1) (fo_formula 'b0 'b1) (fo_formula_list 'b0 'b1)

  function nat_size_tableau (t:tableau 'b0 'b1) : nat =
    match t with | Root -> let s = one_nat in s
      | Node v0 v1 v2 ->
        let s = one_nat in let s = add_nat (nat_size_fo_formula_list v2) s in
          let s = add_nat (nat_size_fo_formula v1) s in
          let s = add_nat (nat_size_tableau v0) s in s
    end

  with size_tableau (t:tableau 'b0 'b1) : int =
    match t with | Root -> let s = 1 in s
      | Node v0 v1 v2 ->
        let s = 1 in let s = size_fo_formula_list v2 + s in
          let s = size_fo_formula v1 + s in let s = size_tableau v0 + s in s
    end

  let rec lemma size_positive_lemma_tableau (t:tableau 'b0 'b1) : unit
    ensures { size_tableau t > 0 }
    variant { nat_to_int (nat_size_tableau t) } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        size_positive_lemma_tableau v0 ; size_positive_lemma_fo_formula v1 ;
          size_positive_lemma_fo_formula_list v2 ; ()
    end

  function rename_tableau (t:tableau 'b0 'b1) (s0:'b0 -> 'c0)
    (s1:'b1 -> 'c1) : tableau 'c0 'c1 =
    match t with | Root -> Root
      | Node v0 v1 v2 ->
        Node (rename_tableau v0 s0 s1) (rename_fo_formula v1 s0 s1)
          (rename_fo_formula_list v2 s0 s1)
    end

  let rec lemma renaming_composition_lemma_tableau (t:tableau 'b0 'b1)
    (s10:'b0 -> 'c0) (s11:'b1 -> 'c1) (s20:'c0 -> 'd0) (s21:'c1 -> 'd1) :
    unit
    ensures { rename_tableau (rename_tableau t s10 s11) s20 s21 =
      rename_tableau t (rcompose s10 s20) (rcompose s11 s21) }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        renaming_composition_lemma_tableau v0 s10 s11 s20 s21 ;
          renaming_composition_lemma_fo_formula v1 s10 s11 s20 s21 ;
          renaming_composition_lemma_fo_formula_list v2 s10 s11 s20 s21 ; ()
    end

  let rec lemma renaming_identity_lemma_tableau (t:tableau 'b0 'b1) : unit
    ensures { rename_tableau t identity identity = t }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        renaming_identity_lemma_tableau v0 ;
          renaming_identity_lemma_fo_formula v1 ;
          renaming_identity_lemma_fo_formula_list v2 ; ()
    end

  function subst_tableau (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)) : tableau 'c0 'c1 =
    match t with | Root -> Root
      | Node v0 v1 v2 ->
        Node
          (subst_tableau v0 (rename_subst_symbol s0 identity)
            (rename_subst_fo_term s1 identity identity))
          (subst_fo_formula v1 (rename_subst_symbol s0 identity)
            (rename_subst_fo_term s1 identity identity))
          (subst_fo_formula_list v2 (rename_subst_symbol s0 identity)
            (rename_subst_fo_term s1 identity identity))
    end

  let rec lemma rename_then_subst_composition_lemma_tableau
    (t:tableau 'b0 'b1) (s10:'b0 -> 'c0) (s11:'b1 -> 'c1)
    (s20:'c0 -> (symbol 'd0)) (s21:'c1 -> (fo_term 'd0 'd1)) : unit
    ensures { subst_tableau (rename_tableau t s10 s11) s20 s21 =
      subst_tableau t (rcompose s10 s20) (rcompose s11 s21) }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        rename_then_subst_composition_lemma_tableau v0 s10 s11
          (rename_subst_symbol s20 identity)
          (rename_subst_fo_term s21 identity identity) ;
          rename_then_subst_composition_lemma_fo_formula v1 s10 s11
          (rename_subst_symbol s20 identity)
          (rename_subst_fo_term s21 identity identity) ;
          rename_then_subst_composition_lemma_fo_formula_list v2 s10 s11
          (rename_subst_symbol s20 identity)
          (rename_subst_fo_term s21 identity identity) ; ()
    end

  let rec lemma subst_then_rename_composition_lemma_tableau
    (t:tableau 'b0 'b1) (s10:'b0 -> (symbol 'c0))
    (s11:'b1 -> (fo_term 'c0 'c1)) (s20:'c0 -> 'd0) (s21:'c1 -> 'd1) : unit
    ensures { rename_tableau (subst_tableau t s10 s11) s20 s21 =
      subst_tableau t (rename_subst_symbol s10 s20)
        (rename_subst_fo_term s11 s20 s21)
      }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        subst_then_rename_composition_lemma_tableau v0
          (rename_subst_symbol s10 identity)
          (rename_subst_fo_term s11 identity identity) s20 s21 ;
          subst_then_rename_composition_lemma_fo_formula v1
          (rename_subst_symbol s10 identity)
          (rename_subst_fo_term s11 identity identity) s20 s21 ;
          subst_then_rename_composition_lemma_fo_formula_list v2
          (rename_subst_symbol s10 identity)
          (rename_subst_fo_term s11 identity identity) s20 s21 ; ()
    end

  let rec lemma subst_composition_lemma_tableau (t:tableau 'b0 'b1)
    (s10:'b0 -> (symbol 'c0)) (s11:'b1 -> (fo_term 'c0 'c1))
    (s20:'c0 -> (symbol 'd0)) (s21:'c1 -> (fo_term 'd0 'd1)) : unit
    ensures { subst_tableau (subst_tableau t s10 s11) s20 s21 =
      subst_tableau t (subst_compose_symbol s10 s20)
        (subst_compose_fo_term s11 s20 s21)
      }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        subst_composition_lemma_tableau v0 (rename_subst_symbol s10 identity)
          (rename_subst_fo_term s11 identity identity)
          (rename_subst_symbol s20 identity)
          (rename_subst_fo_term s21 identity identity) ;
          subst_composition_lemma_fo_formula v1
          (rename_subst_symbol s10 identity)
          (rename_subst_fo_term s11 identity identity)
          (rename_subst_symbol s20 identity)
          (rename_subst_fo_term s21 identity identity) ;
          subst_composition_lemma_fo_formula_list v2
          (rename_subst_symbol s10 identity)
          (rename_subst_fo_term s11 identity identity)
          (rename_subst_symbol s20 identity)
          (rename_subst_fo_term s21 identity identity) ; ()
    end

  let rec lemma subst_identity_lemma_tableau (t:tableau 'b0 'b1) : unit
    ensures { subst_tableau t subst_id_symbol subst_id_fo_term = t }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        subst_identity_lemma_tableau v0 ; subst_identity_lemma_fo_formula
          v1 ; subst_identity_lemma_fo_formula_list v2 ; ()
    end

  let rec lemma renaming_preserve_size_tableau (t:tableau 'b0 'b1)
    (s0:'b0 -> 'c0) (s1:'b1 -> 'c1) : unit
    ensures { size_tableau (rename_tableau t s0 s1) = size_tableau t }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        renaming_preserve_size_tableau v0 (s0) (s1) ;
          renaming_preserve_size_fo_formula v1 (s0) (s1) ;
          renaming_preserve_size_fo_formula_list v2 (s0) (s1) ; ()
    end

  predicate is_symbol_free_var_in_tableau (x:'b0) (t:tableau 'b0 'b1) =
    match t with | Root -> false
      | Node v0 v1 v2 ->
        is_symbol_free_var_in_tableau x v0 \/
          is_symbol_free_var_in_fo_formula x v1 \/
          is_symbol_free_var_in_fo_formula_list x v2
    end

  with is_fo_term_free_var_in_tableau (x:'b1) (t:tableau 'b0 'b1) =
    match t with | Root -> false
      | Node v0 v1 v2 ->
        is_fo_term_free_var_in_tableau x v0 \/
          is_fo_term_free_var_in_fo_formula x v1 \/
          is_fo_term_free_var_in_fo_formula_list x v2
    end

  let rec ghost rename_free_var_constructive_inversion_symbol_tableau (x:'c0)
    (t:tableau 'b0 'b1) (s0:'b0 -> 'c0) (s1:'b1 -> 'c1) : 'b0
    requires { is_symbol_free_var_in_tableau x (rename_tableau t s0 s1) }
    ensures { is_symbol_free_var_in_tableau result t /\ s0 result = x }
    variant { size_tableau t } =
    match t with | Root -> absurd
      | Node v0 v1 v2 ->
        if is_symbol_free_var_in_tableau x (rename_tableau v0 s0 s1)
          then
            let sumx =
              rename_free_var_constructive_inversion_symbol_tableau x v0 s0
                s1
            in sumx
          else
            if is_symbol_free_var_in_fo_formula x
                 (rename_fo_formula v1 s0 s1)
              then
                let sumx =
                  rename_free_var_constructive_inversion_symbol_fo_formula x
                    v1 s0 s1
                in sumx
              else
                if is_symbol_free_var_in_fo_formula_list x
                     (rename_fo_formula_list v2 s0 s1)
                  then
                    let sumx =
                      rename_free_var_constructive_inversion_symbol_fo_formula_list
                        x v2 s0 s1
                    in sumx
                  else absurd
    end

  with lemma rename_free_var_inversion_symbol_tableau (x:'c0)
    (t:tableau 'b0 'b1) (s0:'b0 -> 'c0) (s1:'b1 -> 'c1) : unit
    requires { is_symbol_free_var_in_tableau x (rename_tableau t s0 s1) }
    ensures { exists y:'b0. is_symbol_free_var_in_tableau y t /\ s0 y = x }
    variant { 1 + size_tableau t } =
    let sumx =
      rename_free_var_constructive_inversion_symbol_tableau x t s0 s1
    in ()

  with ghost rename_free_var_constructive_inversion_fo_term_tableau (x:'c1)
    (t:tableau 'b0 'b1) (s0:'b0 -> 'c0) (s1:'b1 -> 'c1) : 'b1
    requires { is_fo_term_free_var_in_tableau x (rename_tableau t s0 s1) }
    ensures { is_fo_term_free_var_in_tableau result t /\ s1 result = x }
    variant { size_tableau t } =
    match t with | Root -> absurd
      | Node v0 v1 v2 ->
        if is_fo_term_free_var_in_tableau x (rename_tableau v0 s0 s1)
          then
            let sumx =
              rename_free_var_constructive_inversion_fo_term_tableau x v0 s0
                s1
            in sumx
          else
            if is_fo_term_free_var_in_fo_formula x
                 (rename_fo_formula v1 s0 s1)
              then
                let sumx =
                  rename_free_var_constructive_inversion_fo_term_fo_formula x
                    v1 s0 s1
                in sumx
              else
                if is_fo_term_free_var_in_fo_formula_list x
                     (rename_fo_formula_list v2 s0 s1)
                  then
                    let sumx =
                      rename_free_var_constructive_inversion_fo_term_fo_formula_list
                        x v2 s0 s1
                    in sumx
                  else absurd
    end

  with lemma rename_free_var_inversion_fo_term_tableau (x:'c1)
    (t:tableau 'b0 'b1) (s0:'b0 -> 'c0) (s1:'b1 -> 'c1) : unit
    requires { is_fo_term_free_var_in_tableau x (rename_tableau t s0 s1) }
    ensures { exists y:'b1. is_fo_term_free_var_in_tableau y t /\ s1 y = x }
    variant { 1 + size_tableau t } =
    let sumx =
      rename_free_var_constructive_inversion_fo_term_tableau x t s0 s1
    in ()

  let rec lemma rename_free_var_propagation_symbol_tableau (x:'b0)
    (t:tableau 'b0 'b1) (s0:'b0 -> 'c0) (s1:'b1 -> 'c1) : unit
    ensures { is_symbol_free_var_in_tableau x t ->
      is_symbol_free_var_in_tableau (s0 x) (rename_tableau t s0 s1) }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        rename_free_var_propagation_symbol_tableau x v0 (s0) (s1) ;
          rename_free_var_propagation_symbol_fo_formula x v1 (s0) (s1) ;
          rename_free_var_propagation_symbol_fo_formula_list x v2 (s0) (s1) ;
          ()
    end

  with lemma rename_free_var_propagation_fo_term_tableau (x:'b1)
    (t:tableau 'b0 'b1) (s0:'b0 -> 'c0) (s1:'b1 -> 'c1) : unit
    ensures { is_fo_term_free_var_in_tableau x t ->
      is_fo_term_free_var_in_tableau (s1 x) (rename_tableau t s0 s1) }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        rename_free_var_propagation_fo_term_tableau x v0 (s0) (s1) ;
          rename_free_var_propagation_fo_term_fo_formula x v1 (s0) (s1) ;
          rename_free_var_propagation_fo_term_fo_formula_list x v2 (s0)
          (s1) ; ()
    end

  let rec ghost subst_free_var_constructive_inversion_symbol_tableau (x:'c0)
    (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)) : sum ('b0) ('b1)
    requires { is_symbol_free_var_in_tableau x (subst_tableau t s0 s1) }
    ensures { let sumx = result in
      match sumx with | Left sumx ->
        is_symbol_free_var_in_tableau sumx t /\
          is_symbol_free_var_in_symbol x (s0 sumx)
        | Right sumx ->
        is_fo_term_free_var_in_tableau sumx t /\
          is_symbol_free_var_in_fo_term x (s1 sumx)
      end }
    variant { size_tableau t } =
    match t with | Root -> absurd
      | Node v0 v1 v2 ->
        if is_symbol_free_var_in_tableau x
             (subst_tableau v0 (rename_subst_symbol s0 identity)
                (rename_subst_fo_term s1 identity identity))
          then
            let sumx =
              subst_free_var_constructive_inversion_symbol_tableau x v0
                (rename_subst_symbol s0 identity)
                (rename_subst_fo_term s1 identity identity)
            in
            match sumx with | Left sumx ->
              let y =
                rename_free_var_constructive_inversion_symbol_symbol x
                  (eval s0 sumx) identity
                in assert { y = x } ; Left sumx
              | Right sumx ->
              Right
                (let y =
                   rename_free_var_constructive_inversion_symbol_fo_term x
                     (eval s1 sumx) identity identity
                in assert { y = x } ; sumx)
            end
          else
            if is_symbol_free_var_in_fo_formula x
                 (subst_fo_formula v1 (rename_subst_symbol s0 identity)
                    (rename_subst_fo_term s1 identity identity))
              then
                let sumx =
                  subst_free_var_constructive_inversion_symbol_fo_formula x
                    v1 (rename_subst_symbol s0 identity)
                    (rename_subst_fo_term s1 identity identity)
                in
                match sumx with | Left sumx ->
                  let y =
                    rename_free_var_constructive_inversion_symbol_symbol x
                      (eval s0 sumx) identity
                    in assert { y = x } ; Left sumx
                  | Right sumx ->
                  Right
                    (let y =
                       rename_free_var_constructive_inversion_symbol_fo_term
                         x (eval s1 sumx) identity identity
                    in assert { y = x } ; sumx)
                end
              else
                if is_symbol_free_var_in_fo_formula_list x
                     (subst_fo_formula_list v2
                        (rename_subst_symbol s0 identity)
                        (rename_subst_fo_term s1 identity identity))
                  then
                    let sumx =
                      subst_free_var_constructive_inversion_symbol_fo_formula_list
                        x v2 (rename_subst_symbol s0 identity)
                        (rename_subst_fo_term s1 identity identity)
                    in
                    match sumx with | Left sumx ->
                      let y =
                        rename_free_var_constructive_inversion_symbol_symbol
                          x (eval s0 sumx) identity
                        in assert { y = x } ; Left sumx
                      | Right sumx ->
                      Right
                        (let y =
                           rename_free_var_constructive_inversion_symbol_fo_term
                             x (eval s1 sumx) identity identity
                        in assert { y = x } ; sumx)
                    end
                  else absurd
    end

  with lemma subst_free_var_inversion_symbol_tableau (x:'c0)
    (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)) : unit
    requires { is_symbol_free_var_in_tableau x (subst_tableau t s0 s1) }
    ensures {
      (exists y:'b0. is_symbol_free_var_in_tableau y t /\
         is_symbol_free_var_in_symbol x (s0 y))
      \/
      (exists y:'b1. is_fo_term_free_var_in_tableau y t /\
         is_symbol_free_var_in_fo_term x (s1 y))
      }
    variant { 1 + size_tableau t } =
    let sumx = subst_free_var_constructive_inversion_symbol_tableau x t s0 s1
    in match sumx with | Left sumx -> () | Right sumx -> () end

  with ghost subst_free_var_constructive_inversion_fo_term_tableau (x:'c1)
    (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)) : 'b1
    requires { is_fo_term_free_var_in_tableau x (subst_tableau t s0 s1) }
    ensures { let sumx = result in is_fo_term_free_var_in_tableau sumx t /\
      is_fo_term_free_var_in_fo_term x (s1 sumx) }
    variant { size_tableau t } =
    match t with | Root -> absurd
      | Node v0 v1 v2 ->
        if is_fo_term_free_var_in_tableau x
             (subst_tableau v0 (rename_subst_symbol s0 identity)
                (rename_subst_fo_term s1 identity identity))
          then
            let sumx =
              subst_free_var_constructive_inversion_fo_term_tableau x v0
                (rename_subst_symbol s0 identity)
                (rename_subst_fo_term s1 identity identity)
            in
            let y =
              rename_free_var_constructive_inversion_fo_term_fo_term x
                (eval s1 sumx) identity identity
            in assert { y = x } ; sumx
          else
            if is_fo_term_free_var_in_fo_formula x
                 (subst_fo_formula v1 (rename_subst_symbol s0 identity)
                    (rename_subst_fo_term s1 identity identity))
              then
                let sumx =
                  subst_free_var_constructive_inversion_fo_term_fo_formula x
                    v1 (rename_subst_symbol s0 identity)
                    (rename_subst_fo_term s1 identity identity)
                in
                let y =
                  rename_free_var_constructive_inversion_fo_term_fo_term x
                    (eval s1 sumx) identity identity
                in assert { y = x } ; sumx
              else
                if is_fo_term_free_var_in_fo_formula_list x
                     (subst_fo_formula_list v2
                        (rename_subst_symbol s0 identity)
                        (rename_subst_fo_term s1 identity identity))
                  then
                    let sumx =
                      subst_free_var_constructive_inversion_fo_term_fo_formula_list
                        x v2 (rename_subst_symbol s0 identity)
                        (rename_subst_fo_term s1 identity identity)
                    in
                    let y =
                      rename_free_var_constructive_inversion_fo_term_fo_term
                        x (eval s1 sumx) identity identity
                    in assert { y = x } ; sumx
                  else absurd
    end

  with lemma subst_free_var_inversion_fo_term_tableau (x:'c1)
    (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)) : unit
    requires { is_fo_term_free_var_in_tableau x (subst_tableau t s0 s1) }
    ensures {
      (exists y:'b1. is_fo_term_free_var_in_tableau y t /\
         is_fo_term_free_var_in_fo_term x (s1 y))
      }
    variant { 1 + size_tableau t } =
    let sumx =
      subst_free_var_constructive_inversion_fo_term_tableau x t s0 s1
    in ()

  let rec lemma subst_free_var_propagation_symbol_symbol_tableau (x:'b0)
    (y:'c0) (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)):  unit
    ensures {
      is_symbol_free_var_in_tableau x t /\
        is_symbol_free_var_in_symbol y (s0 x) ->
      is_symbol_free_var_in_tableau y (subst_tableau t s0 s1) }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        subst_free_var_propagation_symbol_symbol_tableau x y v0
          ((rename_subst_symbol s0 identity))
          ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_symbol_symbol y (eval s0 x) identity ;
          assert { is_symbol_free_var_in_symbol y (s0 x) ->
            is_symbol_free_var_in_symbol y
              (eval ((rename_subst_symbol s0 identity)) x)
            } ;
          subst_free_var_propagation_symbol_symbol_fo_formula x y v1
            ((rename_subst_symbol s0 identity))
            ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_symbol_symbol y (eval s0 x) identity ;
          assert { is_symbol_free_var_in_symbol y (s0 x) ->
            is_symbol_free_var_in_symbol y
              (eval ((rename_subst_symbol s0 identity)) x)
            } ;
          subst_free_var_propagation_symbol_symbol_fo_formula_list x y v2
            ((rename_subst_symbol s0 identity))
            ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_symbol_symbol y (eval s0 x) identity ;
          assert { is_symbol_free_var_in_symbol y (s0 x) ->
            is_symbol_free_var_in_symbol y
              (eval ((rename_subst_symbol s0 identity)) x)
            } ;
          ()
    end

  with lemma subst_free_var_propagation_fo_term_symbol_tableau (x:'b1)
    (y:'c0) (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)):  unit
    ensures {
      is_fo_term_free_var_in_tableau x t /\
        is_symbol_free_var_in_fo_term y (s1 x) ->
      is_symbol_free_var_in_tableau y (subst_tableau t s0 s1) }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        subst_free_var_propagation_fo_term_symbol_tableau x y v0
          ((rename_subst_symbol s0 identity))
          ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_symbol_fo_term y (eval s1 x) identity
            identity ;
          assert { is_symbol_free_var_in_fo_term y (s1 x) ->
            is_symbol_free_var_in_fo_term y
              (eval ((rename_subst_fo_term s1 identity identity)) x)
            } ;
          subst_free_var_propagation_fo_term_symbol_fo_formula x y v1
            ((rename_subst_symbol s0 identity))
            ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_symbol_fo_term y (eval s1 x) identity
            identity ;
          assert { is_symbol_free_var_in_fo_term y (s1 x) ->
            is_symbol_free_var_in_fo_term y
              (eval ((rename_subst_fo_term s1 identity identity)) x)
            } ;
          subst_free_var_propagation_fo_term_symbol_fo_formula_list x y v2
            ((rename_subst_symbol s0 identity))
            ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_symbol_fo_term y (eval s1 x) identity
            identity ;
          assert { is_symbol_free_var_in_fo_term y (s1 x) ->
            is_symbol_free_var_in_fo_term y
              (eval ((rename_subst_fo_term s1 identity identity)) x)
            } ;
          ()
    end

  with lemma subst_free_var_propagation_fo_term_fo_term_tableau (x:'b1)
    (y:'c1) (t:tableau 'b0 'b1) (s0:'b0 -> (symbol 'c0))
    (s1:'b1 -> (fo_term 'c0 'c1)):  unit
    ensures {
      is_fo_term_free_var_in_tableau x t /\
        is_fo_term_free_var_in_fo_term y (s1 x) ->
      is_fo_term_free_var_in_tableau y (subst_tableau t s0 s1) }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        subst_free_var_propagation_fo_term_fo_term_tableau x y v0
          ((rename_subst_symbol s0 identity))
          ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_fo_term_fo_term y (eval s1 x) identity
            identity ;
          assert { is_fo_term_free_var_in_fo_term y (s1 x) ->
            is_fo_term_free_var_in_fo_term y
              (eval ((rename_subst_fo_term s1 identity identity)) x)
            } ;
          subst_free_var_propagation_fo_term_fo_term_fo_formula x y v1
            ((rename_subst_symbol s0 identity))
            ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_fo_term_fo_term y (eval s1 x) identity
            identity ;
          assert { is_fo_term_free_var_in_fo_term y (s1 x) ->
            is_fo_term_free_var_in_fo_term y
              (eval ((rename_subst_fo_term s1 identity identity)) x)
            } ;
          subst_free_var_propagation_fo_term_fo_term_fo_formula_list x y v2
            ((rename_subst_symbol s0 identity))
            ((rename_subst_fo_term s1 identity identity)) ;
          rename_free_var_propagation_fo_term_fo_term y (eval s1 x) identity
            identity ;
          assert { is_fo_term_free_var_in_fo_term y (s1 x) ->
            is_fo_term_free_var_in_fo_term y
              (eval ((rename_subst_fo_term s1 identity identity)) x)
            } ;
          ()
    end

  let rec lemma free_var_equivalence_of_subst_tableau (t:tableau 'b0 'b1)
    (s10:'b0 -> (symbol 'c0)) (s20:'b0 -> (symbol 'c0))
    (s11:'b1 -> (fo_term 'c0 'c1)) (s21:'b1 -> (fo_term 'c0 'c1)) : unit
    requires {
      forall x:'b0. is_symbol_free_var_in_tableau x t -> s10 x = s20 x }
    requires {
      forall x:'b1. is_fo_term_free_var_in_tableau x t -> s11 x = s21 x }
    ensures { subst_tableau t s10 s11 = subst_tableau t s20 s21 }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        assert {
          forall x:'b0. is_symbol_free_var_in_tableau x v0 ->
            is_symbol_free_var_in_tableau x t
          } ;
          assert {
            forall x:'b1. is_fo_term_free_var_in_tableau x v0 ->
              is_fo_term_free_var_in_tableau x t
            } ;
          free_var_equivalence_of_subst_tableau v0
            ((rename_subst_symbol s10 identity))
            ((rename_subst_symbol s20 identity))
            ((rename_subst_fo_term s11 identity identity))
            ((rename_subst_fo_term s21 identity identity)) ;
          assert {
            forall x:'b0. is_symbol_free_var_in_fo_formula x v1 ->
              is_symbol_free_var_in_tableau x t
            } ;
          assert {
            forall x:'b1. is_fo_term_free_var_in_fo_formula x v1 ->
              is_fo_term_free_var_in_tableau x t
            } ;
          free_var_equivalence_of_subst_fo_formula v1
            ((rename_subst_symbol s10 identity))
            ((rename_subst_symbol s20 identity))
            ((rename_subst_fo_term s11 identity identity))
            ((rename_subst_fo_term s21 identity identity)) ;
          assert {
            forall x:'b0. is_symbol_free_var_in_fo_formula_list x v2 ->
              is_symbol_free_var_in_tableau x t
            } ;
          assert {
            forall x:'b1. is_fo_term_free_var_in_fo_formula_list x v2 ->
              is_fo_term_free_var_in_tableau x t
            } ;
          free_var_equivalence_of_subst_fo_formula_list v2
            ((rename_subst_symbol s10 identity))
            ((rename_subst_symbol s20 identity))
            ((rename_subst_fo_term s11 identity identity))
            ((rename_subst_fo_term s21 identity identity)) ;
          ()
    end

  let lemma free_var_equivalence_of_rename_tableau (t:tableau 'b0 'b1)
    (s10:'b0 -> 'c0) (s20:'b0 -> 'c0) (s11:'b1 -> 'c1) (s21:'b1 -> 'c1) :
    unit
    requires {
      forall x:'b0. is_symbol_free_var_in_tableau x t -> s10 x = s20 x }
    requires {
      forall x:'b1. is_fo_term_free_var_in_tableau x t -> s11 x = s21 x }
    ensures { rename_tableau t s10 s11 = rename_tableau t s20 s21 } =
    free_var_equivalence_of_subst_tableau t (subst_of_rename_symbol s10)
      (subst_of_rename_symbol s20) (subst_of_rename_fo_term s11)
      (subst_of_rename_fo_term s21)

  let rec lemma free_var_derive_equivalence_of_subst_tableau
    (t:tableau 'b0 'b1) (s10:'b0 -> (symbol 'c0)) (s20:'b0 -> (symbol 'c0))
    (s11:'b1 -> (fo_term 'c0 'c1)) (s21:'b1 -> (fo_term 'c0 'c1)) : unit
    ensures {
      forall x:'b0. is_symbol_free_var_in_tableau x t -> s10 x = s20 x }
    ensures {
      forall x:'b1. is_fo_term_free_var_in_tableau x t -> s11 x = s21 x }
    requires { subst_tableau t s10 s11 = subst_tableau t s20 s21 }
    variant { size_tableau t } =
    match t with | Root -> ()
      | Node v0 v1 v2 ->
        free_var_derive_equivalence_of_subst_tableau v0
          ((rename_subst_symbol s10 identity))
          ((rename_subst_symbol s20 identity))
          ((rename_subst_fo_term s11 identity identity))
          ((rename_subst_fo_term s21 identity identity));
          assert {
            (forall x:'b0, y0:'c0. is_symbol_free_var_in_tableau x v0 ->
               rename_symbol (s10 x) identity =
               eval ((rename_subst_symbol s10 identity)) x =
               eval ((rename_subst_symbol s20 identity)) x =
               rename_symbol (s20 x) identity && s10 x =
               rename_symbol (rename_symbol (s10 x) identity) identity =
               rename_symbol (rename_symbol (s20 x) identity) identity =
               s20 x && s10 x = s20 x)
            &&
            forall x:'b0. is_symbol_free_var_in_tableau x v0 -> s10 x = s20 x
            };
          assert {
            (forall x:'b1, y0:'c0, y1:'c1.
               is_fo_term_free_var_in_tableau x v0 ->
               rename_fo_term (s11 x) identity identity =
               eval ((rename_subst_fo_term s11 identity identity)) x =
               eval ((rename_subst_fo_term s21 identity identity)) x =
               rename_fo_term (s21 x) identity identity && s11 x =
               rename_fo_term (rename_fo_term (s11 x) identity identity)
                 identity identity
               =
               rename_fo_term (rename_fo_term (s21 x) identity identity)
                 identity identity
               = s21 x && s11 x = s21 x)
            &&
            forall x:'b1. is_fo_term_free_var_in_tableau x v0 -> s11 x =
              s21 x
            } ;
          free_var_derive_equivalence_of_subst_fo_formula v1
            ((rename_subst_symbol s10 identity))
            ((rename_subst_symbol s20 identity))
            ((rename_subst_fo_term s11 identity identity))
            ((rename_subst_fo_term s21 identity identity));
          assert {
            (forall x:'b0, y0:'c0. is_symbol_free_var_in_fo_formula x v1 ->
               rename_symbol (s10 x) identity =
               eval ((rename_subst_symbol s10 identity)) x =
               eval ((rename_subst_symbol s20 identity)) x =
               rename_symbol (s20 x) identity && s10 x =
               rename_symbol (rename_symbol (s10 x) identity) identity =
               rename_symbol (rename_symbol (s20 x) identity) identity =
               s20 x && s10 x = s20 x)
            &&
            forall x:'b0. is_symbol_free_var_in_fo_formula x v1 -> s10 x =
              s20 x
            };
          assert {
            (forall x:'b1, y0:'c0, y1:'c1.
               is_fo_term_free_var_in_fo_formula x v1 ->
               rename_fo_term (s11 x) identity identity =
               eval ((rename_subst_fo_term s11 identity identity)) x =
               eval ((rename_subst_fo_term s21 identity identity)) x =
               rename_fo_term (s21 x) identity identity && s11 x =
               rename_fo_term (rename_fo_term (s11 x) identity identity)
                 identity identity
               =
               rename_fo_term (rename_fo_term (s21 x) identity identity)
                 identity identity
               = s21 x && s11 x = s21 x)
            &&
            forall x:'b1. is_fo_term_free_var_in_fo_formula x v1 -> s11 x =
              s21 x
            } ;
          free_var_derive_equivalence_of_subst_fo_formula_list v2
            ((rename_subst_symbol s10 identity))
            ((rename_subst_symbol s20 identity))
            ((rename_subst_fo_term s11 identity identity))
            ((rename_subst_fo_term s21 identity identity));
          assert {
            (forall x:'b0, y0:'c0. is_symbol_free_var_in_fo_formula_list x v2
               -> rename_symbol (s10 x) identity =
               eval ((rename_subst_symbol s10 identity)) x =
               eval ((rename_subst_symbol s20 identity)) x =
               rename_symbol (s20 x) identity && s10 x =
               rename_symbol (rename_symbol (s10 x) identity) identity =
               rename_symbol (rename_symbol (s20 x) identity) identity =
               s20 x && s10 x = s20 x)
            &&
            forall x:'b0. is_symbol_free_var_in_fo_formula_list x v2 -> s10 x
              = s20 x
            };
          assert {
            (forall x:'b1, y0:'c0, y1:'c1.
               is_fo_term_free_var_in_fo_formula_list x v2 ->
               rename_fo_term (s11 x) identity identity =
               eval ((rename_subst_fo_term s11 identity identity)) x =
               eval ((rename_subst_fo_term s21 identity identity)) x =
               rename_fo_term (s21 x) identity identity && s11 x =
               rename_fo_term (rename_fo_term (s11 x) identity identity)
                 identity identity
               =
               rename_fo_term (rename_fo_term (s21 x) identity identity)
                 identity identity
               = s21 x && s11 x = s21 x)
            &&
            forall x:'b1. is_fo_term_free_var_in_fo_formula_list x v2 ->
              s11 x = s21 x
            } ;
          ()
    end

  let lemma free_var_derive_equivalence_of_rename_tableau (t:tableau 'b0 'b1)
    (s10:'b0 -> 'c0) (s20:'b0 -> 'c0) (s11:'b1 -> 'c1) (s21:'b1 -> 'c1) :
    unit
    ensures {
      forall x:'b0. is_symbol_free_var_in_tableau x t -> s10 x = s20 x }
    ensures {
      forall x:'b1. is_fo_term_free_var_in_tableau x t -> s11 x = s21 x }
    requires { rename_tableau t s10 s11 = rename_tableau t s20 s21 } =
    free_var_derive_equivalence_of_subst_tableau t
      (subst_of_rename_symbol s10) (subst_of_rename_symbol s20)
      (subst_of_rename_fo_term s11) (subst_of_rename_fo_term s21);
    assert {
      forall x:'b0. (subst_of_rename_symbol s10 x:symbol 'c0) =
        (subst_of_rename_symbol s20 x:symbol 'c0) ->
        (Var_symbol (s10 x):symbol 'c0) = (Var_symbol (s20 x):symbol 'c0) &&
        s10 x = s20 x
      };
    assert {
      forall x:'b1. (subst_of_rename_fo_term s11 x:fo_term 'c0 'c1) =
        (subst_of_rename_fo_term s21 x:fo_term 'c0 'c1) ->
        (Var_fo_term (s11 x):fo_term 'c0 'c1) =
        (Var_fo_term (s21 x):fo_term 'c0 'c1) && s11 x = s21 x
      }

end