# Theory SyntaxN

chapter‹Syntax of Terms and Formulas using Nominal Logic›

theory SyntaxN
imports Nominal2.Nominal2 HereditarilyFinite.OrdArith
begin

section‹Terms and Formulas›

subsection‹Hf is a pure permutation type›

instantiation hf :: pt
begin
definition "p ∙ (s::hf) = s"
instance
end

instance hf :: pure
proof qed (rule permute_hf_def)

atom_decl name

declare fresh_set_empty [simp]

lemma supp_name [simp]: fixes i::name shows "supp i = {atom i}"
by (rule supp_at_base)

subsection‹The datatypes›

nominal_datatype tm = Zero | Var name | Eats tm tm

nominal_datatype fm =
Mem tm tm    (infixr "IN" 150)
| Eq tm tm     (infixr "EQ" 150)
| Disj fm fm   (infixr "OR" 130)
| Neg fm
| Ex x::name f::fm binds x in f

text ‹Mem, Eq are atomic formulas; Disj, Neg, Ex are non-atomic›

declare tm.supp [simp] fm.supp [simp]

subsection‹Substitution›

nominal_function subst :: "name ⇒ tm ⇒ tm ⇒ tm"
where
"subst i x Zero       = Zero"
| "subst i x (Var k)    = (if i=k then x else Var k)"
| "subst i x (Eats t u) = Eats (subst i x t) (subst i x u)"
by (auto simp: eqvt_def subst_graph_aux_def) (metis tm.strong_exhaust)

nominal_termination (eqvt)
by lexicographic_order

lemma fresh_subst_if [simp]:
"j ♯ subst i x t ⟷ (atom i ♯ t ∧ j ♯ t) ∨ (j ♯ x ∧ (j ♯ t ∨ j = atom i))"
by (induct t rule: tm.induct) (auto simp: fresh_at_base)

lemma forget_subst_tm [simp]: "atom a ♯ tm ⟹ subst a x tm = tm"
by (induct tm rule: tm.induct) (simp_all add: fresh_at_base)

lemma subst_tm_id [simp]: "subst a (Var a) tm = tm"
by (induct tm rule: tm.induct) simp_all

lemma subst_tm_commute [simp]:
"atom j ♯ tm ⟹ subst j u (subst i t tm) = subst i (subst j u t) tm"
by (induct tm rule: tm.induct) (auto simp: fresh_Pair)

lemma subst_tm_commute2 [simp]:
"atom j ♯ t ⟹ atom i ♯ u ⟹ i ≠ j ⟹ subst j u (subst i t tm) = subst i t (subst j u tm)"
by (induct tm rule: tm.induct) auto

lemma repeat_subst_tm [simp]: "subst i u (subst i t tm) = subst i (subst i u t) tm"
by (induct tm rule: tm.induct) auto

nominal_function  subst_fm :: "fm ⇒ name ⇒ tm ⇒ fm" ("_'(_::=_')" [1000, 0, 0] 200)
where
Mem:  "(Mem t u)(i::=x)  = Mem (subst i x t) (subst i x u)"
| Eq:   "(Eq t u)(i::=x)   = Eq  (subst i x t) (subst i x u)"
| Disj: "(Disj A B)(i::=x) = Disj (A(i::=x)) (B(i::=x))"
| Neg:  "(Neg A)(i::=x)    = Neg  (A(i::=x))"
| Ex:   "atom j ♯ (i, x) ⟹ (Ex j A)(i::=x) = Ex j (A(i::=x))"
apply auto [16]
apply (rule_tac y=a and c="(aa, b)" in fm.strong_exhaust)
apply (auto simp: eqvt_at_def fresh_star_def fresh_Pair fresh_at_base)
apply (metis flip_at_base_simps(3) flip_fresh_fresh)
done

nominal_termination (eqvt)
by lexicographic_order

lemma size_subst_fm [simp]: "size (A(i::=x)) = size A"
by (nominal_induct A avoiding: i x rule: fm.strong_induct) auto

lemma forget_subst_fm [simp]: "atom a ♯ A ⟹ A(a::=x) = A"
by (nominal_induct A avoiding: a x rule: fm.strong_induct) (auto simp: fresh_at_base)

lemma subst_fm_id [simp]: "A(a::=Var a) = A"
by (nominal_induct A avoiding: a rule: fm.strong_induct) (auto simp: fresh_at_base)

lemma fresh_subst_fm_if [simp]:
"j ♯ (A(i::=x)) ⟷ (atom i ♯ A ∧ j ♯ A) ∨ (j ♯ x ∧ (j ♯ A ∨ j = atom i))"
by (nominal_induct A avoiding: i x rule: fm.strong_induct) (auto simp: fresh_at_base)

lemma subst_fm_commute [simp]:
"atom j ♯ A ⟹ (A(i::=t))(j::=u) = A(i ::= subst j u t)"
by (nominal_induct A avoiding: i j t u rule: fm.strong_induct) (auto simp: fresh_at_base)

lemma repeat_subst_fm [simp]: "(A(i::=t))(i::=u) = A(i ::= subst i u t)"
by (nominal_induct A avoiding: i t u rule: fm.strong_induct) auto

lemma subst_fm_Ex_with_renaming:
"atom i' ♯ (A, i, j, t) ⟹ (Ex i A)(j ::= t) = Ex i' (((i ↔ i') ∙ A)(j ::= t))"
by (rule subst [of "Ex i' ((i ↔ i') ∙ A)" "Ex i A"])
(auto simp: Abs1_eq_iff flip_def swap_commute)

text ‹the simplifier cannot apply the rule above, because
it introduces a new variable at the right hand side.›

simproc_setup subst_fm_renaming ("(Ex i A)(j ::= t)") = ‹fn _ => fn ctxt => fn ctrm =>
let
val _ $(_$ i $A)$ j $t = Thm.term_of ctrm val atoms = Simplifier.prems_of ctxt |> map_filter (fn thm => case Thm.prop_of thm of _$ (Const (@{const_name fresh}, _) $atm$ _) => SOME (atm) | _ => NONE)
|> distinct ((=))

fun get_thm atm =
let
val goal = HOLogic.mk_Trueprop (mk_fresh atm (HOLogic.mk_tuple [A, i, j, t]))
in
SOME ((Goal.prove ctxt [] [] goal (K (asm_full_simp_tac ctxt 1)))
RS @{thm subst_fm_Ex_with_renaming} RS eq_reflection)
handle ERROR _ => NONE
end
in
get_first get_thm atoms
end
›

subsection‹Semantics›

definition e0 :: "(name, hf) finfun"    ― ‹the null environment›
where "e0 ≡ finfun_const 0"

nominal_function eval_tm :: "(name, hf) finfun ⇒ tm ⇒ hf"
where
"eval_tm e Zero = 0"
| "eval_tm e (Var k) = finfun_apply e k"
| "eval_tm e (Eats t u) = eval_tm e t ◃ eval_tm e u"
by (auto simp: eqvt_def eval_tm_graph_aux_def) (metis tm.strong_exhaust)

nominal_termination (eqvt)
by lexicographic_order

syntax
"_EvalTm" :: "tm ⇒ (name, hf) finfun ⇒ hf"      ("⟦_⟧_" [0,1000]1000)

translations
"⟦tm⟧e"    == "CONST eval_tm e tm"

nominal_function eval_fm :: "(name, hf) finfun ⇒ fm ⇒ bool"
where
"eval_fm e (t IN u) ⟷ ⟦t⟧e ❙∈ ⟦u⟧e"
| "eval_fm e (t EQ u) ⟷ ⟦t⟧e = ⟦u⟧e"
| "eval_fm e (A OR B) ⟷ eval_fm e A ∨ eval_fm e B"
| "eval_fm e (Neg A) ⟷ (~ eval_fm e A)"
| "atom k ♯ e ⟹ eval_fm e (Ex k A) ⟷ (∃x. eval_fm (finfun_update e k x) A)"
supply [[simproc del: defined_all]]
apply(auto del: iffI)[16]
apply(rule_tac y=b and c="(a)" in fm.strong_exhaust)
apply(auto simp: fresh_star_def)[5]
using [[simproc del: alpha_lst]] apply clarsimp
apply(erule_tac c="(ea)" in Abs_lst1_fcb2')
apply(rule pure_fresh)
done

nominal_termination (eqvt)
by lexicographic_order

lemma eval_tm_rename:
assumes "atom k' ♯ t"
shows "⟦t⟧(finfun_update e k x) = ⟦(k' ↔ k) ∙ t⟧(finfun_update e k' x)"
using assms
by (induct t rule: tm.induct) (auto simp: permute_flip_at)

lemma eval_fm_rename:
assumes "atom k' ♯ A"
shows "eval_fm (finfun_update e k x) A = eval_fm (finfun_update e k' x) ((k' ↔ k) ∙ A)"
using assms
apply (nominal_induct A avoiding: e k k' x rule: fm.strong_induct)
apply (simp add: fresh_finfun_update fresh_at_base finfun_update_twist)
done

lemma better_ex_eval_fm[simp]:
"eval_fm e (Ex k A) ⟷ (∃x. eval_fm (finfun_update e k x) A)"
proof -
obtain k'::name where k': "atom k' ♯ (k, e, A)"
by (rule obtain_fresh)
then have eq: "Ex k' ((k' ↔ k) ∙ A) = Ex k A"
have "eval_fm e (Ex k' ((k' ↔ k) ∙ A)) = (∃x. eval_fm (finfun_update e k' x) ((k' ↔ k) ∙ A))"
using k' by simp
also have "... = (∃x. eval_fm (finfun_update e k x) A)"
by (metis eval_fm_rename k' fresh_Pair)
finally show ?thesis
by (metis eq)
qed

lemma forget_eval_tm [simp]: "atom i ♯ t ⟹  ⟦t⟧(finfun_update e i x) = ⟦t⟧e"
by (induct t rule: tm.induct) (simp_all add: fresh_at_base)

lemma forget_eval_fm [simp]:
"atom k ♯ A ⟹ eval_fm (finfun_update e k x) A = eval_fm e A"
by (nominal_induct A avoiding: k e rule: fm.strong_induct)

lemma eval_subst_tm: "⟦subst i t u⟧e = ⟦u⟧(finfun_update e i ⟦t⟧e)"
by (induct u rule: tm.induct) (auto)

lemma eval_subst_fm: "eval_fm e (fm(i::= t)) = eval_fm (finfun_update e i ⟦t⟧e) fm"
by (nominal_induct fm avoiding: i t e rule: fm.strong_induct)

subsection‹Derived syntax›

subsubsection‹Ordered pairs›

definition HPair :: "tm ⇒ tm ⇒ tm"
where "HPair a b = Eats (Eats Zero (Eats (Eats Zero b) a)) (Eats (Eats Zero a) a)"

lemma HPair_eqvt [eqvt]: "(p ∙ HPair a b) = HPair (p ∙ a) (p ∙ b)"
by (auto simp: HPair_def)

lemma fresh_HPair [simp]: "x ♯ HPair a b ⟷ (x ♯ a ∧ x ♯ b)"
by (auto simp: HPair_def)

lemma HPair_injective_iff [iff]: "HPair a b = HPair a' b' ⟷ (a = a' ∧ b = b')"
by (auto simp: HPair_def)

lemma subst_tm_HPair [simp]: "subst i x (HPair a b) = HPair (subst i x a) (subst i x b)"
by (auto simp: HPair_def)

lemma eval_tm_HPair [simp]: "⟦HPair a b⟧e = hpair ⟦a⟧e ⟦b⟧e"
by (auto simp: HPair_def hpair_def)

subsubsection‹Ordinals›

definition
SUCC  :: "tm ⇒ tm"  where
"SUCC x ≡ Eats x x"

fun ORD_OF :: "nat ⇒ tm"
where
"ORD_OF 0 = Zero"
| "ORD_OF (Suc k) = SUCC (ORD_OF k)"

lemma eval_tm_SUCC [simp]: "⟦SUCC t⟧e = succ ⟦t⟧e"

lemma SUCC_fresh_iff [simp]: "a ♯ SUCC t ⟷ a ♯ t"

lemma SUCC_eqvt [eqvt]: "(p ∙ SUCC a) = SUCC (p ∙ a)"

lemma SUCC_subst [simp]: "subst i t (SUCC k) = SUCC (subst i t k)"

lemma eval_tm_ORD_OF [simp]: "⟦ORD_OF n⟧e = ord_of n"
by (induct n) auto

lemma ORD_OF_fresh [simp]: "a ♯ ORD_OF n"
by (induct n) (auto simp: SUCC_def)

lemma ORD_OF_eqvt [eqvt]: "(p ∙ ORD_OF n) = ORD_OF (p ∙ n)"
by (induct n) (auto simp: permute_pure SUCC_eqvt)

subsection‹Derived logical connectives›

abbreviation Imp :: "fm ⇒ fm ⇒ fm"   (infixr "IMP" 125)
where "Imp A B ≡ Disj (Neg A) B"

abbreviation All :: "name ⇒ fm ⇒ fm"
where "All i A ≡ Neg (Ex i (Neg A))"

abbreviation All2 :: "name ⇒ tm ⇒ fm ⇒ fm" ― ‹bounded universal quantifier, for Sigma formulas›
where "All2 i t A ≡ All i ((Var i IN t) IMP A)"

subsubsection‹Conjunction›

definition Conj :: "fm ⇒ fm ⇒ fm"   (infixr "AND" 135)
where "Conj A B ≡ Neg (Disj (Neg A) (Neg B))"

lemma Conj_eqvt [eqvt]: "p ∙ (A AND B) = (p ∙ A) AND (p ∙ B)"

lemma fresh_Conj [simp]: "a ♯ A AND B ⟷ (a ♯ A ∧ a ♯ B)"
by (auto simp: Conj_def)

lemma supp_Conj [simp]: "supp (A AND B) = supp A ∪ supp B"
by (auto simp: Conj_def)

lemma size_Conj [simp]: "size (A AND B) = size A + size B + 4"

lemma Conj_injective_iff [iff]: "(A AND B) = (A' AND B') ⟷ (A = A' ∧ B = B')"
by (auto simp: Conj_def)

lemma subst_fm_Conj [simp]: "(A AND B)(i::=x) = (A(i::=x)) AND (B(i::=x))"
by (auto simp: Conj_def)

lemma eval_fm_Conj [simp]: "eval_fm e (Conj A B) ⟷ (eval_fm e A ∧ eval_fm e B)"
by (auto simp: Conj_def)

subsubsection‹If and only if›

definition Iff :: "fm ⇒ fm ⇒ fm"   (infixr "IFF" 125)
where "Iff A B = Conj (Imp A B) (Imp B A)"

lemma Iff_eqvt [eqvt]: "p ∙ (A IFF B) = (p ∙ A) IFF (p ∙ B)"

lemma fresh_Iff [simp]: "a ♯ A IFF B ⟷ (a ♯ A ∧ a ♯ B)"
by (auto simp: Conj_def Iff_def)

lemma size_Iff [simp]: "size (A IFF B) = 2*(size A + size B) + 8"

lemma Iff_injective_iff [iff]: "(A IFF B) = (A' IFF B') ⟷ (A = A' ∧ B = B')"
by (auto simp: Iff_def)

lemma subst_fm_Iff [simp]: "(A IFF B)(i::=x) = (A(i::=x)) IFF (B(i::=x))"
by (auto simp: Iff_def)

lemma eval_fm_Iff [simp]: "eval_fm e (Iff A B) ⟷ (eval_fm e A  ⟷  eval_fm e B)"
by (auto simp: Iff_def)

section‹Axioms and Theorems›

subsection‹Logical axioms›

inductive_set boolean_axioms :: "fm set"
where
Ident:     "A IMP A ∈ boolean_axioms"
| DisjI1:    "A IMP (A OR B) ∈ boolean_axioms"
| DisjCont:  "(A OR A) IMP A ∈ boolean_axioms"
| DisjAssoc: "(A OR (B OR C)) IMP ((A OR B) OR C) ∈ boolean_axioms"
| DisjConj:  "(C OR A) IMP (((Neg C) OR B) IMP (A OR B)) ∈ boolean_axioms"

lemma boolean_axioms_hold: "A ∈ boolean_axioms ⟹ eval_fm e A"
by (induct rule: boolean_axioms.induct, auto)

inductive_set special_axioms :: "fm set" where
I: "A(i::=x) IMP (Ex i A) ∈ special_axioms"

lemma special_axioms_hold: "A ∈ special_axioms ⟹ eval_fm e A"
by (induct rule: special_axioms.induct, auto) (metis eval_subst_fm)

inductive_set induction_axioms :: "fm set" where
ind:
"atom (j::name) ♯ (i,A)
⟹ A(i::=Zero) IMP ((All i (All j (A IMP (A(i::= Var j) IMP A(i::= Eats(Var i)(Var j))))))
IMP (All i A))
∈ induction_axioms"

lemma twist_forget_eval_fm [simp]:
"atom j ♯ (i, A)
⟹ eval_fm (finfun_update (finfun_update (finfun_update e i x) j y) i z) A =
eval_fm (finfun_update e i z) A"
by (metis finfun_update_twice finfun_update_twist forget_eval_fm fresh_Pair)

lemma induction_axioms_hold: "A ∈ induction_axioms ⟹ eval_fm e A"
by (induction rule: induction_axioms.induct) (auto simp: eval_subst_fm intro: hf_induct_ax)

subsection ‹Concrete variables›

declare Abs_name_inject[simp]

abbreviation
"X0 ≡ Abs_name (Atom (Sort ''SyntaxN.name'' []) 0)"

abbreviation
"X1 ≡ Abs_name (Atom (Sort ''SyntaxN.name'' []) (Suc 0))"
― ‹We prefer @{term "Suc 0"} because simplification will transform 1 to that form anyway.›

abbreviation
"X2 ≡ Abs_name (Atom (Sort ''SyntaxN.name'' []) 2)"

abbreviation
"X3 ≡ Abs_name (Atom (Sort ''SyntaxN.name'' []) 3)"

abbreviation
"X4 ≡ Abs_name (Atom (Sort ''SyntaxN.name'' []) 4)"

subsection‹The HF axioms›

definition HF1 :: fm where  ― ‹the axiom @{term"z=0 ⟷ (∀x. ¬ x ❙∈ z)"}›
"HF1 = (Var X0 EQ Zero) IFF (All X1 (Neg (Var X1 IN Var X0)))"

lemma HF1_holds: "eval_fm e HF1"
by (auto simp: HF1_def)

definition HF2 :: fm where  ― ‹the axiom @{term"z = x ◃ y ⟷ (∀u. u ❙∈ z ⟷ u ❙∈ x | u=y)"}›
"HF2 ≡ Var X0 EQ Eats (Var X1) (Var X2) IFF
All X3 (Var X3 IN Var X0 IFF Var X3 IN Var X1 OR Var X3 EQ Var X2)"

lemma HF2_holds: "eval_fm e HF2"
by (auto simp: HF2_def)

definition HF_axioms where "HF_axioms = {HF1, HF2}"

lemma HF_axioms_hold: "A ∈ HF_axioms ⟹ eval_fm e A"
by (auto simp: HF_axioms_def HF1_holds HF2_holds)

subsection‹Equality axioms›

definition refl_ax :: fm where
"refl_ax = Var X1 EQ Var X1"

lemma refl_ax_holds: "eval_fm e refl_ax"
by (auto simp: refl_ax_def)

definition eq_cong_ax :: fm where
"eq_cong_ax = ((Var X1 EQ Var X2) AND (Var X3 EQ Var X4)) IMP
((Var X1 EQ Var X3) IMP (Var X2 EQ Var X4))"

lemma eq_cong_ax_holds: "eval_fm e eq_cong_ax"
by (auto simp: Conj_def eq_cong_ax_def)

definition mem_cong_ax :: fm where
"mem_cong_ax = ((Var X1 EQ Var X2) AND (Var X3 EQ Var X4)) IMP
((Var X1 IN Var X3) IMP (Var X2 IN Var X4))"

lemma mem_cong_ax_holds: "eval_fm e mem_cong_ax"
by (auto simp: Conj_def mem_cong_ax_def)

definition eats_cong_ax :: fm where
"eats_cong_ax = ((Var X1 EQ Var X2) AND (Var X3 EQ Var X4)) IMP
((Eats (Var X1) (Var X3)) EQ (Eats (Var X2) (Var X4)))"

lemma eats_cong_ax_holds: "eval_fm e eats_cong_ax"
by (auto simp: Conj_def eats_cong_ax_def)

definition equality_axioms :: "fm set" where
"equality_axioms = {refl_ax, eq_cong_ax, mem_cong_ax, eats_cong_ax}"

lemma equality_axioms_hold: "A ∈ equality_axioms ⟹ eval_fm e A"
by (auto simp: equality_axioms_def refl_ax_holds eq_cong_ax_holds mem_cong_ax_holds eats_cong_ax_holds)

subsection‹The proof system›

text‹This arbitrary additional axiom generalises the statements of the incompleteness theorems
and other results to any formal system stronger than the HF theory. The additional axiom
could be the conjunction of any finite number of assertions. Any more general extension
must be a form that can be formalised for the proof predicate.›
consts  extra_axiom :: fm

specification (extra_axiom)
extra_axiom_holds:  "eval_fm e extra_axiom"
by (rule exI [where x = "Zero IN Eats Zero Zero"], auto)

inductive hfthm :: "fm set ⇒ fm ⇒ bool" (infixl "⊢" 55)
where
Hyp:    "A ∈ H ⟹ H ⊢ A"
| Extra:  "H ⊢ extra_axiom"
| Bool:   "A ∈ boolean_axioms ⟹ H ⊢ A"
| Eq:     "A ∈ equality_axioms ⟹ H ⊢ A"
| Spec:   "A ∈ special_axioms ⟹ H ⊢ A"
| HF:     "A ∈ HF_axioms ⟹ H ⊢ A"
| Ind:    "A ∈ induction_axioms ⟹ H ⊢ A"
| MP:     "H ⊢ A IMP B ⟹ H' ⊢ A ⟹ H ∪ H' ⊢ B"
| Exists: "H ⊢ A IMP B ⟹ atom i ♯ B ⟹ ∀C ∈ H. atom i ♯ C ⟹ H ⊢ (Ex i A) IMP B"

text‹Soundness theorem!›
theorem hfthm_sound: assumes "H ⊢ A" shows "(∀B∈H. eval_fm e B) ⟹ eval_fm e A"
using assms
proof (induct arbitrary: e)
case (Hyp A H) thus ?case
by auto
next
case (Extra H) thus ?case
by (metis extra_axiom_holds)
next
case (Bool A H) thus ?case
by (metis boolean_axioms_hold)
next
case (Eq A H) thus ?case
by (metis equality_axioms_hold)
next
case (Spec A H) thus ?case
by (metis special_axioms_hold)
next
case (HF A H) thus ?case
by (metis HF_axioms_hold)
next
case (Ind A H) thus ?case
by (metis induction_axioms_hold)
next
case (MP H A B H') thus ?case
by auto
next
case (Exists H A B i e) thus ?case
by auto (metis forget_eval_fm)
qed

subsection‹Derived rules of inference›

lemma contraction: "insert A (insert A H) ⊢ B ⟹ insert A H ⊢ B"
by (metis insert_absorb2)

lemma thin_Un: "H ⊢ A ⟹ H ∪ H' ⊢ A"
by (metis Bool MP boolean_axioms.Ident sup_commute)

lemma thin: "H ⊢ A ⟹ H ⊆ H' ⟹ H' ⊢ A"
by (metis Un_absorb1 thin_Un)

lemma thin0: "{} ⊢ A ⟹ H ⊢ A"
by (metis sup_bot_left thin_Un)

lemma thin1: "H ⊢ B ⟹ insert A H ⊢ B"
by (metis subset_insertI thin)

lemma thin2: "insert A1 H ⊢ B ⟹ insert A1 (insert A2 H) ⊢ B"
by (blast intro: thin)

lemma thin3: "insert A1 (insert A2 H) ⊢ B ⟹ insert A1 (insert A2 (insert A3 H)) ⊢ B"
by (blast intro: thin)

lemma thin4:
"insert A1 (insert A2 (insert A3 H)) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 H))) ⊢ B"
by (blast intro: thin)

lemma rotate2: "insert A2 (insert A1 H) ⊢ B ⟹ insert A1 (insert A2 H) ⊢ B"
by (blast intro: thin)

lemma rotate3: "insert A3 (insert A1 (insert A2 H)) ⊢ B ⟹ insert A1 (insert A2 (insert A3 H)) ⊢ B"
by (blast intro: thin)

lemma rotate4:
"insert A4 (insert A1 (insert A2 (insert A3 H))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 H))) ⊢ B"
by (blast intro: thin)

lemma rotate5:
"insert A5 (insert A1 (insert A2 (insert A3 (insert A4 H)))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 H)))) ⊢ B"
by (blast intro: thin)

lemma rotate6:
"insert A6 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 H))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H))))) ⊢ B"
by (blast intro: thin)

lemma rotate7:
"insert A7 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H)))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H)))))) ⊢ B"
by (blast intro: thin)

lemma rotate8:
"insert A8 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H))))))) ⊢ B"
by (blast intro: thin)

lemma rotate9:
"insert A9 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H)))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H)))))))) ⊢ B"
by (blast intro: thin)

lemma rotate10:
"insert A10 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H))))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H))))))))) ⊢ B"
by (blast intro: thin)

lemma rotate11:
"insert A11 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H)))))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H)))))))))) ⊢ B"
by (blast intro: thin)

lemma rotate12:
"insert A12 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H))))))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H))))))))))) ⊢ B"
by (blast intro: thin)

lemma rotate13:
"insert A13 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H)))))))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H)))))))))))) ⊢ B"
by (blast intro: thin)

lemma rotate14:
"insert A14 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H))))))))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 H))))))))))))) ⊢ B"
by (blast intro: thin)

lemma rotate15:
"insert A15 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 H)))))))))))))) ⊢ B
⟹ insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 (insert A15 H)))))))))))))) ⊢ B"
by (blast intro: thin)

lemma MP_same: "H ⊢ A IMP B ⟹ H ⊢ A ⟹ H ⊢ B"
by (metis MP Un_absorb)

lemma MP_thin: "HA ⊢ A IMP B ⟹ HB ⊢ A ⟹ HA ∪ HB ⊆ H ⟹ H ⊢ B"
by (metis MP_same le_sup_iff thin)

lemma MP_null: "{} ⊢ A IMP B ⟹ H ⊢ A ⟹ H ⊢ B"
by (metis MP_same thin0)

lemma Disj_commute: "H ⊢ B OR A ⟹ H ⊢ A OR B"
using DisjConj [of B A B] Ident [of B]
by (metis Bool MP_same)

lemma S: assumes  "H ⊢ A IMP (B IMP C)" "H' ⊢ A IMP B" shows "H ∪ H' ⊢ A IMP C"
proof -
have "H' ∪ H ⊢ (Neg A) OR (C OR (Neg A))"
by (metis Bool MP MP_same boolean_axioms.DisjConj Disj_commute DisjAssoc assms)
thus ?thesis
by (metis Bool Disj_commute Un_commute MP_same DisjAssoc DisjCont DisjI1)
qed

lemma Assume: "insert A H ⊢ A"
by (metis Hyp insertI1)

lemmas AssumeH = Assume Assume [THEN rotate2] Assume [THEN rotate3] Assume [THEN rotate4] Assume [THEN rotate5]
Assume [THEN rotate6] Assume [THEN rotate7] Assume [THEN rotate8] Assume [THEN rotate9] Assume [THEN rotate10]
Assume [THEN rotate11] Assume [THEN rotate12]
declare AssumeH [intro!]

lemma Imp_triv_I: "H ⊢ B ⟹ H ⊢ A IMP B"
by (metis Bool Disj_commute MP_same boolean_axioms.DisjI1)

lemma DisjAssoc1: "H ⊢ A OR (B OR C) ⟹ H ⊢ (A OR B) OR C"
by (metis Bool MP_same boolean_axioms.DisjAssoc)

lemma DisjAssoc2: "H ⊢ (A OR B) OR C ⟹ H ⊢ A OR (B OR C)"
by (metis DisjAssoc1 Disj_commute)

lemma Disj_commute_Imp: "H ⊢ (B OR A) IMP (A OR B)"
using DisjConj [of B A B] Ident [of B]
by (metis Bool DisjAssoc2 Disj_commute MP_same)

lemma Disj_Semicong_1: "H ⊢ A OR C ⟹ H ⊢ A IMP B ⟹ H ⊢ B OR C"
using DisjConj [of A C B]
by (metis Bool Disj_commute MP_same)

lemma Imp_Imp_commute: "H ⊢ B IMP (A IMP C) ⟹ H ⊢ A IMP (B IMP C)"
by (metis DisjAssoc1 DisjAssoc2 Disj_Semicong_1 Disj_commute_Imp)

subsection‹The Deduction Theorem›

lemma deduction_Diff: assumes "H ⊢ B" shows "H - {C} ⊢ C IMP B"
using assms
proof (induct)
case (Hyp A H) thus ?case
by (metis Bool Imp_triv_I boolean_axioms.Ident hfthm.Hyp member_remove remove_def)
next
case (Extra H) thus ?case
by (metis Imp_triv_I hfthm.Extra)
next
case (Bool A H) thus ?case
by (metis Imp_triv_I hfthm.Bool)
next
case (Eq A H) thus ?case
by (metis Imp_triv_I hfthm.Eq)
next
case (Spec A H) thus ?case
by (metis Imp_triv_I hfthm.Spec)
next
case (HF A H) thus ?case
by (metis Imp_triv_I hfthm.HF)
next
case (Ind A H) thus ?case
by (metis Imp_triv_I hfthm.Ind)
next
case (MP H A B H')
hence "(H-{C}) ∪ (H'-{C}) ⊢ Imp C B"
thus ?case
by (metis Un_Diff)
next
case (Exists H A B i) show ?case
proof (cases "C ∈ H")
case True
hence "atom i ♯ C" using Exists by auto
moreover have "H - {C} ⊢ A IMP C IMP B" using Exists
by (metis Imp_Imp_commute)
ultimately have "H - {C} ⊢ (Ex i A) IMP C IMP B" using Exists
by (metis fm.fresh(3) fm.fresh(4) hfthm.Exists member_remove remove_def)
thus ?thesis
by (metis Imp_Imp_commute)
next
case False
hence "H - {C} = H" by auto
thus ?thesis using Exists
by (metis Imp_triv_I hfthm.Exists)
qed
qed

theorem Imp_I [intro!]: "insert A H ⊢ B ⟹ H ⊢ A IMP B"
by (metis Diff_insert_absorb Imp_triv_I deduction_Diff insert_absorb)

lemma anti_deduction: "H ⊢ A IMP B ⟹ insert A H ⊢ B"
by (metis Assume MP_same thin1)

subsection‹Cut rules›

lemma cut:  "H ⊢ A ⟹ insert A H' ⊢ B ⟹ H ∪ H' ⊢ B"
by (metis MP Un_commute Imp_I)

lemma cut_same: "H ⊢ A ⟹ insert A H ⊢ B ⟹ H ⊢ B"
by (metis Un_absorb cut)

lemma cut_thin: "HA ⊢ A ⟹ insert A HB ⊢ B ⟹ HA ∪ HB ⊆ H ⟹ H ⊢ B"
by (metis thin cut)

lemma cut0: "{} ⊢ A ⟹ insert A H ⊢ B ⟹ H ⊢ B"
by (metis cut_same thin0)

lemma cut1: "{A} ⊢ B ⟹ H ⊢ A ⟹ H ⊢ B"
by (metis cut sup_bot_right)

lemma rcut1: "{A} ⊢ B ⟹ insert B H ⊢ C ⟹ insert A H ⊢ C"
by (metis Assume cut1 cut_same rotate2 thin1)

lemma cut2: "⟦{A,B} ⊢ C; H ⊢ A; H ⊢ B⟧ ⟹ H ⊢ C"
by (metis Un_empty_right Un_insert_right cut cut_same)

lemma rcut2: "{A,B} ⊢ C ⟹ insert C H ⊢ D ⟹ H ⊢ B ⟹ insert A H ⊢ D"
by (metis Assume cut2 cut_same insert_commute thin1)

lemma cut3: "⟦{A,B,C} ⊢ D; H ⊢ A; H ⊢ B; H ⊢ C⟧ ⟹ H ⊢ D"
by (metis MP_same cut2 Imp_I)

lemma cut4: "⟦{A,B,C,D} ⊢ E; H ⊢ A; H ⊢ B; H ⊢ C; H ⊢ D⟧ ⟹ H ⊢ E"
by (metis MP_same cut3 [of B C D] Imp_I)

section‹Miscellaneous logical rules›

lemma Disj_I1: "H ⊢ A ⟹ H ⊢ A OR B"
by (metis Bool MP_same boolean_axioms.DisjI1)

lemma Disj_I2: "H ⊢ B ⟹ H ⊢ A OR B"
by (metis Disj_commute Disj_I1)

lemma Peirce: "H ⊢ (Neg A) IMP A ⟹ H ⊢ A"
using DisjConj [of "Neg A" A A]  DisjCont [of A]
by (metis Bool MP_same boolean_axioms.Ident)

lemma Contra: "insert (Neg A) H ⊢ A ⟹ H ⊢ A"
by (metis Peirce Imp_I)

lemma Imp_Neg_I: "H ⊢ A IMP B ⟹ H ⊢ A IMP (Neg B) ⟹ H ⊢ Neg A"
by (metis DisjConj [of B "Neg A" "Neg A"] DisjCont Bool Disj_commute MP_same)

lemma NegNeg_I: "H ⊢ A ⟹ H ⊢ Neg (Neg A)"
using DisjConj [of "Neg (Neg A)" "Neg A" "Neg (Neg A)"]
by (metis Bool Ident MP_same)

lemma NegNeg_D: "H ⊢ Neg (Neg A) ⟹ H ⊢ A"
by (metis Disj_I1 Peirce)

lemma Neg_D: "H ⊢ Neg A ⟹ H ⊢ A ⟹ H ⊢ B"
by (metis Imp_Neg_I Imp_triv_I NegNeg_D)

lemma Disj_Neg_1: "H ⊢ A OR B ⟹ H ⊢ Neg B ⟹ H ⊢ A"
by (metis Disj_I1 Disj_Semicong_1 Disj_commute Peirce)

lemma Disj_Neg_2: "H ⊢ A OR B ⟹ H ⊢ Neg A ⟹ H ⊢ B"
by (metis Disj_Neg_1 Disj_commute)

lemma Neg_Disj_I: "H ⊢ Neg A ⟹ H ⊢ Neg B ⟹ H ⊢ Neg (A OR B)"
by (metis Bool Disj_Neg_1 MP_same boolean_axioms.Ident DisjAssoc)

lemma Conj_I [intro!]: "H ⊢ A ⟹ H ⊢ B ⟹ H ⊢ A AND B"
by (metis Conj_def NegNeg_I Neg_Disj_I)

lemma Conj_E1: "H ⊢ A AND B ⟹ H ⊢ A"
by (metis Conj_def Bool Disj_Neg_1 NegNeg_D boolean_axioms.DisjI1)

lemma Conj_E2: "H ⊢ A AND B ⟹ H ⊢ B"
by (metis Conj_def Bool Disj_I2 Disj_Neg_2 MP_same DisjAssoc Ident)

lemma Conj_commute: "H ⊢ B AND A ⟹ H ⊢ A AND B"
by (metis Conj_E1 Conj_E2 Conj_I)

lemma Conj_E: assumes "insert A (insert B H) ⊢ C" shows "insert (A AND B) H ⊢ C"
apply (rule cut_same [where A=A], metis Conj_E1 Hyp insertI1)
by (metis (full_types) AssumeH(2) Conj_E2 assms cut_same [where A=B] insert_commute thin2)

lemmas Conj_EH = Conj_E Conj_E [THEN rotate2] Conj_E [THEN rotate3] Conj_E [THEN rotate4] Conj_E [THEN rotate5]
Conj_E [THEN rotate6] Conj_E [THEN rotate7] Conj_E [THEN rotate8] Conj_E [THEN rotate9] Conj_E [THEN rotate10]
declare Conj_EH [intro!]

lemma Neg_I0: assumes "(⋀B. atom i ♯ B ⟹ insert A H ⊢ B)" shows "H ⊢ Neg A"
by (rule Imp_Neg_I [where B = "Zero IN Zero"]) (auto simp: assms)

lemma Neg_mono: "insert A H ⊢ B ⟹ insert (Neg B) H ⊢ Neg A"
by (rule Neg_I0) (metis Hyp Neg_D insert_commute insertI1 thin1)

lemma Conj_mono: "insert A H ⊢ B ⟹ insert C H ⊢ D ⟹ insert (A AND C) H ⊢ B AND D"
by (metis Conj_E1 Conj_E2 Conj_I Hyp Un_absorb2 cut insertI1 subset_insertI)

lemma Disj_mono:
assumes "insert A H ⊢ B" "insert C H ⊢ D" shows "insert (A OR C) H ⊢ B OR D"
proof -
{ fix A B C H
have "insert (A OR C) H ⊢ (A IMP B) IMP C OR B"
by (metis Bool Hyp MP_same boolean_axioms.DisjConj insertI1)
hence "insert A H ⊢ B ⟹ insert (A OR C) H ⊢ C OR B"
by (metis MP_same Un_absorb Un_insert_right Imp_I thin_Un)
}
thus ?thesis
by (metis cut_same assms thin2)
qed

lemma Disj_E:
assumes A: "insert A H ⊢ C" and B: "insert B H ⊢ C" shows "insert (A OR B) H ⊢ C"
by (metis A B Disj_mono NegNeg_I Peirce)

lemmas Disj_EH = Disj_E Disj_E [THEN rotate2] Disj_E [THEN rotate3] Disj_E [THEN rotate4] Disj_E [THEN rotate5]
Disj_E [THEN rotate6] Disj_E [THEN rotate7] Disj_E [THEN rotate8] Disj_E [THEN rotate9] Disj_E [THEN rotate10]
declare Disj_EH [intro!]

lemma Contra': "insert A H ⊢ Neg A ⟹ H ⊢ Neg A"
by (metis Contra Neg_mono)

lemma NegNeg_E [intro!]: "insert A H ⊢ B ⟹ insert (Neg (Neg A)) H ⊢ B"
by (metis NegNeg_D Neg_mono)

declare NegNeg_E [THEN rotate2, intro!]
declare NegNeg_E [THEN rotate3, intro!]
declare NegNeg_E [THEN rotate4, intro!]
declare NegNeg_E [THEN rotate5, intro!]
declare NegNeg_E [THEN rotate6, intro!]
declare NegNeg_E [THEN rotate7, intro!]
declare NegNeg_E [THEN rotate8, intro!]

lemma Imp_E:
assumes A: "H ⊢ A" and B: "insert B H ⊢ C" shows "insert (A IMP B) H ⊢ C"
proof -
have "insert (A IMP B) H ⊢ B"
by (metis Hyp A thin1 MP_same insertI1)
thus ?thesis
by (metis cut [where B=C] Un_insert_right sup_commute sup_idem B)
qed

lemma Imp_cut:
assumes "insert C H ⊢ A IMP B" "{A} ⊢ C"
shows "H ⊢ A IMP B"
by (metis Contra Disj_I1 Neg_mono assms rcut1)

lemma Iff_I [intro!]: "insert A H ⊢ B ⟹ insert B H ⊢ A ⟹ H ⊢ A IFF B"
by (metis Iff_def Conj_I Imp_I)

lemma Iff_MP_same: "H ⊢ A IFF B ⟹ H ⊢ A ⟹ H ⊢ B"
by (metis Iff_def Conj_E1 MP_same)

lemma Iff_MP2_same: "H ⊢ A IFF B ⟹ H ⊢ B ⟹ H ⊢ A"
by (metis Iff_def Conj_E2 MP_same)

lemma Iff_refl [intro!]: "H ⊢ A IFF A"
by (metis Hyp Iff_I insertI1)

lemma Iff_sym: "H ⊢ A IFF B ⟹ H ⊢ B IFF A"
by (metis Iff_def Conj_commute)

lemma Iff_trans: "H ⊢ A IFF B ⟹ H ⊢ B IFF C ⟹ H ⊢ A IFF C"
unfolding Iff_def
by (metis Conj_E1 Conj_E2 Conj_I Disj_Semicong_1 Disj_commute)

lemma Iff_E:
"insert A (insert B H) ⊢ C ⟹ insert (Neg A) (insert (Neg B) H) ⊢ C ⟹ insert (A IFF B) H ⊢ C"
apply (auto simp: Iff_def insert_commute)
apply (metis Disj_I1 Hyp anti_deduction insertCI)
apply (metis Assume Disj_I1 anti_deduction)
done

lemma Iff_E1:
assumes A: "H ⊢ A" and B: "insert B H ⊢ C" shows "insert (A IFF B) H ⊢ C"
by (metis Iff_def A B Conj_E Imp_E insert_commute thin1)

lemma Iff_E2:
assumes A: "H ⊢ A" and B: "insert B H ⊢ C" shows "insert (B IFF A) H ⊢ C"
by (metis Iff_def A B Bool Conj_E2 Conj_mono Imp_E boolean_axioms.Ident)

lemma Iff_MP_left: "H ⊢ A IFF B ⟹ insert A H ⊢ C ⟹ insert B H ⊢ C"
by (metis Hyp Iff_E2 cut_same insertI1 insert_commute thin1)

lemma Iff_MP_left': "H ⊢ A IFF B ⟹ insert B H ⊢ C ⟹ insert A H ⊢ C"
by (metis Iff_MP_left Iff_sym)

lemma Swap: "insert (Neg B) H ⊢ A ⟹ insert (Neg A) H ⊢ B"
by (metis NegNeg_D Neg_mono)

lemma Cases: "insert A H ⊢ B ⟹ insert (Neg A) H ⊢ B ⟹ H ⊢ B"
by (metis Contra Neg_D Neg_mono)

lemma Neg_Conj_E: "H ⊢ B ⟹ insert (Neg A) H ⊢ C ⟹ insert (Neg (A AND B)) H ⊢ C"
by (metis Conj_I Swap thin1)

lemma Disj_CI: "insert (Neg B) H ⊢ A ⟹ H ⊢ A OR B"
by (metis Contra Disj_I1 Disj_I2 Swap)

lemma Disj_3I: "insert (Neg A) (insert (Neg C) H) ⊢ B ⟹ H ⊢ A OR B OR C"
by (metis Disj_CI Disj_commute insert_commute)

lemma Contrapos1: "H ⊢ A IMP B ⟹ H ⊢ Neg B IMP Neg A"
by (metis Bool MP_same boolean_axioms.DisjConj boolean_axioms.Ident)

lemma Contrapos2: "H ⊢ (Neg B) IMP (Neg A) ⟹ H ⊢ A IMP B"
by (metis Bool MP_same boolean_axioms.DisjConj boolean_axioms.Ident)

lemma ContraAssumeN [intro]: "B ∈ H ⟹ insert (Neg B) H ⊢ A"
by (metis Hyp Swap thin1)

lemma ContraAssume: "Neg B ∈ H ⟹ insert B H ⊢ A"
by (metis Disj_I1 Hyp anti_deduction)

lemma ContraProve: "H ⊢ B ⟹ insert (Neg B) H ⊢ A"
by (metis Swap thin1)

lemma Disj_IE1: "insert B H ⊢ C ⟹ insert (A OR B) H ⊢ A OR C"
by (metis Assume Disj_mono)

lemmas Disj_IE1H = Disj_IE1 Disj_IE1 [THEN rotate2] Disj_IE1 [THEN rotate3] Disj_IE1 [THEN rotate4] Disj_IE1 [THEN rotate5]
Disj_IE1 [THEN rotate6] Disj_IE1 [THEN rotate7] Disj_IE1 [THEN rotate8]
declare Disj_IE1H [intro!]

subsection‹Quantifier reasoning›

lemma Ex_I: "H ⊢ A(i::=x) ⟹ H ⊢ Ex i A"
by (metis MP_same Spec special_axioms.intros)

lemma Ex_E:
assumes "insert A H ⊢ B" "atom i ♯ B" "∀C ∈ H. atom i ♯ C"
shows "insert (Ex i A) H ⊢ B"
by (metis Exists Imp_I anti_deduction assms)

lemma Ex_E_with_renaming:
assumes "insert ((i ↔ i') ∙ A) H ⊢ B" "atom i' ♯ (A,i,B)" "∀C ∈ H. atom i' ♯ C"
shows "insert (Ex i A) H ⊢ B"
proof -
have "Ex i A = Ex i' ((i ↔ i') ∙ A)" using assms
apply (auto simp: Abs1_eq_iff fresh_Pair)
apply (metis flip_at_simps(2) fresh_at_base_permute_iff)+
done
thus ?thesis
by (metis Ex_E assms fresh_Pair)
qed

lemmas Ex_EH = Ex_E Ex_E [THEN rotate2] Ex_E [THEN rotate3] Ex_E [THEN rotate4] Ex_E [THEN rotate5]
Ex_E [THEN rotate6] Ex_E [THEN rotate7] Ex_E [THEN rotate8] Ex_E [THEN rotate9] Ex_E [THEN rotate10]
declare Ex_EH [intro!]

lemma Ex_mono: "insert A H ⊢ B ⟹ ∀C ∈ H. atom i ♯ C ⟹ insert (Ex i A) H ⊢ (Ex i B)"
by (auto simp add: intro: Ex_I [where x="Var i"])

lemma All_I [intro!]: "H ⊢ A ⟹ ∀C ∈ H. atom i ♯ C ⟹ H ⊢ All i A"
by (auto intro: ContraProve Neg_I0)

lemma All_D: "H ⊢ All i A ⟹ H ⊢ A(i::=x)"
by (metis Assume Ex_I NegNeg_D Neg_mono SyntaxN.Neg cut_same)

lemma All_E: "insert (A(i::=x)) H ⊢ B ⟹ insert (All i A) H ⊢ B"
by (metis Ex_I NegNeg_D Neg_mono SyntaxN.Neg)

lemma All_E': "H ⊢ All i A ⟹ insert (A(i::=x)) H ⊢ B ⟹ H ⊢ B"
by (metis All_D cut_same)

lemma All2_E: "⟦atom i ♯ t; H ⊢ x IN t; insert (A(i::=x)) H ⊢ B⟧ ⟹ insert (All2 i t A) H ⊢ B"
apply (rule All_E [where x=x], auto)
by (metis Swap thin1)

lemma All2_E': "⟦H ⊢ All2 i t A; H ⊢ x IN t; insert (A(i::=x)) H ⊢ B; atom i ♯ t⟧ ⟹ H ⊢ B"
by (metis All2_E cut_same)

subsection‹Congruence rules›

lemma Neg_cong: "H ⊢ A IFF A' ⟹ H ⊢ Neg A IFF Neg A'"
by (metis Iff_def Conj_E1 Conj_E2 Conj_I Contrapos1)

lemma Disj_cong: "H ⊢ A IFF A' ⟹ H ⊢ B IFF B' ⟹ H ⊢ A OR B IFF A' OR B'"
by (metis Conj_E1 Conj_E2 Disj_mono Iff_I Iff_def anti_deduction)

lemma Conj_cong: "H ⊢ A IFF A' ⟹ H ⊢ B IFF B' ⟹ H ⊢ A AND B IFF A' AND B'"
by (metis Conj_def Disj_cong Neg_cong)

lemma Imp_cong: "H ⊢ A IFF A' ⟹ H ⊢ B IFF B' ⟹ H ⊢ (A IMP B) IFF (A' IMP B')"
by (metis Disj_cong Neg_cong)

lemma Iff_cong: "H ⊢ A IFF A' ⟹ H ⊢ B IFF B' ⟹ H ⊢ (A IFF B) IFF (A' IFF B')"
by (metis Iff_def Conj_cong Imp_cong)

lemma Ex_cong: "H ⊢ A IFF A' ⟹ ∀C ∈ H. atom i ♯ C ⟹ H ⊢ (Ex i A) IFF (Ex i A')"
apply (rule Iff_I)
apply (metis Ex_mono Hyp Iff_MP_same Un_absorb Un_insert_right insertI1 thin_Un)
apply (metis Ex_mono Hyp Iff_MP2_same Un_absorb Un_insert_right insertI1 thin_Un)
done

lemma All_cong: "H ⊢ A IFF A' ⟹ ∀C ∈ H. atom i ♯ C ⟹ H ⊢ (All i A) IFF (All i A')"
by (metis Ex_cong Neg_cong)

lemma Subst: "H ⊢ A ⟹ ∀B ∈ H. atom i ♯ B ⟹ H ⊢ A (i::=x)"
by (metis All_D All_I)

section‹Equality reasoning›

subsection‹The congruence property for @{term Eq}, and other basic properties of equality›

lemma Eq_cong1: "{} ⊢ (t EQ t' AND u EQ u') IMP (t EQ u IMP t' EQ u')"
proof -
obtain v2::name and v3::name and v4::name
where v2: "atom v2 ♯ (t,X1,X3,X4)"
and v3: "atom v3 ♯ (t,t',X1,v2,X4)"
and v4: "atom v4 ♯ (t,t',u,X1,v2,v3)"
by (metis obtain_fresh)
have "{} ⊢ (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Var X1 EQ Var X3 IMP Var X2 EQ Var X4)"
by (rule Eq) (simp add: eq_cong_ax_def equality_axioms_def)
hence "{} ⊢ (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Var X1 EQ Var X3 IMP Var X2 EQ Var X4)"
by (drule_tac i=X1 and x="Var X1" in Subst) simp_all
hence "{} ⊢ (Var X1 EQ Var v2 AND Var X3 EQ Var X4) IMP (Var X1 EQ Var X3 IMP Var v2 EQ Var X4)"
by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
hence "{} ⊢ (Var X1 EQ Var v2 AND Var v3 EQ Var X4) IMP (Var X1 EQ Var v3 IMP Var v2 EQ Var X4)"
using v2
by (drule_tac i=X3 and x="Var v3" in Subst) simp_all
hence "{} ⊢ (Var X1 EQ Var v2 AND Var v3 EQ Var v4) IMP (Var X1 EQ Var v3 IMP Var v2 EQ Var v4)"
using v2 v3
by (drule_tac i=X4 and x="Var v4" in Subst) simp_all
hence "{} ⊢ (t EQ Var v2 AND Var v3 EQ Var v4) IMP (t EQ Var v3 IMP Var v2 EQ Var v4)"
using v2 v3 v4
by (drule_tac i=X1 and x=t in Subst) simp_all
hence "{} ⊢ (t EQ t' AND Var v3 EQ Var v4) IMP (t EQ Var v3 IMP t' EQ Var v4)"
using v2 v3 v4
by (drule_tac i=v2 and x="t'" in Subst) simp_all
hence "{} ⊢ (t EQ t' AND u EQ Var v4) IMP (t EQ u IMP t' EQ Var v4)"
using v3 v4
by (drule_tac i=v3 and x=u in Subst) simp_all
thus ?thesis
using v4
by (drule_tac i=v4 and x="u'" in Subst) simp_all
qed

lemma Refl [iff]: "H ⊢ t EQ t"
proof -
have "{} ⊢ Var X1 EQ Var X1"
by (rule Eq) (simp add: equality_axioms_def refl_ax_def)
hence "{} ⊢ t EQ t"
by (drule_tac i=X1 and x=t in Subst) simp_all
thus ?thesis
by (metis empty_subsetI thin)
qed

text‹Apparently necessary in order to prove the congruence property.›
lemma Sym: assumes "H ⊢ t EQ u" shows "H ⊢ u EQ t"
proof -
have "{} ⊢  (t EQ u AND t EQ t) IMP (t EQ t IMP u EQ t)"
by (rule Eq_cong1)
moreover have "{t EQ u} ⊢ t EQ u AND t EQ t"
by (metis Assume Conj_I Refl)
ultimately have "{t EQ u} ⊢ u EQ t"
by (metis MP_same MP Refl sup_bot_left)
thus "H ⊢ u EQ t" by (metis assms cut1)
qed

lemma Sym_L: "insert (t EQ u) H ⊢ A ⟹ insert (u EQ t) H ⊢ A"
by (metis Assume Sym Un_empty_left Un_insert_left cut)

lemma Trans: assumes "H ⊢ x EQ y" "H ⊢ y EQ z" shows "H ⊢ x EQ z"
proof -
have "⋀H. H ⊢ (x EQ x AND y EQ z) IMP (x EQ y IMP x EQ z)"
by (metis Eq_cong1 bot_least thin)
moreover have "{x EQ y, y EQ z} ⊢ x EQ x AND y EQ z"
by (metis Assume Conj_I Refl thin1)
ultimately have "{x EQ y, y EQ z} ⊢ x EQ z"
by (metis Hyp MP_same insertI1)
thus ?thesis
by (metis assms cut2)
qed

lemma Eq_cong:
assumes "H ⊢ t EQ t'" "H ⊢ u EQ u'" shows "H ⊢ t EQ u IFF t' EQ u'"
proof -
{ fix t t' u u'
assume  "H ⊢ t EQ t'" "H ⊢ u EQ u'"
moreover have "{t EQ t', u EQ u'} ⊢ t EQ u IMP t' EQ u'" using Eq_cong1
by (metis Assume Conj_I MP_null insert_commute)
ultimately have "H ⊢ t EQ u IMP t' EQ u'"
by (metis cut2)
}
thus ?thesis
by (metis Iff_def Conj_I assms Sym)
qed

lemma Eq_Trans_E: "H ⊢ x EQ u ⟹ insert (t EQ u) H ⊢ A ⟹ insert (x EQ t) H ⊢ A"
by (metis Assume Sym_L Trans cut_same thin1 thin2)

subsection‹The congruence property for @{term Mem}›

lemma Mem_cong1: "{} ⊢ (t EQ t' AND u EQ u') IMP (t IN u IMP t' IN u')"
proof -
obtain v2::name and v3::name and v4::name
where v2: "atom v2 ♯ (t,X1,X3,X4)"
and v3: "atom v3 ♯ (t,t',X1,v2,X4)"
and v4: "atom v4 ♯ (t,t',u,X1,v2,v3)"
by (metis obtain_fresh)
have "{} ⊢ (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Var X1 IN Var X3 IMP Var X2 IN Var X4)"
by (metis mem_cong_ax_def equality_axioms_def insert_iff Eq)
hence "{} ⊢ (Var X1 EQ Var v2 AND Var X3 EQ Var X4) IMP (Var X1 IN Var X3 IMP Var v2 IN Var X4)"
by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
hence "{} ⊢ (Var X1 EQ Var v2 AND Var v3 EQ Var X4) IMP (Var X1 IN Var v3 IMP Var v2 IN Var X4)"
using v2
by (drule_tac i=X3 and x="Var v3" in Subst) simp_all
hence "{} ⊢ (Var X1 EQ Var v2 AND Var v3 EQ Var v4) IMP (Var X1 IN Var v3 IMP Var v2 IN Var v4)"
using v2 v3
by (drule_tac i=X4 and x="Var v4" in Subst) simp_all
hence "{} ⊢ (t EQ Var v2 AND Var v3 EQ Var v4) IMP (t IN Var v3 IMP Var v2 IN Var v4)"
using v2 v3 v4
by (drule_tac i=X1 and x=t in Subst) simp_all
hence "{} ⊢ (t EQ t' AND Var v3 EQ Var v4) IMP (t IN Var v3 IMP t' IN Var v4)"
using v2 v3 v4
by (drule_tac i=v2 and x=t' in Subst) simp_all
hence "{} ⊢ (t EQ t' AND u EQ Var v4) IMP (t IN u IMP t' IN Var v4)"
using v3 v4
by (drule_tac i=v3 and x=u in Subst) simp_all
thus ?thesis
using v4
by (drule_tac i=v4 and x=u' in Subst) simp_all
qed

lemma Mem_cong:
assumes "H ⊢ t EQ t'" "H ⊢ u EQ u'" shows "H ⊢ t IN u IFF t' IN u'"
proof -
{ fix t t' u u'
have cong: "{t EQ t', u EQ u'} ⊢ t IN u IMP t' IN u'"
by (metis AssumeH(2) Conj_I MP_null Mem_cong1 insert_commute)
}
thus ?thesis
by (metis Iff_def Conj_I cut2 assms Sym)
qed

subsection‹The congruence properties for @{term Eats} and @{term HPair}›

lemma Eats_cong1: "{} ⊢ (t EQ t' AND u EQ u') IMP (Eats t u EQ Eats t' u')"
proof -
obtain v2::name and v3::name and v4::name
where v2: "atom v2 ♯ (t,X1,X3,X4)"
and v3: "atom v3 ♯ (t,t',X1,v2,X4)"
and v4: "atom v4 ♯ (t,t',u,X1,v2,v3)"
by (metis obtain_fresh)
have "{} ⊢ (Var X1 EQ Var X2 AND Var X3 EQ Var X4) IMP (Eats (Var X1) (Var X3) EQ Eats (Var X2) (Var X4))"
by (metis eats_cong_ax_def equality_axioms_def insert_iff Eq)
hence "{} ⊢ (Var X1 EQ Var v2 AND Var X3 EQ Var X4) IMP (Eats (Var X1) (Var X3) EQ Eats (Var v2) (Var X4))"
by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
hence "{} ⊢ (Var X1 EQ Var v2 AND Var v3 EQ Var X4) IMP (Eats (Var X1) (Var v3) EQ Eats (Var v2) (Var X4))"
using v2
by (drule_tac i=X3 and x="Var v3" in Subst) simp_all
hence "{} ⊢ (Var X1 EQ Var v2 AND Var v3 EQ Var v4) IMP (Eats (Var X1) (Var v3) EQ Eats (Var v2) (Var v4))"
using v2 v3
by (drule_tac i=X4 and x="Var v4" in Subst) simp_all
hence "{} ⊢ (t EQ Var v2 AND Var v3 EQ Var v4) IMP (Eats t (Var v3) EQ Eats (Var v2) (Var v4))"
using v2 v3 v4
by (drule_tac i=X1 and x=t in Subst) simp_all
hence "{} ⊢ (t EQ t' AND Var v3 EQ Var v4) IMP (Eats t (Var v3) EQ Eats t' (Var v4))"
using v2 v3 v4
by (drule_tac i=v2 and x=t' in Subst) simp_all
hence "{} ⊢ (t EQ t' AND u EQ Var v4) IMP (Eats t u EQ Eats t' (Var v4))"
using v3 v4
by (drule_tac i=v3 and x=u in Subst) simp_all
thus ?thesis
using v4
by (drule_tac i=v4 and x=u' in Subst) simp_all
qed

lemma Eats_cong: "⟦H ⊢ t EQ t'; H ⊢ u EQ u'⟧ ⟹ H ⊢ Eats t u EQ Eats t' u'"
by (metis Conj_I anti_deduction Eats_cong1 cut1)

lemma HPair_cong: "⟦H ⊢ t EQ t'; H ⊢ u EQ u'⟧ ⟹ H ⊢ HPair t u EQ HPair t' u'"
by (metis HPair_def Eats_cong Refl)

lemma SUCC_cong: "H ⊢ t EQ t' ⟹ H ⊢ SUCC t EQ SUCC t'"
by (metis Eats_cong SUCC_def)

subsection‹Substitution for Equalities›

lemma Eq_subst_tm_Iff: "{t EQ u} ⊢ subst i t tm EQ subst i u tm"
by (induct tm rule: tm.induct) (auto simp: Eats_cong)

lemma Eq_subst_fm_Iff: "insert (t EQ u) H ⊢ A(i::=t) IFF A(i::=u)"
proof -
have "{ t EQ u } ⊢ A(i::=t) IFF A(i::=u)"
by (nominal_induct A avoiding: i t u rule: fm.strong_induct)
(auto simp: Disj_cong Neg_cong Ex_cong Mem_cong Eq_cong Eq_subst_tm_Iff)
thus ?thesis
by (metis Assume cut1)
qed

lemma Var_Eq_subst_Iff: "insert (Var i EQ t) H ⊢ A(i::=t) IFF A"
by (metis Eq_subst_fm_Iff Iff_sym subst_fm_id)

lemma Var_Eq_imp_subst_Iff: "H ⊢ Var i EQ t ⟹ H ⊢ A(i::=t) IFF A"
by (metis Var_Eq_subst_Iff cut_same)

subsection‹Congruence Rules for Predicates›

lemma P1_cong:
fixes tms :: "tm list"
assumes "⋀i t x. atom i ♯ tms ⟹ (P t)(i::=x) = P (subst i x t)" and "H ⊢ x EQ x'"
shows "H ⊢ P x IFF P x'"
proof -
obtain i::name where i: "atom i ♯ tms"
by (metis obtain_fresh)
have "insert (x EQ x') H  ⊢ (P (Var i))(i::=x) IFF (P(Var i))(i::=x')"
by (rule Eq_subst_fm_Iff)
thus ?thesis using assms i
by (metis cut_same subst.simps(2))
qed

lemma P2_cong:
fixes tms :: "tm list"
assumes sub: "⋀i t u x. atom i ♯ tms ⟹ (P t u)(i::=x) = P (subst i x t) (subst i x u)"
and eq:  "H ⊢ x EQ x'" "H ⊢ y EQ y'"
shows "H ⊢ P x y IFF P x' y'"
proof -
have yy': "{ y EQ y' } ⊢ P x' y IFF P x' y'"
by (rule P1_cong [where tms="[y,x']@tms"]) (auto simp: fresh_Cons sub)
have "{ x EQ x' } ⊢ P x y IFF P x' y"
by (rule P1_cong [where tms="[y,x']@tms"]) (auto simp: fresh_Cons sub)
hence "{x EQ x', y EQ y'} ⊢ P x y IFF P x' y'"
by (metis Assume Iff_trans cut1 rotate2 yy')
thus ?thesis
by (metis cut2 eq)
qed

lemma P3_cong:
fixes tms :: "tm list"
assumes sub: "⋀i t u v x. atom i ♯ tms ⟹
(P t u v)(i::=x) = P (subst i x t) (subst i x u) (subst i x v)"
and eq:  "H ⊢ x EQ x'" "H ⊢ y EQ y'" "H ⊢ z EQ z'"
shows "H ⊢ P x y z IFF P x' y' z'"
proof -
obtain i::name where i: "atom i ♯ (z,z',y,y',x,x')"
by (metis obtain_fresh)
have tl: "{ y EQ y', z EQ z' } ⊢ P x' y z IFF P x' y' z'"
by (rule P2_cong [where tms="[z,z',y,y',x,x']@tms"]) (auto simp: fresh_Cons sub)
have hd: "{ x EQ x' } ⊢ P x y z IFF P x' y z"
by (rule P1_cong [where tms="[z,y,x']@tms"]) (auto simp: fresh_Cons sub)
have "{x EQ x', y EQ y', z EQ z'} ⊢ P x y z IFF P x' y' z'"
by (metis Assume thin1 hd [THEN cut1] tl Iff_trans)
thus ?thesis
by (rule cut3) (rule eq)+
qed

lemma P4_cong:
fixes tms :: "tm list"
assumes sub: "⋀i t1 t2 t3 t4 x. atom i ♯ tms ⟹
(P t1 t2 t3 t4)(i::=x) = P (subst i x t1) (subst i x t2) (subst i x t3) (subst i x t4)"
and eq:  "H ⊢ x1 EQ x1'" "H ⊢ x2 EQ x2'" "H ⊢ x3 EQ x3'" "H ⊢ x4 EQ x4'"
shows "H ⊢ P x1 x2 x3 x4 IFF P x1' x2' x3' x4'"
proof -
obtain i::name where i: "atom i ♯ (x4,x4',x3,x3',x2,x2',x1,x1')"
by (metis obtain_fresh)
have tl: "{ x2 EQ x2', x3 EQ x3', x4 EQ x4' } ⊢ P x1' x2 x3 x4 IFF P x1' x2' x3' x4'"
by (rule P3_cong [where tms="[x4,x4',x3,x3',x2,x2',x1,x1']@tms"]) (auto simp: fresh_Cons sub)
have hd: "{ x1 EQ x1' } ⊢ P x1 x2 x3 x4 IFF P x1' x2 x3 x4"
by (auto simp: fresh_Cons sub intro!: P1_cong [where tms="[x4,x3,x2,x1']@tms"])
have "{x1 EQ x1', x2 EQ x2', x3 EQ x3', x4 EQ x4'} ⊢ P x1 x2 x3 x4 IFF P x1' x2' x3' x4'"
by (metis Assume thin1 hd [THEN cut1] tl Iff_trans)
thus ?thesis
by (rule cut4) (rule eq)+
qed

section‹Zero and Falsity›

lemma Mem_Zero_iff:
assumes "atom i ♯ t" shows "H ⊢ (t EQ Zero) IFF (All i (Neg ((Var i) IN t)))"
proof -
obtain i'::name where i': "atom i' ♯ (t, X0, X1, i)"
by (rule obtain_fresh)
have "{} ⊢ ((Var X0) EQ Zero) IFF (All X1 (Neg ((Var X1) IN (Var X0))))"
by (simp add: HF HF_axioms_def HF1_def)
then have "{} ⊢ (((Var X0) EQ Zero) IFF (All X1 (Neg ((Var X1) IN (Var X0)))))(X0 ::= t)"
by (rule Subst) simp
hence "{} ⊢ (t EQ Zero) IFF (All i' (Neg ((Var i') IN t)))" using i'
by simp
also have "... = (FRESH i'. (t EQ Zero) IFF (All i' (Neg ((Var i') IN t))))"
using i' by simp
also have "... = (t EQ Zero) IFF (All i (Neg ((Var i) IN t)))"
using assms by simp
finally show ?thesis
by (metis empty_subsetI thin)
qed

lemma Mem_Zero_E [intro!]: "insert (x IN Zero) H ⊢ A"
proof -
obtain i::name where "atom i ♯ Zero"
by (rule obtain_fresh)
hence "{} ⊢ All i (Neg ((Var i) IN Zero))"
by (metis Mem_Zero_iff Iff_MP_same Refl)
hence "{} ⊢ Neg (x IN Zero)"
by (drule_tac x=x in All_D) simp
thus ?thesis
by (metis Contrapos2 Hyp Imp_triv_I MP_same empty_subsetI insertI1 thin)
qed

declare Mem_Zero_E [THEN rotate2, intro!]
declare Mem_Zero_E [THEN rotate3, intro!]
declare Mem_Zero_E [THEN rotate4, intro!]
declare Mem_Zero_E [THEN rotate5, intro!]
declare Mem_Zero_E [THEN rotate6, intro!]
declare Mem_Zero_E [THEN rotate7, intro!]
declare Mem_Zero_E [THEN rotate8, intro!]

subsection‹The Formula @{term Fls}; Consistency of the Calculus›

definition Fls  where "Fls ≡ Zero IN Zero"

lemma Fls_eqvt [eqvt]: "(p ∙ Fls) = Fls"

lemma Fls_fresh [simp]: "a ♯ Fls"

lemma Neg_I [intro!]: "insert A H ⊢ Fls ⟹ H ⊢ Neg A"
unfolding Fls_def
by (rule Neg_I0) (metis Mem_Zero_E cut_same)

lemma Neg_E [intro!]: "H ⊢ A ⟹ insert (Neg A) H ⊢ Fls"
by (rule ContraProve)

declare Neg_E [THEN rotate2, intro!]
declare Neg_E [THEN rotate3, intro!]
declare Neg_E [THEN rotate4, intro!]
declare Neg_E [THEN rotate5, intro!]
declare Neg_E [THEN rotate6, intro!]
declare Neg_E [THEN rotate7, intro!]
declare Neg_E [THEN rotate8, intro!]

text‹We need these because Neg (A IMP B) doesn't have to be syntactically a conjunction.›
lemma Neg_Imp_I [intro!]: "H ⊢ A ⟹ insert B H ⊢ Fls ⟹ H ⊢ Neg (A IMP B)"
by (metis NegNeg_I Neg_Disj_I Neg_I)

lemma Neg_Imp_E [intro!]: "insert (Neg B) (insert A H) ⊢ C ⟹ insert (Neg (A IMP B)) H ⊢ C"
apply (rule cut_same [where A=A])
apply (metis Assume Disj_I1 NegNeg_D Neg_mono)
apply (metis Swap Imp_I rotate2 thin1)
done

declare Neg_Imp_E [THEN rotate2, intro!]
declare Neg_Imp_E [THEN rotate3, intro!]
declare Neg_Imp_E [THEN rotate4, intro!]
declare Neg_Imp_E [THEN rotate5, intro!]
declare Neg_Imp_E [THEN rotate6, intro!]
declare Neg_Imp_E [THEN rotate7, intro!]
declare Neg_Imp_E [THEN rotate8, intro!]

lemma Fls_E [intro!]: "insert Fls H ⊢ A"
by (metis Mem_Zero_E Fls_def)

declare Fls_E [THEN rotate2, intro!]
declare Fls_E [THEN rotate3, intro!]
declare Fls_E [THEN rotate4, intro!]
declare Fls_E [THEN rotate5, intro!]
declare Fls_E [THEN rotate6, intro!]
declare Fls_E [THEN rotate7, intro!]
declare Fls_E [THEN rotate8, intro!]

lemma truth_provable: "H ⊢ (Neg Fls)"
by (metis Fls_E Neg_I)

lemma ExFalso: "H ⊢ Fls ⟹ H ⊢ A"
by (metis Neg_D truth_provable)

text‹Thanks to Andrei Popescu for pointing out that consistency was provable here.›
proposition consistent: "¬ {} ⊢ Fls"
by (meson empty_iff eval_fm.simps(4) hfthm_sound truth_provable)

subsection‹More properties of @{term Zero}›

lemma Eq_Zero_D:
assumes "H ⊢ t EQ Zero" "H ⊢ u IN t" shows "H ⊢ A"
proof -
obtain i::name where  i: "atom i ♯ t"
by (rule obtain_fresh)
with assms have an: "H ⊢ (All i (Neg ((Var i) IN t)))"
by (metis Iff_MP_same Mem_Zero_iff)
have "H ⊢ Neg (u IN t)" using All_D [OF an, of u] i
by simp
thus ?thesis using assms
by (metis Neg_D)
qed

lemma Eq_Zero_thm:
assumes "atom i ♯ t" shows "{All i (Neg ((Var i) IN t))} ⊢ t EQ Zero"
by (metis Assume Iff_MP2_same Mem_Zero_iff assms)

lemma Eq_Zero_I:
assumes insi: "insert ((Var i) IN t) H ⊢ Fls" and i1: "atom i ♯ t" and i2: "∀B ∈ H. atom i ♯ B"
shows "H ⊢ t EQ Zero"
proof -
have "H ⊢ All i (Neg ((Var i) IN t))"
by (metis All_I Neg_I i2 insi)
thus ?thesis
by (metis cut_same  cut [OF Eq_Zero_thm [OF i1] Hyp] insertCI insert_is_Un)
qed

subsection‹Basic properties of @{term Eats}›

lemma Eq_Eats_iff:
assumes "atom i ♯ (z,t,u)"
shows "H ⊢ (z EQ Eats t u) IFF (All i (Var i IN z IFF Var i IN t OR Var i EQ u))"
proof -
obtain v1::name and v2::name and i'::name
where v1: "atom v1 ♯ (z,X0,X2,X3)"
and v2: "atom v2 ♯ (t,z,X0,v1,X3)"
and i': "atom i' ♯ (t,u,z,X0,v1,v2,X3)"
by (metis obtain_fresh)
have "{} ⊢ ((Var X0) EQ (Eats (Var X1) (Var X2))) IFF
(All X3 (Var X3 IN Var X0 IFF Var X3 IN Var X1 OR Var X3 EQ Var X2))"
by (simp add: HF HF_axioms_def HF2_def)
hence "{} ⊢ ((Var X0) EQ (Eats (Var X1) (Var X2))) IFF
(All X3 (Var X3 IN Var X0 IFF Var X3 IN Var X1 OR Var X3 EQ Var X2))"
by (drule_tac i=X0 and x="Var X0" in Subst) simp_all
hence "{} ⊢ ((Var X0) EQ (Eats (Var v1) (Var X2))) IFF
(All X3 (Var X3 IN Var X0 IFF Var X3 IN Var v1 OR Var X3 EQ Var X2))"
using v1 by (drule_tac i=X1 and x="Var v1" in Subst) simp_all
hence "{} ⊢ ((Var X0) EQ (Eats (Var v1) (Var v2))) IFF
(All X3 (Var X3 IN Var X0 IFF Var X3 IN Var v1 OR Var X3 EQ Var v2))"
using v1 v2 by (drule_tac i=X2 and x="Var v2" in Subst) simp_all
hence "{} ⊢ (((Var X0) EQ (Eats (Var v1) (Var v2))) IFF
(All X3 (Var X3 IN Var X0 IFF Var X3 IN Var v1 OR Var X3 EQ Var v2)))(X0 ::= z)"
by (rule Subst) simp
hence "{} ⊢ ((z EQ (Eats (Var v1) (Var v2))) IFF
(All i' (Var i' IN z IFF Var i' IN Var v1 OR Var i' EQ Var v2)))"
using v1 v2 i' by (simp add: Conj_def Iff_def)
hence "{} ⊢ (z EQ (Eats t (Var v2))) IFF
(All i' (Var i' IN z IFF Var i' IN t OR Var i' EQ Var v2))"
using v1 v2 i' by (drule_tac i=v1 and x=t in Subst) simp_all
hence "{} ⊢ (z EQ Eats t u) IFF
(All i' (Var i' IN z IFF Var i' IN t OR Var i' EQ u))"
using v1 v2 i' by (drule_tac i=v2 and x=u in Subst) simp_all
also have "... = (FRESH i'. (z EQ Eats t u) IFF (All i' (Var i' IN z IFF Var i' IN t OR Var i' EQ u)))"
using i' by simp
also have "... = (z EQ Eats t u) IFF (All i (Var i IN z IFF Var i IN t OR Var i EQ u))"
using assms i' by simp
finally show ?thesis
by (rule thin0)
qed

lemma Eq_Eats_I:
"H ⊢ All i (Var i IN z IFF Var i IN t OR Var i EQ u) ⟹ atom i ♯ (z,t,u) ⟹ H ⊢ z EQ Eats t u"
by (metis Iff_MP2_same Eq_Eats_iff)

lemma Mem_Eats_Iff:
"H ⊢ x IN (Eats t u) IFF x IN t OR x EQ u"
proof -
obtain i::name where "atom i ♯ (Eats t u, t, u)"
by (rule obtain_fresh)
thus ?thesis
using Iff_MP_same [OF Eq_Eats_iff, THEN All_D]
by auto
qed

lemma Mem_Eats_I1: "H ⊢ u IN t ⟹ H ⊢ u IN Eats t z"
by (metis Disj_I1 Iff_MP2_same Mem_Eats_Iff)

lemma Mem_Eats_I2: "H ⊢ u EQ z ⟹ H ⊢ u IN Eats t z"
by (metis Disj_I2 Iff_MP2_same Mem_Eats_Iff)

lemma Mem_Eats_E:
assumes A: "insert (u IN t) H ⊢ C" and B: "insert (u EQ z) H ⊢ C"
shows "insert (u IN Eats t z) H ⊢ C"
by (rule Mem_Eats_Iff [of _ u t z, THEN Iff_MP_left']) (metis A B Disj_E)

lemmas Mem_Eats_EH = Mem_Eats_E Mem_Eats_E [THEN rotate2] Mem_Eats_E [THEN rotate3] Mem_Eats_E [THEN rotate4] Mem_Eats_E [THEN rotate5]
Mem_Eats_E [THEN rotate6] Mem_Eats_E [THEN rotate7] Mem_Eats_E [THEN rotate8]
declare Mem_Eats_EH [intro!]

lemma Mem_SUCC_I1: "H ⊢ u IN t ⟹ H ⊢ u IN SUCC t"
by (metis Mem_Eats_I1 SUCC_def)

lemma Mem_SUCC_I2: "H ⊢ u EQ t ⟹ H ⊢ u IN SUCC t"
by (metis Mem_Eats_I2 SUCC_def)

lemma Mem_SUCC_Refl [simp]: "H ⊢ k IN SUCC k"
by (metis Mem_SUCC_I2 Refl)

lemma Mem_SUCC_E:
assumes "insert (u IN t) H ⊢ C" "insert (u EQ t) H ⊢ C" shows "insert (u IN SUCC t) H ⊢ C"
by (metis assms Mem_Eats_E SUCC_def)

lemmas Mem_SUCC_EH = Mem_SUCC_E Mem_SUCC_E [THEN rotate2] Mem_SUCC_E [THEN rotate3] Mem_SUCC_E [THEN rotate4] Mem_SUCC_E [THEN rotate5]
Mem_SUCC_E [THEN rotate6] Mem_SUCC_E [THEN rotate7] Mem_SUCC_E [THEN rotate8]

lemma Eats_EQ_Zero_E: "insert (Eats t u EQ Zero) H ⊢ A"
by (metis Assume Eq_Zero_D Mem_Eats_I2 Refl)

lemmas Eats_EQ_Zero_EH = Eats_EQ_Zero_E Eats_EQ_Zero_E [THEN rotate2] Eats_EQ_Zero_E [THEN rotate3] Eats_EQ_Zero_E [THEN rotate4] Eats_EQ_Zero_E [THEN rotate5]
Eats_EQ_Zero_E [THEN rotate6] Eats_EQ_Zero_E [THEN rotate7] Eats_EQ_Zero_E [THEN rotate8]
declare Eats_EQ_Zero_EH [intro!]

lemma Eats_EQ_Zero_E2: "insert (Zero EQ Eats t u) H ⊢ A"
by (metis Eats_EQ_Zero_E Sym_L)

lemmas Eats_EQ_Zero_E2H = Eats_EQ_Zero_E2 Eats_EQ_Zero_E2 [THEN rotate2] Eats_EQ_Zero_E2 [THEN rotate3] Eats_EQ_Zero_E2 [THEN rotate4] Eats_EQ_Zero_E2 [THEN rotate5]
Eats_EQ_Zero_E2 [THEN rotate6] Eats_EQ_Zero_E2 [THEN rotate7] Eats_EQ_Zero_E2 [THEN rotate8]
declare Eats_EQ_Zero_E2H [intro!]

section‹Bounded Quantification involving @{term Eats}›

lemma All2_cong: "H ⊢ t EQ t' ⟹ H ⊢ A IFF A' ⟹ ∀C ∈ H. atom i ♯ C ⟹ H ⊢ (All2 i t A) IFF (All2 i t' A')"
by (metis All_cong Imp_cong Mem_cong Refl)

lemma All2_Zero_E [intro!]: "H  ⊢ B ⟹ insert (All2 i Zero A) H ⊢ B"
by (rule thin1)

lemma All2_Eats_I_D:
"atom i ♯ (t,u) ⟹ { All2 i t A, A(i::=u)} ⊢ (All2 i (Eats t u) A)"
apply (auto, auto intro!: Ex_I [where x="Var i"])
apply (metis Assume thin1 Var_Eq_subst_Iff [THEN Iff_MP_same])
done

lemma All2_Eats_I:
"⟦atom i ♯ (t,u); H ⊢ All2 i t A; H ⊢ A(i::=u)⟧ ⟹ H ⊢ (All2 i (Eats t u) A)"
by (rule cut2 [OF All2_Eats_I_D], auto)

lemma All2_Eats_E1:
"⟦atom i ♯ (t,u); ∀C ∈ H. atom i ♯ C⟧ ⟹ insert (All2 i (Eats t u) A) H ⊢ All2 i t A"
by auto (metis Assume Ex_I Imp_E Mem_Eats_I1 Neg_mono subst_fm_id)

lemma All2_Eats_E2:
"⟦atom i ♯ (t,u); ∀C ∈ H. atom i ♯ C⟧ ⟹ insert (All2 i (Eats t u) A) H ⊢ A(i::=u)"
by (rule All_E [where x=u]) (auto intro: ContraProve Mem_Eats_I2)

lemma All2_Eats_E:
assumes i: "atom i ♯ (t,u)"
and B: "insert (All2 i t A) (insert (A(i::=u)) H) ⊢ B"
shows "insert (All2 i (Eats t u) A) H ⊢ B"
using i
apply (rule cut_thin [OF All2_Eats_E2, where HB = "insert (All2 i (Eats t u) A) H"], auto)
apply (rule cut_thin [OF All2_Eats_E1 B], auto)
done

lemma All2_SUCC_I:
"atom i ♯ t ⟹ H ⊢ All2 i t A ⟹ H ⊢ A(i::=t) ⟹ H ⊢ (All2 i (SUCC t) A)"

lemma All2_SUCC_E:
assumes "atom i ♯ t"
and "insert (All2 i t A) (insert (A(i::=t)) H) ⊢ B"
shows "insert (All2 i (SUCC t) A) H ⊢ B"
by (simp add: SUCC_def All2_Eats_E assms)

lemma All2_SUCC_E':
assumes "H ⊢ u EQ SUCC t"
and "atom i ♯ t" "∀C ∈ H. atom i ♯ C"
and "insert (All2 i t A) (insert (A(i::=t)) H) ⊢ B"
shows "insert (All2 i u A) H ⊢ B"
by (metis All2_SUCC_E Iff_MP_left' Iff_refl All2_cong assms)

section‹Induction›

lemma Ind:
assumes j: "atom (j::name) ♯ (i,A)"
and prems: "H ⊢ A(i::=Zero)" "H ⊢ All i (All j (A IMP (A(i::= Var j) IMP A(i::= Eats(Var i)(Var j)))))"
shows "H ⊢ A"
proof -
have  "{A(i::=Zero), All i (All j (A IMP (A(i::= Var j) IMP A(i::= Eats(Var i)(Var j)))))} ⊢ All i A"
by (metis j hfthm.Ind ind anti_deduction insert_commute)
hence "H ⊢ (All i A)"
by (metis cut2 prems)
thus ?thesis
by (metis All_E' Assume subst_fm_id)
qed

end


# Theory Coding

chapter‹De Bruijn Syntax, Quotations, Codes, V-Codes›

theory Coding
imports SyntaxN
begin

declare fresh_Nil [iff]

section ‹de Bruijn Indices (locally-nameless version)›

nominal_datatype dbtm = DBZero | DBVar name | DBInd nat | DBEats dbtm dbtm

nominal_datatype dbfm =
DBMem dbtm dbtm
| DBEq dbtm dbtm
| DBDisj dbfm dbfm
| DBNeg dbfm
| DBEx dbfm

declare dbtm.supp [simp]
declare dbfm.supp [simp]

fun lookup :: "name list ⇒ nat ⇒ name ⇒ dbtm"
where
"lookup [] n x = DBVar x"
| "lookup (y # ys) n x = (if x = y then DBInd n else (lookup ys (Suc n) x))"

lemma fresh_imp_notin_env: "atom name ♯ e ⟹ name ∉ set e"
by (metis List.finite_set fresh_finite_set_at_base fresh_set)

lemma lookup_notin: "x ∉ set e ⟹ lookup e n x = DBVar x"
by (induct e arbitrary: n) auto

lemma lookup_in:
"x ∈ set e ⟹ ∃k. lookup e n x = DBInd k ∧ n ≤ k ∧ k < n + length e"
apply (induct e arbitrary: n)
apply (auto intro: Suc_leD)
done

lemma lookup_fresh: "x ♯ lookup e n y ⟷ y ∈ set e ∨ x ≠ atom y"
by (induct arbitrary: n rule: lookup.induct) (auto simp: pure_fresh fresh_at_base)

lemma lookup_eqvt[eqvt]: "(p ∙ lookup xs n x) = lookup (p ∙ xs) (p ∙ n) (p ∙ x)"
by (induct xs arbitrary: n) (simp_all add: permute_pure)

lemma lookup_inject [iff]: "(lookup e n x = lookup e n y) ⟷ x = y"
apply (induct e n x arbitrary: y rule: lookup.induct, force, simp)
by (metis Suc_n_not_le_n dbtm.distinct(7) dbtm.eq_iff(3) lookup_in lookup_notin)

nominal_function trans_tm :: "name list ⇒ tm ⇒ dbtm"
where
"trans_tm e Zero = DBZero"
| "trans_tm e (Var k) = lookup e 0 k"
| "trans_tm e (Eats t u) = DBEats (trans_tm e t) (trans_tm e u)"
by (auto simp: eqvt_def trans_tm_graph_aux_def) (metis tm.strong_exhaust)

nominal_termination (eqvt)
by lexicographic_order

lemma fresh_trans_tm_iff [simp]: "i ♯ trans_tm e t ⟷ i ♯ t ∨ i ∈ atom  set e"
by (induct t rule: tm.induct, auto simp: lookup_fresh fresh_at_base)

lemma trans_tm_forget: "atom i ♯ t ⟹ trans_tm [i] t = trans_tm [] t"
by (induct t rule: tm.induct, auto simp: fresh_Pair)

nominal_function (invariant "λ(xs, _) y. atom  set xs ♯* y")
trans_fm :: "name list ⇒ fm ⇒ dbfm"
where
"trans_fm e (Mem t u) = DBMem (trans_tm e t) (trans_tm e u)"
| "trans_fm e (Eq t u)  = DBEq (trans_tm e t) (trans_tm e u)"
| "trans_fm e (Disj A B) = DBDisj (trans_fm e A) (trans_fm e B)"
| "trans_fm e (Neg A)   = DBNeg (trans_fm e A)"
| "atom k ♯ e ⟹ trans_fm e (Ex k A) = DBEx (trans_fm (k#e) A)"
supply [[simproc del: defined_all]]
apply(erule trans_fm_graph.induct)
using [[simproc del: alpha_lst]]
apply(auto simp: fresh_star_def)
apply(rule_tac y=b and c=a in fm.strong_exhaust)
apply(auto simp: fresh_star_def)
apply(erule_tac c=ea in Abs_lst1_fcb2')
done

nominal_termination (eqvt)
by lexicographic_order

lemma fresh_trans_fm [simp]: "i ♯ trans_fm e A ⟷ i ♯ A ∨ i ∈ atom  set e"
by (nominal_induct A avoiding: e rule: fm.strong_induct, auto simp: fresh_at_base)

abbreviation DBConj :: "dbfm ⇒ dbfm ⇒ dbfm"
where "DBConj t u ≡ DBNeg (DBDisj (DBNeg t) (DBNeg u))"

lemma trans_fm_Conj [simp]: "trans_fm e (Conj A B) = DBConj (trans_fm e A) (trans_fm e B)"

lemma trans_tm_inject [iff]: "(trans_tm e t = trans_tm e u) ⟷ t = u"
proof (induct t arbitrary: e u rule: tm.induct)
case Zero show ?case
apply (cases u rule: tm.exhaust, auto)
apply (metis dbtm.distinct(1) dbtm.distinct(3) lookup_in lookup_notin)
done
next
case (Var i) show ?case
apply (cases u rule: tm.exhaust, auto)
apply (metis dbtm.distinct(1) dbtm.distinct(3) lookup_in lookup_notin)
apply (metis dbtm.distinct(10) dbtm.distinct(11) lookup_in lookup_notin)
done
next
case (Eats tm1 tm2) thus ?case
apply (cases u rule: tm.exhaust, auto)
apply (metis dbtm.distinct(12) dbtm.distinct(9) lookup_in lookup_notin)
done
qed

lemma trans_fm_inject [iff]: "(trans_fm e A = trans_fm e B) ⟷ A = B"
proof (nominal_induct A avoiding: e B rule: fm.strong_induct)
case (Mem tm1 tm2) thus ?case
by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: fresh_star_def)
next
case (Eq tm1 tm2) thus ?case
by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: fresh_star_def)
next
case (Disj fm1 fm2) show ?case
by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: Disj fresh_star_def)
next
case (Neg fm) show ?case
by (rule fm.strong_exhaust [where y=B and c=e]) (auto simp: Neg fresh_star_def)
next
case (Ex name fm)
thus ?case  using [[simproc del: alpha_lst]]
proof (cases rule: fm.strong_exhaust [where y=B and c="(e, name)"], simp_all add: fresh_star_def)
fix name'::name and fm'::fm
assume name': "atom name' ♯ (e, name)"
assume "atom name ♯ fm' ∨ name = name'"
thus "(trans_fm (name # e) fm = trans_fm (name' # e) fm') = ([[atom name]]lst. fm = [[atom name']]lst. fm')"
(is "?lhs = ?rhs")
proof (rule disjE)
assume "name = name'"
thus "?lhs = ?rhs"
by (metis fresh_Pair fresh_at_base(2) name')
next
assume name: "atom name ♯ fm'"
have eq1: "(name ↔ name') ∙ trans_fm (name' # e) fm' = trans_fm (name' # e) fm'"
have eq2: "(name ↔ name') ∙ ([[atom name']]lst. fm') = [[atom name']]lst. fm'"
by (rule flip_fresh_fresh) (auto simp: Abs_fresh_iff name)
show "?lhs = ?rhs" using name' eq1 eq2 Ex(1) Ex(3) [of "name#e" "(name ↔ name') ∙ fm'"]
by (simp add: flip_fresh_fresh) (metis Abs1_eq(3))
qed
qed
qed

lemma trans_fm_perm:
assumes c: "atom c ♯ (i,j,A,B)"
and     t: "trans_fm [i] A = trans_fm [j] B"
shows "(i ↔ c) ∙ A = (j ↔ c) ∙ B"
proof -
have c_fresh1: "atom c ♯ trans_fm [i] A"
using c by (auto simp: supp_Pair)
moreover
have i_fresh: "atom i ♯ trans_fm [i] A"
by auto
moreover
have c_fresh2: "atom c ♯ trans_fm [j] B"
using c by (auto simp: supp_Pair)
moreover
have j_fresh: "atom j ♯ trans_fm [j] B"
by auto
ultimately have "((i ↔ c) ∙ (trans_fm [i] A)) = ((j ↔ c) ∙ trans_fm [j] B)"
by (simp only: flip_fresh_fresh t)
then have "trans_fm [c] ((i ↔ c) ∙ A) = trans_fm [c] ((j ↔ c) ∙ B)"
by simp
then show "(i ↔ c) ∙ A = (j ↔ c) ∙ B" by simp
qed

section‹Characterising the Well-Formed de Bruijn Formulas›

subsection‹Well-Formed Terms›

inductive wf_dbtm :: "dbtm ⇒ bool"
where
Zero:  "wf_dbtm DBZero"
| Var:   "wf_dbtm (DBVar name)"
| Eats:  "wf_dbtm t1 ⟹ wf_dbtm t2 ⟹ wf_dbtm (DBEats t1 t2)"

equivariance wf_dbtm

inductive_cases Zero_wf_dbtm [elim!]: "wf_dbtm DBZero"
inductive_cases Var_wf_dbtm [elim!]:  "wf_dbtm (DBVar name)"
inductive_cases Ind_wf_dbtm [elim!]:  "wf_dbtm (DBInd i)"
inductive_cases Eats_wf_dbtm [elim!]: "wf_dbtm (DBEats t1 t2)"

declare wf_dbtm.intros [intro]

lemma wf_dbtm_imp_is_tm:
assumes "wf_dbtm x"
shows "∃t::tm. x = trans_tm [] t"
using assms
proof (induct rule: wf_dbtm.induct)
case Zero thus ?case
by (metis trans_tm.simps(1))
next
case (Var i) thus ?case
by (metis lookup.simps(1) trans_tm.simps(2))
next
case (Eats dt1 dt2) thus ?case
by (metis trans_tm.simps(3))
qed

lemma wf_dbtm_trans_tm: "wf_dbtm (trans_tm [] t)"
by (induct t rule: tm.induct) auto

theorem wf_dbtm_iff_is_tm: "wf_dbtm x ⟷ (∃t::tm. x = trans_tm [] t)"
by (metis wf_dbtm_imp_is_tm wf_dbtm_trans_tm)

nominal_function abst_dbtm :: "name ⇒ nat ⇒ dbtm ⇒ dbtm"
where
"abst_dbtm name i DBZero = DBZero"
| "abst_dbtm name i (DBVar name') = (if name = name' then DBInd i else DBVar name')"
| "abst_dbtm name i (DBInd j) = DBInd j"
| "abst_dbtm name i (DBEats t1 t2) = DBEats (abst_dbtm name i t1) (abst_dbtm name i t2)"
apply (simp add: eqvt_def abst_dbtm_graph_aux_def, auto)
apply (metis dbtm.exhaust)
done

nominal_termination (eqvt)
by lexicographic_order

nominal_function subst_dbtm :: "dbtm ⇒ name ⇒ dbtm ⇒ dbtm"
where
"subst_dbtm u i DBZero = DBZero"
| "subst_dbtm u i (DBVar name) = (if i = name then u else DBVar name)"
| "subst_dbtm u i (DBInd j) = DBInd j"
| "subst_dbtm u i (DBEats t1 t2) = DBEats (subst_dbtm u i t1) (subst_dbtm u i t2)"
by (auto simp: eqvt_def subst_dbtm_graph_aux_def) (metis dbtm.exhaust)

nominal_termination (eqvt)
by lexicographic_order

lemma fresh_iff_non_subst_dbtm: "subst_dbtm DBZero i t = t ⟷ atom i ♯ t"
by (induct t rule: dbtm.induct) (auto simp: pure_fresh fresh_at_base(2))

lemma lookup_append: "lookup (e @ [i]) n j = abst_dbtm i (length e + n) (lookup e n j)"
by (induct e arbitrary: n) (auto simp: fresh_Cons)

lemma trans_tm_abs: "trans_tm (e@[name]) t = abst_dbtm name (length e) (trans_tm e t)"
by (induct t rule: tm.induct) (auto simp: lookup_notin lookup_append)

subsection‹Well-Formed Formulas›

nominal_function abst_dbfm :: "name ⇒ nat ⇒ dbfm ⇒ dbfm"
where
"abst_dbfm name i (DBMem t1 t2) = DBMem (abst_dbtm name i t1) (abst_dbtm name i t2)"
| "abst_dbfm name i (DBEq t1 t2) =  DBEq (abst_dbtm name i t1) (abst_dbtm name i t2)"
| "abst_dbfm name i (DBDisj A1 A2) = DBDisj (abst_dbfm name i A1) (abst_dbfm name i A2)"
| "abst_dbfm name i (DBNeg A) = DBNeg (abst_dbfm name i A)"
| "abst_dbfm name i (DBEx A) = DBEx (abst_dbfm name (i+1) A)"
apply (simp add: eqvt_def abst_dbfm_graph_aux_def, auto)
apply (metis dbfm.exhaust)
done

nominal_termination (eqvt)
by lexicographic_order

nominal_function subst_dbfm :: "dbtm ⇒ name ⇒ dbfm ⇒ dbfm"
where
"subst_dbfm u i (DBMem t1 t2) = DBMem (subst_dbtm u i t1) (subst_dbtm u i t2)"
| "subst_dbfm u i (DBEq t1 t2) =  DBEq (subst_dbtm u i t1) (subst_dbtm u i t2)"
| "subst_dbfm u i (DBDisj A1 A2) = DBDisj (subst_dbfm u i A1) (subst_dbfm u i A2)"
| "subst_dbfm u i (DBNeg A) = DBNeg (subst_dbfm u i A)"
| "subst_dbfm u i (DBEx A) = DBEx (subst_dbfm u i A)"
by (auto simp: eqvt_def subst_dbfm_graph_aux_def) (metis dbfm.exhaust)

nominal_termination (eqvt)
by lexicographic_order

lemma fresh_iff_non_subst_dbfm: "subst_dbfm DBZero i t = t ⟷ atom i ♯ t"
by (induct t rule: dbfm.induct) (auto simp: fresh_iff_non_subst_dbtm)

section‹Well formed terms and formulas (de Bruijn representation)›

inductive wf_dbfm :: "dbfm ⇒ bool"
where
Mem:   "wf_dbtm t1 ⟹ wf_dbtm t2 ⟹ wf_dbfm (DBMem t1 t2)"
| Eq:    "wf_dbtm t1 ⟹ wf_dbtm t2 ⟹ wf_dbfm (DBEq t1 t2)"
| Disj:  "wf_dbfm A1 ⟹ wf_dbfm A2 ⟹ wf_dbfm (DBDisj A1 A2)"
| Neg:   "wf_dbfm A ⟹ wf_dbfm (DBNeg A)"
| Ex:    "wf_dbfm A ⟹ wf_dbfm (DBEx (abst_dbfm name 0 A))"

equivariance wf_dbfm

lemma atom_fresh_abst_dbtm [simp]: "atom i ♯ abst_dbtm i n t"
by (induct t rule: dbtm.induct) (auto simp: pure_fresh)

lemma atom_fresh_abst_dbfm [simp]: "atom i ♯ abst_dbfm i n A"
by (nominal_induct A arbitrary: n rule: dbfm.strong_induct) auto

text‹Setting up strong induction: "avoiding" for name. Necessary to allow some proofs to go through›
nominal_inductive wf_dbfm
avoids Ex: name
by (auto simp: fresh_star_def)

inductive_cases Mem_wf_dbfm [elim!]:  "wf_dbfm (DBMem t1 t2)"
inductive_cases Eq_wf_dbfm [elim!]:   "wf_dbfm (DBEq t1 t2)"
inductive_cases Disj_wf_dbfm [elim!]: "wf_dbfm (DBDisj A1 A2)"
inductive_cases Neg_wf_dbfm [elim!]:  "wf_dbfm (DBNeg A)"
inductive_cases Ex_wf_dbfm [elim!]:   "wf_dbfm (DBEx z)"

declare wf_dbfm.intros [intro]

lemma trans_fm_abs: "trans_fm (e@[name]) A = abst_dbfm name (length e) (trans_fm e A)"
apply (nominal_induct A avoiding: name e rule: fm.strong_induct)
apply (auto simp: trans_tm_abs fresh_Cons fresh_append)
apply (metis One_nat_def Suc_eq_plus1 append_Cons list.size(4))
done

lemma abst_trans_fm: "abst_dbfm name 0 (trans_fm [] A) = trans_fm [name] A"
by (metis append_Nil list.size(3) trans_fm_abs)

lemma abst_trans_fm2: "i ≠ j ⟹ abst_dbfm i (Suc 0) (trans_fm [j] A) = trans_fm [j,i] A"
using trans_fm_abs [where e="[j]" and name=i]
by auto

lemma wf_dbfm_imp_is_fm:
assumes "wf_dbfm x" shows "∃A::fm. x = trans_fm [] A"
using assms
proof (induct rule: wf_dbfm.induct)
case (Mem t1 t2) thus ?case
by (metis trans_fm.simps(1) wf_dbtm_imp_is_tm)
next
case (Eq t1 t2) thus ?case
by (metis trans_fm.simps(2) wf_dbtm_imp_is_tm)
next
case (Disj fm1 fm2) thus ?case
by (metis trans_fm.simps(3))
next
case (Neg fm) thus ?case
by (metis trans_fm.simps(4))
next
case (Ex fm name) thus ?case
apply auto
apply (rule_tac x="Ex name A" in exI)
apply (auto simp: abst_trans_fm)
done
qed

lemma wf_dbfm_trans_fm: "wf_dbfm (trans_fm [] A)"
apply (nominal_induct A rule: fm.strong_induct)
apply (auto simp: wf_dbtm_trans_tm abst_trans_fm)
apply (metis abst_trans_fm wf_dbfm.Ex)
done

lemma wf_dbfm_iff_is_fm: "wf_dbfm x ⟷ (∃A::fm. x = trans_fm [] A)"
by (metis wf_dbfm_imp_is_fm wf_dbfm_trans_fm)

lemma dbtm_abst_ignore [simp]:
"abst_dbtm name i (abst_dbtm name j t) = abst_dbtm name j t"
by (induct t rule: dbtm.induct) auto

lemma abst_dbtm_fresh_ignore [simp]: "atom name ♯ u ⟹ abst_dbtm name j u = u"
by (induct u rule: dbtm.induct) auto

lemma dbtm_subst_ignore [simp]:
"subst_dbtm u name (abst_dbtm name j t) = abst_dbtm name j t"
by (induct t rule: dbtm.induct) auto

lemma dbtm_abst_swap_subst:
"name ≠ name' ⟹ atom name' ♯ u ⟹
subst_dbtm u name (abst_dbtm name' j t) = abst_dbtm name' j (subst_dbtm u name t)"
by (induct t rule: dbtm.induct) auto

lemma dbfm_abst_swap_subst:
"name ≠ name' ⟹ atom name' ♯ u ⟹
subst_dbfm u name (abst_dbfm name' j A) = abst_dbfm name' j (subst_dbfm u name A)"
by (induct A arbitrary: j rule: dbfm.induct) (auto simp: dbtm_abst_swap_subst)

lemma subst_trans_commute [simp]:
"atom i ♯ e ⟹ subst_dbtm (trans_tm e u) i (trans_tm e t) = trans_tm e (subst i u t)"
apply (induct t rule: tm.induct)
apply (auto simp: lookup_notin fresh_imp_notin_env)
apply (metis abst_dbtm_fresh_ignore dbtm_subst_ignore lookup_fresh lookup_notin subst_dbtm.simps(2))
done

lemma subst_fm_trans_commute [simp]:
"subst_dbfm (trans_tm [] u) name (trans_fm [] A) = trans_fm [] (A (name::= u))"
apply (nominal_induct A avoiding: name u rule: fm.strong_induct)
apply (auto simp: lookup_notin abst_trans_fm [symmetric])
apply (metis dbfm_abst_swap_subst fresh_at_base(2) fresh_trans_tm_iff)
done

lemma subst_fm_trans_commute_eq:
"du = trans_tm [] u ⟹ subst_dbfm du i (trans_fm [] A) = trans_fm [] (A(i::=u))"
by (metis subst_fm_trans_commute)

section‹Quotations›

fun htuple :: "nat ⇒ hf"  where
"htuple 0 = ⟨0,0⟩"
| "htuple (Suc k) = ⟨0, htuple k⟩"

fun HTuple :: "nat ⇒ tm"  where
"HTuple 0 = HPair Zero Zero"
| "HTuple (Suc k) = HPair Zero (HTuple k)"

lemma eval_tm_HTuple [simp]:  "⟦HTuple n⟧e = htuple n"
by (induct n) auto

lemma fresh_HTuple [simp]: "x ♯ HTuple n"
by (induct n) auto

lemma HTuple_eqvt[eqvt]: "(p ∙ HTuple n) = HTuple (p ∙ n)"
by (induct n, auto simp: HPair_eqvt permute_pure)

lemma htuple_nonzero [simp]: "htuple k ≠ 0"
by (induct k) auto

lemma htuple_inject [iff]: "htuple i = htuple j ⟷ i=j"
proof (induct i arbitrary: j)
case 0 show ?case
by (cases j) auto
next
case (Suc i) show ?case
by (cases j) (auto simp: Suc)
qed

subsection ‹Quotations of de Bruijn terms›

definition nat_of_name :: "name ⇒ nat"
where "nat_of_name x = nat_of (atom x)"

lemma nat_of_name_inject [simp]: "nat_of_name n1 = nat_of_name n2 ⟷ n1 = n2"
by (metis nat_of_name_def atom_components_eq_iff atom_eq_iff sort_of_atom_eq)

definition name_of_nat :: "nat ⇒ name"
where "name_of_nat n ≡ Abs_name (Atom (Sort ''SyntaxN.name'' []) n)"

lemma nat_of_name_Abs_eq [simp]: "nat_of_name (Abs_name (Atom (Sort ''SyntaxN.name'' []) n)) = n"
by (auto simp: nat_of_name_def atom_name_def Abs_name_inverse)

lemma nat_of_name_name_eq [simp]: "nat_of_name (name_of_nat n) = n"

lemma name_of_nat_nat_of_name [simp]: "name_of_nat (nat_of_name i) = i"
by (metis nat_of_name_inject nat_of_name_name_eq)

lemma HPair_neq_ORD_OF [simp]: "HPair x y ≠ ORD_OF i"
by (metis Not_Ord_hpair Ord_ord_of eval_tm_HPair eval_tm_ORD_OF)

text‹Infinite support, so we cannot use nominal primrec.›
function quot_dbtm :: "dbtm ⇒ tm"
where
"quot_dbtm DBZero = Zero"
| "quot_dbtm (DBVar name) = ORD_OF (Suc (nat_of_name name))"
| "quot_dbtm (DBInd k) = HPair (HTuple 6) (ORD_OF k)"
| "quot_dbtm (DBEats t u) = HPair (HTuple 1) (HPair (quot_dbtm t) (quot_dbtm u))"
by (rule dbtm.exhaust) auto

termination
by lexicographic_order

lemma quot_dbtm_inject_lemma [simp]: "⟦quot_dbtm t⟧e = ⟦quot_dbtm u⟧e ⟷ t=u"
proof (induct t arbitrary: u rule: dbtm.induct)
case DBZero show ?case
by (induct u rule: dbtm.induct) auto
next
case (DBVar name) show ?case
by (induct u rule: dbtm.induct) (auto simp: hpair_neq_Ord')
next
case (DBInd k) show ?case
by (induct u rule: dbtm.induct) (auto simp: hpair_neq_Ord hpair_neq_Ord')
next
case (DBEats t1 t2) thus ?case
by (induct u rule: dbtm.induct) (simp_all add: hpair_neq_Ord)
qed

lemma quot_dbtm_inject [iff]: "quot_dbtm t = quot_dbtm u ⟷ t=u"
by (metis quot_dbtm_inject_lemma)

subsection ‹Quotations of de Bruijn formulas›

text‹Infinite support, so we cannot use nominal primrec.›
function quot_dbfm :: "dbfm ⇒ tm"
where
"quot_dbfm (DBMem t u) = HPair (HTuple 0) (HPair (quot_dbtm t) (quot_dbtm u))"
| "quot_dbfm (DBEq t u) = HPair (HTuple 2) (HPair (quot_dbtm t) (quot_dbtm u))"
| "quot_dbfm (DBDisj A B) = HPair (HTuple 3) (HPair (quot_dbfm A) (quot_dbfm B))"
| "quot_dbfm (DBNeg A) = HPair (HTuple 4) (quot_dbfm A)"
| "quot_dbfm (DBEx A) = HPair (HTuple 5) (quot_dbfm A)"
by (rule_tac y=x in dbfm.exhaust, auto)

termination
by lexicographic_order

lemma htuple_minus_1: "n > 0 ⟹ htuple n = ⟨0, htuple (n - 1)⟩"
by (metis Suc_diff_1 htuple.simps(2))

lemma HTuple_minus_1: "n > 0 ⟹ HTuple n = HPair Zero (HTuple (n - 1))"
by (metis Suc_diff_1 HTuple.simps(2))

lemma quot_dbfm_inject_lemma [simp]: "⟦quot_dbfm A⟧e = ⟦quot_dbfm B⟧e ⟷ A=B"
proof (induct A arbitrary: B rule: dbfm.induct)
case (DBMem t u) show ?case
by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
next
case (DBEq t u) show ?case
by (induct B rule: dbfm.induct) (auto simp: htuple_minus_1)
next
case (DBDisj A B') thus ?case
by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
next
case (DBNeg A) thus ?case
by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
next
case (DBEx A) thus ?case
by (induct B rule: dbfm.induct) (simp_all add: htuple_minus_1)
qed

class quot =
fixes quot :: "'a ⇒ tm"  ("«_»")

instantiation tm :: quot
begin
definition quot_tm :: "tm ⇒ tm"
where "quot_tm t = quot_dbtm (trans_tm [] t)"

instance ..
end

lemma quot_dbtm_fresh [simp]: "s ♯ (quot_dbtm t)"
by (induct t rule: dbtm.induct) auto

lemma quot_tm_fresh [simp]: fixes t::tm shows "s ♯ «t»"

lemma quot_Zero [simp]: "«Zero» = Zero"

lemma quot_Var: "«Var x» = SUCC (ORD_OF (nat_of_name x))"

lemma quot_Eats: "«Eats x y» = HPair (HTuple 1) (HPair «x» «y»)"

text‹irrelevance of the environment for quotations, because they are ground terms›
lemma eval_quot_dbtm_ignore:
"⟦quot_dbtm t⟧e = ⟦quot_dbtm t⟧e'"
by (induct t rule: dbtm.induct) auto

lemma eval_quot_dbfm_ignore:
"⟦quot_dbfm A⟧e = ⟦quot_dbfm A⟧e'"
by (induct A rule: dbfm.induct) (auto intro: eval_quot_dbtm_ignore)

instantiation fm :: quot
begin
definition quot_fm :: "fm ⇒ tm"
where "quot_fm A = quot_dbfm (trans_fm [] A)"

instance ..
end

lemma quot_dbfm_fresh [simp]: "s ♯ (quot_dbfm A)"
by (induct A rule: dbfm.induct) auto

lemma quot_fm_fresh [simp]: fixes A::fm shows "s ♯ «A»"

lemma quot_fm_permute [simp]: fixes A:: fm shows "p ∙ «A» = «A»"
by (metis fresh_star_def perm_supp_eq quot_fm_fresh)

lemma quot_Mem: "«x IN y» = HPair (HTuple 0) (HPair («x») («y»))"

lemma quot_Eq: "«x EQ y» = HPair (HTuple 2) (HPair («x») («y»))"

lemma quot_Disj: "«A OR B» = HPair (HTuple 3) (HPair («A») («B»))"

lemma quot_Neg: "«Neg A» = HPair (HTuple 4) («A»)"

lemma quot_Ex: "«Ex i A» = HPair (HTuple 5) (quot_dbfm (trans_fm [i] A))"

lemma eval_quot_fm_ignore: fixes A:: fm shows "⟦«A»⟧e = ⟦«A»⟧e'"
by (metis eval_quot_dbfm_ignore quot_fm_def)

lemmas quot_simps = quot_Var quot_Eats quot_Eq quot_Mem quot_Disj quot_Neg quot_Ex

section‹Definitions Involving Coding›

definition q_Var :: "name ⇒ hf"
where "q_Var i ≡ succ (ord_of (nat_of_name i))"

definition q_Ind :: "hf ⇒ hf"
where "q_Ind k ≡  ⟨htuple 6, k⟩"

abbreviation Q_Eats :: "tm ⇒ tm ⇒ tm"
where "Q_Eats t u ≡ HPair (HTuple (Suc 0)) (HPair t u)"

definition q_Eats :: "hf ⇒ hf ⇒ hf"
where "q_Eats x y ≡ ⟨htuple 1, x, y⟩"

abbreviation Q_Succ :: "tm ⇒ tm"
where "Q_Succ t ≡ Q_Eats t t"

definition q_Succ :: "hf ⇒ hf"
where "q_Succ x ≡ q_Eats x x"

lemma quot_Succ: "«SUCC x» = Q_Succ «x»"
by (auto simp: SUCC_def quot_Eats)

abbreviation Q_HPair :: "tm ⇒ tm ⇒ tm"
where "Q_HPair t u ≡
Q_Eats (Q_Eats Zero (Q_Eats (Q_Eats Zero u) t))
(Q_Eats (Q_Eats Zero t) t)"

definition q_HPair :: "hf ⇒ hf ⇒ hf"
where "q_HPair x y ≡
q_Eats (q_Eats 0 (q_Eats (q_Eats 0 y) x))
(q_Eats (q_Eats 0 x) x)"

abbreviation Q_Mem :: "tm ⇒ tm ⇒ tm"
where "Q_Mem t u ≡ HPair (HTuple 0) (HPair t u)"

definition q_Mem :: "hf ⇒ hf ⇒ hf"
where "q_Mem x y ≡ ⟨htuple 0, x, y⟩"

abbreviation Q_Eq :: "tm ⇒ tm ⇒ tm"
where "Q_Eq t u ≡ HPair (HTuple 2) (HPair t u)"

definition q_Eq :: "hf ⇒ hf ⇒ hf"
where "q_Eq x y ≡ ⟨htuple 2, x, y⟩"

abbreviation Q_Disj :: "tm ⇒ tm ⇒ tm"
where "Q_Disj t u ≡ HPair (HTuple 3) (HPair t u)"

definition q_Disj :: "hf ⇒ hf ⇒ hf"
where "q_Disj x y ≡ ⟨htuple 3, x, y⟩"

abbreviation Q_Neg :: "tm ⇒ tm"
where "Q_Neg t ≡ HPair (HTuple 4) t"

definition q_Neg :: "hf ⇒ hf"
where "q_Neg x ≡ ⟨htuple 4, x⟩"

abbreviation Q_Conj :: "tm ⇒ tm ⇒ tm"
where "Q_Conj t u ≡ Q_Neg (Q_Disj (Q_Neg t) (Q_Neg u))"

definition q_Conj :: "hf ⇒ hf ⇒ hf"
where "q_Conj t u ≡ q_Neg (q_Disj (q_Neg t) (q_Neg u))"

abbreviation Q_Imp :: "tm ⇒ tm ⇒ tm"
where "Q_Imp t u ≡ Q_Disj (Q_Neg t) u"

definition q_Imp :: "hf ⇒ hf ⇒ hf"
where "q_Imp t u ≡ q_Disj (q_Neg t) u"

abbreviation Q_Ex :: "tm ⇒ tm"
where "Q_Ex t ≡ HPair (HTuple 5) t"

definition q_Ex :: "hf ⇒ hf"
where "q_Ex x ≡ ⟨htuple 5, x⟩"

abbreviation Q_All :: "tm ⇒ tm"
where "Q_All t ≡ Q_Neg (Q_Ex (Q_Neg t))"

definition q_All :: "hf ⇒ hf"
where "q_All x ≡ q_Neg (q_Ex (q_Neg x))"

lemmas q_defs = q_Var_def q_Ind_def q_Eats_def q_HPair_def q_Eq_def q_Mem_def
q_Disj_def q_Neg_def q_Conj_def q_Imp_def q_Ex_def q_All_def

lemma q_Eats_iff [iff]: "q_Eats x y = q_Eats x' y' ⟷ x=x' ∧ y=y'"
by (metis hpair_iff q_Eats_def)

lemma quot_subst_eq: "«A(i::=t)» = quot_dbfm (subst_dbfm (trans_tm [] t) i (trans_fm [] A))"
by (metis quot_fm_def subst_fm_trans_commute)

lemma Q_Succ_cong: "H ⊢ x EQ x' ⟹ H ⊢ Q_Succ x EQ Q_Succ x'"
by (metis HPair_cong Refl)

section‹Quotations are Injective›

subsection‹Terms›

lemma eval_tm_inject [simp]: fixes t::tm shows "⟦«t»⟧ e = ⟦«u»⟧ e ⟷ t=u"
proof (induct t arbitrary: u rule: tm.induct)
case Zero thus ?case
by (cases u rule: tm.exhaust) (auto simp: quot_Var quot_Eats)
next
case (Var i) thus ?case
apply (cases u rule: tm.exhaust, auto)
apply (auto simp: quot_Var quot_Eats)
done
next
case (Eats t1 t2) thus ?case
apply (cases u rule: tm.exhaust, auto)
apply (auto simp: quot_Eats quot_Var)
done
qed

subsection‹Formulas›

lemma eval_fm_inject [simp]: fixes A::fm shows "⟦«A»⟧ e = ⟦«B»⟧ e ⟷ A=B"
proof (nominal_induct B arbitrary: A rule: fm.strong_induct)
case (Mem tm1 tm2) thus ?case
by (cases A rule: fm.exhaust, auto simp: quot_simps htuple_minus_1)
next
case (Eq tm1 tm2) thus ?case
by (cases A rule: fm.exhaust, auto simp: quot_simps htuple_minus_1)
next
case (Neg α) thus ?case
by (cases A rule: fm.exhaust, auto simp: quot_simps htuple_minus_1)
next
case (Disj fm1 fm2)
thus ?case
by (cases A rule: fm.exhaust, auto simp: quot_simps htuple_minus_1)
next
case (Ex i α)
thus ?case
apply (induct A arbitrary: i rule: fm.induct)
apply (auto simp: trans_fm_perm quot_simps htuple_minus_1 Abs1_eq_iff_all)
by (metis (no_types) Abs1_eq_iff_all(3) dbfm.eq_iff(5) fm.eq_iff(5) fresh_Nil trans_fm.simps(5))
qed

subsection‹The set ‹Γ› of Definition 1.1, constant terms used for coding›

inductive coding_tm :: "tm ⇒ bool"
where
Ord:    "∃i. x = ORD_OF i ⟹ coding_tm x"
| HPair:  "coding_tm x ⟹ coding_tm y ⟹ coding_tm (HPair x y)"

declare coding_tm.intros [intro]

lemma coding_tm_Zero [intro]: "coding_tm Zero"
by (metis ORD_OF.simps(1) Ord)

lemma coding_tm_HTuple [intro]: "coding_tm (HTuple k)"
by (induct k, auto)

inductive_simps coding_tm_HPair [simp]: "coding_tm (HPair x y)"

lemma quot_dbtm_coding [simp]: "coding_tm (quot_dbtm t)"
apply (induct t rule: dbtm.induct, auto)
apply (metis ORD_OF.simps(2) Ord)
done

lemma quot_dbfm_coding [simp]: "coding_tm (quot_dbfm fm)"
by (induct fm rule: dbfm.induct, auto)

lemma quot_fm_coding: fixes A::fm shows "coding_tm «A»"
by (metis quot_dbfm_coding quot_fm_def)

inductive coding_hf :: "hf ⇒ bool"
where
Ord:    "∃i. x = ord_of i ⟹ coding_hf x"
| HPair:  "coding_hf x ⟹ coding_hf y ⟹ coding_hf (⟨x,y⟩)"

declare coding_hf.intros [intro]

lemma coding_hf_0 [intro]: "coding_hf 0"
by (metis coding_hf.Ord ord_of.simps(1))

inductive_simps coding_hf_hpair [simp]: "coding_hf (⟨x,y⟩)"

lemma coding_tm_hf [simp]: "coding_tm t ⟹ coding_hf ⟦t⟧e"
by (induct t rule: coding_tm.induct) auto

section ‹V-Coding for terms and formulas, for the Second Theorem›

text‹Infinite support, so we cannot use nominal primrec.›
function vquot_dbtm :: "name set ⇒ dbtm ⇒ tm"
where
"vquot_dbtm V DBZero = Zero"
| "vquot_dbtm V (DBVar name) = (if name ∈ V then Var name
else ORD_OF (Suc (nat_of_name name)))"
| "vquot_dbtm V (DBInd k) = HPair (HTuple 6) (ORD_OF k)"
| "vquot_dbtm V (DBEats t u) = HPair (HTuple 1) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
by (auto, rule_tac y=b in dbtm.exhaust, auto)

termination
by lexicographic_order

lemma fresh_vquot_dbtm [simp]: "i ♯ vquot_dbtm V tm ⟷ i ♯ tm ∨ i ∉ atom  V"
by (induct tm rule: dbtm.induct) (auto simp: fresh_at_base pure_fresh)

text‹Infinite support, so we cannot use nominal primrec.›
function vquot_dbfm :: "name set ⇒ dbfm ⇒ tm"
where
"vquot_dbfm V (DBMem t u) = HPair (HTuple 0) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
| "vquot_dbfm V (DBEq t u) = HPair (HTuple 2) (HPair (vquot_dbtm V t) (vquot_dbtm V u))"
| "vquot_dbfm V (DBDisj A B) = HPair (HTuple 3) (HPair (vquot_dbfm V A) (vquot_dbfm V B))"
| "vquot_dbfm V (DBNeg A) = HPair (HTuple 4) (vquot_dbfm V A)"
| "vquot_dbfm V (DBEx A) = HPair (HTuple 5) (vquot_dbfm V A)"
by (auto, rule_tac y=b in dbfm.exhaust, auto)

termination
by lexicographic_order

lemma fresh_vquot_dbfm [simp]: "i ♯ vquot_dbfm V fm ⟷ i ♯ fm ∨ i ∉ atom  V"
by (induct fm rule: dbfm.induct) (auto simp: HPair_def HTuple_minus_1)

class vquot =
fixes vquot :: "'a ⇒ name set ⇒ tm"  ("⌊_⌋_"  [0,1000]1000)

instantiation tm :: vquot
begin
definition vquot_tm :: "tm ⇒ name set ⇒ tm"
where "vquot_tm t V = vquot_dbtm V (trans_tm [] t)"
instance ..
end

lemma vquot_dbtm_empty [simp]: "vquot_dbtm {} t = quot_dbtm t"
by (induct t rule: dbtm.induct) auto

lemma vquot_tm_empty [simp]: fixes t::tm shows "⌊t⌋{} = «t»"

lemma vquot_dbtm_eq: "atom  V ∩ supp t = atom  W ∩ supp t ⟹ vquot_dbtm V t = vquot_dbtm W t"
by (induct t rule: dbtm.induct) (auto simp: image_iff, blast+)

instantiation fm :: vquot
begin
definition vquot_fm :: "fm ⇒ name set ⇒ tm"
where "vquot_fm A V = vquot_dbfm V (trans_fm [] A)"
instance ..
end

lemma vquot_fm_fresh [simp]: fixes A::fm shows "i ♯ ⌊A⌋V ⟷ i ♯ A ∨ i ∉ atom  V"

lemma vquot_dbfm_empty [simp]: "vquot_dbfm {} A = quot_dbfm A"
by (induct A rule: dbfm.induct) auto

lemma vquot_fm_empty [simp]: fixes A::fm shows "⌊A⌋{} = «A»"

lemma vquot_dbfm_eq: "atom  V ∩ supp A = atom  W ∩ supp A ⟹ vquot_dbfm V A = vquot_dbfm W A"
by (induct A rule: dbfm.induct) (auto simp: intro!: vquot_dbtm_eq, blast+)

lemma vquot_fm_insert:
fixes A::fm shows "atom i ∉ supp A ⟹ ⌊A⌋(insert i V) = ⌊A⌋V"
by (auto simp: vquot_fm_def supp_conv_fresh intro: vquot_dbfm_eq)

declare HTuple.simps [simp del]

end


# Theory Predicates

chapter‹Basic Predicates›

theory Predicates
imports SyntaxN
begin

section ‹The Subset Relation›

nominal_function Subset :: "tm ⇒ tm ⇒ fm"   (infixr "SUBS" 150)
where "atom z ♯ (t, u) ⟹ t SUBS u = All2 z t ((Var z) IN u)"
by (auto simp: eqvt_def Subset_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)

nominal_termination (eqvt)
by lexicographic_order

declare Subset.simps [simp del]

lemma Subset_fresh_iff [simp]: "a ♯ t SUBS u ⟷ a ♯ t ∧ a ♯ u"
apply (rule obtain_fresh [where x="(t, u)"])
apply (subst Subset.simps, auto)
done

lemma eval_fm_Subset [simp]: "eval_fm e (Subset t u) ⟷ (⟦t⟧e ≤ ⟦u⟧e)"
apply (rule obtain_fresh [where x="(t, u)"])
apply (subst Subset.simps, auto)
done

lemma subst_fm_Subset [simp]: "(t SUBS u)(i::=x) = (subst i x t) SUBS (subst i x u)"
proof -
obtain j::name where "atom j ♯ (i,x,t,u)"
by (rule obtain_fresh)
thus ?thesis
by (auto simp: Subset.simps [of j])
qed

lemma Subset_I:
assumes "insert ((Var i) IN t) H ⊢ (Var i) IN u" "atom i ♯ (t,u)" "∀B ∈ H. atom i ♯ B"
shows "H ⊢ t SUBS u"
by (subst Subset.simps [of i]) (auto simp: assms)

lemma Subset_D:
assumes major: "H ⊢ t SUBS u" and minor: "H ⊢ a IN t" shows "H ⊢ a IN u"
proof -
obtain i::name where i: "atom i ♯ (t, u)"
by (rule obtain_fresh)
hence "H ⊢ (Var i IN t IMP Var i IN u) (i::=a)"
by (metis Subset.simps major All_D)
thus ?thesis
using i  by simp (metis MP_same minor)
qed

lemma Subset_E: "H ⊢ t SUBS u ⟹ H ⊢ a IN t ⟹ insert (a IN u) H ⊢ A ⟹ H ⊢ A"
by (metis Subset_D cut_same)

lemma Subset_cong: "H ⊢ t EQ t' ⟹ H ⊢ u EQ u' ⟹ H ⊢ t SUBS u IFF t' SUBS u'"
by (rule P2_cong) auto

lemma Set_MP: "x SUBS y ∈ H ⟹ z IN x ∈ H ⟹ insert (z IN y) H ⊢ A ⟹ H ⊢ A"
by (metis Assume Subset_D cut_same insert_absorb)

lemma Zero_Subset_I [intro!]: "H ⊢ Zero SUBS t"
proof -
have "{} ⊢ Zero SUBS t"
by (rule obtain_fresh [where x="(Zero,t)"]) (auto intro: Subset_I)
thus ?thesis
by (auto intro: thin)
qed

lemma Zero_SubsetE: "H ⊢ A ⟹ insert (Zero SUBS X) H ⊢ A"
by (rule thin1)

lemma Subset_Zero_D:
assumes "H ⊢ t SUBS Zero" shows "H ⊢ t EQ Zero"
proof -
obtain i::name where i [iff]: "atom i ♯ t"
by (rule obtain_fresh)
have "{t SUBS Zero} ⊢ t EQ Zero"
proof (rule Eq_Zero_I)
fix A
show "{Var i IN t, t SUBS Zero} ⊢ A"
by (metis Hyp Subset_D insertI1 thin1 Mem_Zero_E cut1)
qed auto
thus ?thesis
by (metis assms cut1)
qed

lemma Subset_refl: "H ⊢ t SUBS t"
proof -
obtain i::name where "atom i ♯ t"
by (rule obtain_fresh)
thus ?thesis
by (metis Assume Subset_I empty_iff fresh_Pair thin0)
qed

lemma Eats_Subset_Iff: "H ⊢ Eats x y SUBS z IFF (x SUBS z) AND (y IN z)"
proof -
obtain i::name where i: "atom i ♯ (x,y,z)"
by (rule obtain_fresh)
have "{} ⊢ (Eats x y SUBS z) IFF (x SUBS z AND y IN z)"
proof (rule Iff_I)
show "{Eats x y SUBS z} ⊢ x SUBS z AND y IN z"
proof (rule Conj_I)
show "{Eats x y SUBS z} ⊢ x SUBS z"
apply (rule Subset_I [where i=i]) using i
apply (auto intro: Subset_D Mem_Eats_I1)
done
next
show "{Eats x y SUBS z} ⊢ y IN z"
by (metis Subset_D Assume Mem_Eats_I2 Refl)
qed
next
show "{x SUBS z AND y IN z} ⊢ Eats x y SUBS z" using i
by (auto intro!: Subset_I [where i=i] intro: Subset_D Mem_cong [THEN Iff_MP2_same])
qed
thus ?thesis
by (rule thin0)
qed

lemma Eats_Subset_I [intro!]: "H ⊢ x SUBS z ⟹ H ⊢ y IN z ⟹ H ⊢ Eats x y SUBS z"
by (metis Conj_I Eats_Subset_Iff Iff_MP2_same)

lemma Eats_Subset_E [intro!]:
"insert (x SUBS z) (insert (y IN z) H) ⊢ C ⟹ insert (Eats x y SUBS z) H ⊢ C"
by (metis Conj_E Eats_Subset_Iff Iff_MP_left')

text‹A surprising proof: a consequence of @{thm Eats_Subset_Iff} and reflexivity!›
lemma Subset_Eats_I [intro!]: "H ⊢ x SUBS Eats x y"
by (metis Conj_E1 Eats_Subset_Iff Iff_MP_same Subset_refl)

lemma SUCC_Subset_I [intro!]: "H ⊢ x SUBS z ⟹ H ⊢ x IN z ⟹ H ⊢ SUCC x SUBS z"
by (metis Eats_Subset_I SUCC_def)

lemma SUCC_Subset_E [intro!]:
"insert (x SUBS z) (insert (x IN z) H) ⊢ C ⟹ insert (SUCC x SUBS z) H ⊢ C"
by (metis Eats_Subset_E SUCC_def)

lemma Subset_trans0: "{ a SUBS b, b SUBS c } ⊢ a SUBS c"
proof -
obtain i::name where [simp]: "atom i ♯ (a,b,c)"
by (rule obtain_fresh)
show ?thesis
by (rule Subset_I [of i]) (auto intro: Subset_D)
qed

lemma Subset_trans: "H ⊢ a SUBS b ⟹ H ⊢ b SUBS c ⟹ H ⊢ a SUBS c"
by (metis Subset_trans0 cut2)

lemma Subset_SUCC: "H ⊢ a SUBS (SUCC a)"
by (metis SUCC_def Subset_Eats_I)

lemma All2_Subset_lemma: "atom l ♯ (k',k) ⟹ {P} ⊢ P' ⟹ {All2 l k P, k' SUBS k} ⊢ All2 l k' P'"
apply auto
apply (rule Ex_I [where x = "Var l"])
apply (auto intro: ContraProve Set_MP cut1)
done

lemma All2_Subset: "⟦H ⊢ All2 l k P; H ⊢ k' SUBS k; {P} ⊢ P'; atom l ♯ (k', k)⟧ ⟹ H ⊢ All2 l k' P'"
by (rule cut2 [OF All2_Subset_lemma]) auto

section‹Extensionality›

lemma Extensionality: "H ⊢ x EQ y IFF x SUBS y AND y SUBS x"
proof -
obtain i::name and j::name and k::name
where atoms: "atom i ♯ (x,y)"  "atom j ♯ (i,x,y)"  "atom k ♯ (i,j,y)"
by (metis obtain_fresh)
have "{} ⊢ (Var i EQ y IFF Var i SUBS y AND y SUBS Var i)" (is "{} ⊢ ?scheme")
proof (rule Ind [of j])
show "atom j ♯ (i, ?scheme)" using atoms
by simp
next
show "{} ⊢ ?scheme(i::=Zero)" using atoms
proof auto
show "{Zero EQ y} ⊢ y SUBS Zero"
by (rule Subset_cong [OF Assume Refl, THEN Iff_MP_same]) (rule Subset_refl)
next
show "{Zero SUBS y, y SUBS Zero} ⊢ Zero EQ y"
by (metis AssumeH(2) Subset_Zero_D Sym)
qed
next
show "{} ⊢ All i (All j (?scheme IMP ?scheme(i::=Var j) IMP ?scheme(i::=Eats (Var i) (Var j))))"
using atoms
apply auto
apply (metis Subset_cong [OF Refl Assume, THEN Iff_MP_same] Subset_Eats_I)
apply (metis Mem_cong [OF Refl Assume, THEN Iff_MP_same] Mem_Eats_I2 Refl)
apply (metis Subset_cong [OF Assume Refl, THEN Iff_MP_same] Subset_refl)
apply (rule Eq_Eats_I [of _ k, THEN Sym])
apply (auto intro: Set_MP [where x=y] Subset_D [where t = "Var i"] Disj_I1 Disj_I2)
apply (rule Var_Eq_subst_Iff [THEN Iff_MP_same], auto)
done
qed
hence "{} ⊢ (Var i EQ y IFF Var i SUBS y AND y SUBS Var i)(i::=x)"
by (metis Subst emptyE)
thus ?thesis using atoms
qed

lemma Equality_I: "H ⊢ y SUBS x ⟹ H ⊢ x SUBS y ⟹ H ⊢ x EQ y"
by (metis Conj_I Extensionality Iff_MP2_same)

lemma EQ_imp_SUBS: "insert (t EQ u) H ⊢ (t SUBS u)"
proof -
have "{t EQ u} ⊢ (t SUBS u)"
by (metis Assume Conj_E Extensionality Iff_MP_left')
thus ?thesis
by (metis Assume cut1)
qed

lemma EQ_imp_SUBS2: "insert (u EQ t) H ⊢ (t SUBS u)"
by (metis EQ_imp_SUBS Sym_L)

lemma Equality_E: "insert (t SUBS u) (insert (u SUBS t) H) ⊢ A ⟹ insert (t EQ u) H ⊢ A"
by (metis Conj_E Extensionality Iff_MP_left')

section ‹The Disjointness Relation›

text‹The following predicate is defined in order to prove Lemma 2.3, Foundation›

nominal_function Disjoint :: "tm ⇒ tm ⇒ fm"
where "atom z ♯ (t, u) ⟹ Disjoint t u = All2 z t (Neg ((Var z) IN u))"
by (auto simp: eqvt_def Disjoint_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)

nominal_termination (eqvt)
by lexicographic_order

declare Disjoint.simps [simp del]

lemma Disjoint_fresh_iff [simp]: "a ♯ Disjoint t u ⟷ a ♯ t ∧ a ♯ u"
proof -
obtain j::name where j: "atom j ♯ (a,t,u)"
by (rule obtain_fresh)
thus ?thesis
by (auto simp: Disjoint.simps [of j])
qed

lemma subst_fm_Disjoint [simp]:
"(Disjoint t u)(i::=x) = Disjoint (subst i x t) (subst i x u)"
proof -
obtain j::name where j: "atom j ♯ (i,x,t,u)"
by (rule obtain_fresh)
thus ?thesis
by (auto simp: Disjoint.simps [of j])
qed

lemma Disjoint_cong: "H ⊢ t EQ t' ⟹ H ⊢ u EQ u' ⟹ H ⊢ Disjoint t u IFF Disjoint t' u'"
by (rule P2_cong) auto

lemma Disjoint_I:
assumes "insert ((Var i) IN t) (insert ((Var i) IN u) H) ⊢ Fls"
"atom i ♯ (t,u)" "∀B ∈ H. atom i ♯ B"
shows "H ⊢ Disjoint t u"
by (subst Disjoint.simps [of i]) (auto simp: assms insert_commute)

lemma Disjoint_E:
assumes major: "H ⊢ Disjoint t u" and minor: "H ⊢ a IN t" "H ⊢ a IN u" shows "H ⊢ A"
proof -
obtain i::name where i: "atom i ♯ (t, u)"
by (rule obtain_fresh)
hence "H ⊢ (Var i IN t IMP Neg (Var i IN u)) (i::=a)"
by (metis Disjoint.simps major All_D)
thus ?thesis using i
by simp (metis MP_same Neg_D minor)
qed

lemma Disjoint_commute: "{ Disjoint t u } ⊢ Disjoint u t"
proof -
obtain i::name where "atom i ♯ (t,u)"
by (rule obtain_fresh)
thus ?thesis
by (auto simp: fresh_Pair intro: Disjoint_I Disjoint_E)
qed

lemma Disjoint_commute_I: "H ⊢ Disjoint t u ⟹ H ⊢ Disjoint u t"
by (metis Disjoint_commute cut1)

lemma Disjoint_commute_D: "insert (Disjoint t u) H ⊢ A ⟹ insert (Disjoint u t) H ⊢ A"
by (metis Assume Disjoint_commute_I cut_same insert_commute thin1)

lemma Zero_Disjoint_I1 [iff]: "H ⊢ Disjoint Zero t"
proof -
obtain i::name where i: "atom i ♯ t"
by (rule obtain_fresh)
hence "{} ⊢ Disjoint Zero t"
by (auto intro: Disjoint_I [of i])
thus ?thesis
by (metis thin0)
qed

lemma Zero_Disjoint_I2 [iff]: "H ⊢ Disjoint t Zero"
by (metis Disjoint_commute Zero_Disjoint_I1 cut1)

lemma Disjoint_Eats_D1: "{ Disjoint (Eats x y) z } ⊢ Disjoint x z"
proof -
obtain i::name where i: "atom i ♯ (x,y,z)"
by (rule obtain_fresh)
show ?thesis
apply (rule Disjoint_I [of i])
apply (blast intro: Disjoint_E Mem_Eats_I1)
using i apply auto
done
qed

lemma Disjoint_Eats_D2: "{ Disjoint (Eats x y) z } ⊢ Neg(y IN z)"
proof -
obtain i::name where i: "atom i ♯ (x,y,z)"
by (rule obtain_fresh)
show ?thesis
by (force intro: Disjoint_E [THEN rotate2] Mem_Eats_I2)
qed

lemma Disjoint_Eats_E:
"insert (Disjoint x z) (insert (Neg(y IN z)) H) ⊢ A ⟹ insert (Disjoint (Eats x y) z) H ⊢ A"
apply (rule cut_same [OF cut1 [OF Disjoint_Eats_D2, OF Assume]])
apply (rule cut_same [OF cut1 [OF Disjoint_Eats_D1, OF Hyp]])
apply (auto intro: thin)
done

lemma Disjoint_Eats_E2:
"insert (Disjoint z x) (insert (Neg(y IN z)) H) ⊢ A ⟹ insert (Disjoint z (Eats x y)) H ⊢ A"
by (metis Disjoint_Eats_E Disjoint_commute_D)

lemma Disjoint_Eats_Imp: "{ Disjoint x z, Neg(y IN z) } ⊢ Disjoint (Eats x y) z"
proof -
obtain i::name where"atom i ♯ (x,y,z)"
by (rule obtain_fresh)
then show ?thesis
by (auto intro: Disjoint_I [of i]  Disjoint_E [THEN rotate3]
Mem_cong [OF Assume Refl, THEN Iff_MP_same])
qed

lemma Disjoint_Eats_I [intro!]: "H ⊢ Disjoint x z ⟹ insert (y IN z) H ⊢ Fls ⟹ H ⊢ Disjoint (Eats x y) z"
by (metis Neg_I cut2 [OF Disjoint_Eats_Imp])

lemma Disjoint_Eats_I2 [intro!]: "H ⊢ Disjoint z x ⟹ insert (y IN z) H ⊢ Fls ⟹ H ⊢ Disjoint z (Eats x y)"
by (metis Disjoint_Eats_I Disjoint_commute cut1)

section ‹The Foundation Theorem›

lemma Foundation_lemma:
assumes i: "atom i ♯ z"
shows "{ All2 i z (Neg (Disjoint (Var i) z)) } ⊢ Neg (Var i IN z) AND Disjoint (Var i) z"
proof -
obtain j::name  where j: "atom j ♯ (z,i)"
by (metis obtain_fresh)
show ?thesis
apply (rule Ind [of j]) using i j
apply auto
apply (rule Ex_I [where x=Zero], auto)
apply (rule Ex_I [where x="Eats (Var i) (Var j)"], auto)
apply (metis ContraAssume insertI1 insert_commute)
apply (metis ContraProve Disjoint_Eats_Imp rotate2 thin1)
apply (metis Assume Disj_I1 anti_deduction rotate3)
done
qed

theorem Foundation: "atom i ♯ z ⟹ {} ⊢ All2 i z (Neg (Disjoint (Var i) z)) IMP z EQ Zero"
apply auto
apply (rule Eq_Zero_I)
apply (rule cut_same [where A = "(Neg ((Var i) IN z) AND Disjoint (Var i) z)"])
apply (rule Foundation_lemma [THEN cut1], auto)
done

lemma Mem_Neg_refl: "{} ⊢ Neg (x IN x)"
proof -
obtain i::name where i: "atom i ♯ x"
by (metis obtain_fresh)
have "{} ⊢ Disjoint x (Eats Zero x)"
apply (rule cut_same [OF Foundation [where z = "Eats Zero x"]]) using i
apply auto
apply (rule cut_same [where A = "Disjoint x (Eats Zero x)"])
apply (metis Assume thin1 Disjoint_cong [OF Assume Refl, THEN Iff_MP_same])
apply (metis Assume AssumeH(4) Disjoint_E Mem_Eats_I2 Refl)
done
thus ?thesis
by (metis Disjoint_Eats_D2 Disjoint_commute cut_same)
qed

lemma Mem_refl_E [intro!]: "insert (x IN x) H ⊢ A"
by (metis Disj_I1 Mem_Neg_refl anti_deduction thin0)

lemma Mem_non_refl: assumes "H ⊢ x IN x" shows "H ⊢ A"
by (metis Mem_refl_E assms cut_same)

lemma Mem_Neg_sym: "{ x IN y, y IN x } ⊢ Fls"
proof -
obtain i::name where i: "atom i ♯ (x,y)"
by (metis obtain_fresh)
have "{} ⊢ Disjoint x (Eats Zero y) OR Disjoint y (Eats Zero x)"
apply (rule cut_same [OF Foundation [where i=i and z = "Eats (Eats Zero y) x"]]) using i
apply (auto intro!: Disjoint_Eats_E2 [THEN rotate2])
apply (rule Disj_I2, auto)
apply (metis Assume EQ_imp_SUBS2 Subset_D insert_commute)
apply (blast intro!: Disj_I1 Disjoint_cong [OF Hyp Refl, THEN Iff_MP_same])
done
thus ?thesis
by (auto intro: cut0 Disjoint_Eats_E2)
qed

lemma Mem_not_sym: "insert (x IN y) (insert (y IN x) H) ⊢ A"
by (rule cut_thin [OF Mem_Neg_sym]) auto

section ‹The Ordinal Property›

nominal_function OrdP :: "tm ⇒ fm"
where "⟦atom y ♯ (x, z); atom z ♯ x⟧ ⟹
OrdP x = All2 y x ((Var y) SUBS x AND  All2 z (Var y) ((Var z) SUBS (Var y)))"
by (auto simp: eqvt_def OrdP_graph_aux_def flip_fresh_fresh) (metis obtain_fresh)

nominal_termination (eqvt)
by lexicographic_order

lemma
shows OrdP_fresh_iff [simp]: "a ♯ OrdP x ⟷ a ♯ x"       (is ?thesis1)
and eval_fm_OrdP [simp]:  "eval_fm e (OrdP x) ⟷ Ord ⟦x⟧e"  (is ?thesis2)
proof -
obtain z::name and y::name where "atom z ♯ x" "atom y ♯ (x, z)"
by (metis obtain_fresh)
thus ?thesis1 ?thesis2
by (auto simp: OrdP.simps [of y _ z] Ord_def Transset_def)
qed

lemma subst_fm_OrdP [simp]: "(OrdP t)(i::=x) = OrdP (subst i x t)"
proof -
obtain z::name and y::name where "atom z ♯ (t,i,x)" "atom y ♯ (t,i,x,z)"
by (metis obtain_fresh)
thus ?thesis
by (auto simp: OrdP.simps [of y _ z])
qed

lemma OrdP_cong: "H ⊢ x EQ x' ⟹ H ⊢ OrdP x IFF OrdP x'"
by (rule P1_cong) auto

lemma OrdP_Mem_lemma:
assumes z: "atom z ♯ (k,l)" and l: "insert (OrdP k) H ⊢ l IN k"
shows "insert (OrdP k) H ⊢ l SUBS k AND All2 z l (Var z SUBS l)"
proof -
obtain y::name where y: "atom y ♯ (k,l,z)"
by (metis obtain_fresh)
have "insert (OrdP k) H
⊢ (Var y IN k IMP (Var y SUBS k AND All2 z (Var y) (Var z SUBS Var y)))(y::=l)"
by (rule All_D) (simp add: OrdP.simps [of y _ z] y z Assume)
also have "... = l IN k IMP (l SUBS k AND All2 z l (Var z SUBS l))"
using y z  by simp
finally show ?thesis
by (metis MP_same l)
qed

lemma OrdP_Mem_E:
assumes "atom z ♯ (k,l)"
"insert (OrdP k) H ⊢ l IN k"
"insert (l SUBS k) (insert (All2 z l (Var z SUBS l)) H) ⊢ A"
shows "insert (OrdP k) H ⊢ A"
apply (rule OrdP_Mem_lemma [THEN cut_same])
apply (auto simp: insert_commute)
apply (blast intro: assms thin1)+
done

lemma OrdP_Mem_imp_Subset:
assumes k: "H ⊢ k IN l" and l: "H ⊢ OrdP l" shows "H ⊢ k SUBS l"
apply (rule obtain_fresh [of "(l,k)"])
apply (rule cut_same [OF l])
using k  apply (auto intro: OrdP_Mem_E thin1)
done

lemma SUCC_Subset_Ord_lemma: "{ k' IN k, OrdP k } ⊢ SUCC k' SUBS k"
by auto (metis Assume thin1 OrdP_Mem_imp_Subset)

lemma SUCC_Subset_Ord: "H ⊢ k' IN k ⟹ H ⊢ OrdP k ⟹ H ⊢ SUCC k' SUBS k"
by (blast intro!: cut2 [OF SUCC_Subset_Ord_lemma])

lemma OrdP_Trans_lemma: "{ OrdP k, i IN j, j IN k } ⊢ i IN k"
proof -
obtain m::name where "atom m ♯ (i,j,k)"
by (metis obtain_fresh)
thus ?thesis
by (auto intro: OrdP_Mem_E [of m k j] Subset_D [THEN rotate3])
qed

lemma OrdP_Trans: "H ⊢  OrdP k ⟹ H ⊢ i IN j ⟹ H ⊢ j IN k ⟹ H  ⊢ i IN k"
by (blast intro: cut3 [OF OrdP_Trans_lemma])

lemma Ord_IN_Ord0:
assumes l: "H ⊢ l IN k"
shows "insert (OrdP k) H ⊢ OrdP l"
proof -
obtain z::name and y::name where z: "atom z ♯ (k,l)" and y: "atom y ♯ (k,l,z)"
by (metis obtain_fresh)
have "{Var y IN l, OrdP k, l IN k} ⊢ All2 z (Var y) (Var z SUBS Var y)"  using y z
apply (simp add: insert_commute [of _ "OrdP k"])
apply (auto intro: OrdP_Mem_E [of z k "Var y"] OrdP_Trans_lemma del: All_I Neg_I)
done
hence "{OrdP k, l IN k} ⊢ OrdP l" using z y
apply (auto simp: OrdP.simps [of y l z])
apply (simp add: insert_commute [of _ "OrdP k"])
apply (rule OrdP_Mem_E [of y k l], simp_all)
apply (metis Assume thin1)
apply (rule All_E [where x= "Var y", THEN thin1], simp)
apply (metis Assume anti_deduction insert_commute)
done
thus ?thesis
by (metis (full_types) Assume l cut2 thin1)
qed

lemma Ord_IN_Ord: "H ⊢ l IN k ⟹ H ⊢ OrdP k ⟹ H ⊢ OrdP l"
by (metis Ord_IN_Ord0 cut_same)

lemma OrdP_I:
assumes "insert (Var y IN x) H ⊢ (Var y) SUBS x"
and "insert (Var z IN Var y) (insert (Var y IN x) H) ⊢ (Var z) SUBS (Var y)"
and "atom y ♯ (x, z)" "∀B ∈ H. atom y ♯ B"  "atom z ♯ x" "∀B ∈ H. atom z ♯ B"
shows "H ⊢ OrdP x"
using assms by auto

lemma OrdP_Zero [simp]: "H ⊢ OrdP Zero"
proof -
obtain y::name and z::name where "atom y ♯ z"
by (rule obtain_fresh)
hence "{} ⊢ OrdP Zero"
by (auto intro: OrdP_I [of y _ _ z])
thus ?thesis
by (metis thin0)
qed

lemma OrdP_SUCC_I0: "{ OrdP k } ⊢ OrdP (SUCC k)"
proof -
obtain w::name and y::name and z::name where atoms: "atom w ♯ (k,y,z)" "atom y ♯ (k,z)" "atom z ♯ k"
by (metis obtain_fresh)
have 1: "{Var y IN SUCC k, OrdP k} ⊢ Var y SUBS SUCC k"
apply (rule Mem_SUCC_E)
apply (rule OrdP_Mem_E [of w _ "Var y", THEN rotate2]) using atoms
apply auto
apply (metis Assume Subset_SUCC Subset_trans)
apply (metis EQ_imp_SUBS Subset_SUCC Subset_trans)
done
have in_case: "{Var y IN k, Var z IN Var y, OrdP k} ⊢ Var z SUBS Var y"
apply (rule OrdP_Mem_E [of w _ "Var y", THEN rotate3]) using atoms
apply (auto intro:  All2_E [THEN thin1])
done
have "{Var y EQ k, Var z IN k, OrdP k} ⊢ Var z SUBS Var y"
by (metis AssumeH(2) AssumeH(3) EQ_imp_SUBS2 OrdP_Mem_imp_Subset Subset_trans)
hence eq_case: "{Var y EQ k, Var z IN Var y, OrdP k} ⊢ Var z SUBS Var y"
by (rule cut3) (auto intro: EQ_imp_SUBS [THEN cut1] Subset_D)
have 2: "{Var z IN Var y, Var y IN SUCC k, OrdP k} ⊢ Var z SUBS Var y"
by (metis rotate2 Mem_SUCC_E in_case eq_case)
show ?thesis
apply (rule OrdP_I [OF 1 2])
using atoms apply auto
done
qed

lemma OrdP_SUCC_I: "H ⊢ OrdP k ⟹ H ⊢ OrdP (SUCC k)"
by (metis OrdP_SUCC_I0 cut1)

lemma Zero_In_OrdP: "{ OrdP x } ⊢ x EQ Zero OR Zero IN x"
proof -
obtain i::name and j::name
where i: "atom i ♯ x" and j: "atom j ♯ (x,i)"
by (metis obtain_fresh)
show ?thesis
apply (rule cut_thin [where HB = "{OrdP x}", OF Foundation [where i=i and z = x]])
using i j apply auto
prefer 2 apply (metis Assume Disj_I1)
apply (rule Disj_I2)
apply (rule cut_same [where A = "Var i EQ Zero"])
prefer 2 apply (blast intro: Iff_MP_same [OF Mem_cong [OF Assume Refl]])
apply (auto intro!: Eq_Zero_I [where i=j] Ex_I [where x="Var i"])
apply (blast intro: Disjoint_E Subset_D)
done
qed

lemma OrdP_HPairE: "insert (OrdP (HPair x y)) H ⊢ A"
proof -
have "{ OrdP (HPair x y) } ⊢ A"
by (rule cut_same [OF Zero_In_OrdP]) (auto simp: HPair_def)
thus ?thesis
by (metis Assume cut1)
qed

lemmas OrdP_HPairEH = OrdP_HPairE OrdP_HPairE [THEN rotate2] OrdP_HPairE [THEN rotate3] OrdP_HPairE [THEN rotate4] OrdP_HPairE [THEN rotate5]
OrdP_HPairE [THEN rotate6] OrdP_HPairE [THEN rotate7] OrdP_HPairE [THEN rotate8] OrdP_HPairE [THEN rotate9] OrdP_HPairE [THEN rotate10]
declare OrdP_HPairEH [intro!]

lemma Zero_Eq_HPairE: "insert (Zero EQ HPair x y) H ⊢ A"
by (metis Eats_EQ_Zero_E2 HPair_def)

lemmas Zero_Eq_HPairEH = Zero_Eq_HPairE Zero_Eq_HPairE [THEN rotate2] Zero_Eq_HPairE [THEN rotate3] Zero_Eq_HPairE [THEN rotate4] Zero_Eq_HPairE [THEN rotate5]
Zero_Eq_HPairE [THEN rotate6] Zero_Eq_HPairE [THEN rotate7] Zero_Eq_HPairE [THEN rotate8] Zero_Eq_HPairE [THEN rotate9] Zero_Eq_HPairE [THEN rotate10]
declare Zero_Eq_HPairEH [intro!]

lemma HPair_Eq_ZeroE: "insert (HPair x y EQ Zero) H ⊢ A"
by (metis Sym_L Zero_Eq_HPairE)

lemmas HPair_Eq_ZeroEH = HPair_Eq_ZeroE HPair_Eq_ZeroE [THEN rotate2] HPair_Eq_ZeroE [THEN rotate3] HPair_Eq_ZeroE [THEN rotate4] HPair_Eq_ZeroE [THEN rotate5]
HPair_Eq_ZeroE [THEN rotate6] HPair_Eq_ZeroE [THEN rotate7] HPair_Eq_ZeroE [THEN rotate8] HPair_Eq_ZeroE [THEN rotate9] HPair_Eq_ZeroE [THEN rotate10]
declare HPair_Eq_ZeroEH [intro!]

section ‹Induction on Ordinals›

lemma OrdInd_lemma:
assumes j: "atom (j::name) ♯ (i,A)"
shows "{ OrdP (Var i) } ⊢ (All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A))) IMP A"
proof -
obtain l::name and k::name
where l: "atom l ♯ (i,j,A)" and k: "atom k ♯ (i,j,l,A)"
by (metis obtain_fresh)
have "{ (All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A))) }
⊢ (All2 l (Var i) (OrdP (Var l) IMP A(i::= Var l)))"
apply (rule Ind [of k])
using j k l apply auto
apply (rule All_E [where x="Var l", THEN rotate5], auto)
apply (metis Assume Disj_I1 anti_deduction thin1)
apply (rule Ex_I [where x="Var l"], auto)
apply (rule All_E [where x="Var j", THEN rotate6], auto)
apply (blast intro: ContraProve Iff_MP_same [OF Mem_cong [OF Refl]])
apply (metis Assume Ord_IN_Ord0 ContraProve insert_commute)
apply (metis Assume Neg_D thin1)+
done
hence "{ (All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A))) }
⊢ (All2 l (Var i) (OrdP (Var l) IMP A(i::= Var l)))(i::= Eats Zero (Var i))"
by (rule Subst, auto)
hence indlem: "{ All i (OrdP (Var i) IMP ((All2 j (Var i) (A(i::= Var j))) IMP A)) }
⊢ All2 l (Eats Zero (Var i)) (OrdP (Var l) IMP A(i::=Var l))"
using j l by simp
show ?thesis
apply (rule Imp_I)
apply (rule cut_thin [OF indlem, where HB = "{OrdP (Var i)}"])
apply (rule All2_Eats_E) using j l
apply auto
done
qed

lemma OrdInd:
assumes j: "atom (j::name) ♯ (i,A)"
and x: "H ⊢ OrdP (Var i)" and step: "H ⊢ All i (OrdP (Var i) IMP (All2 j (Var i) (A(i::= Var j)) IMP A))"
shows "H ⊢ A"
apply (rule cut_thin [OF x, where HB=H])
apply (rule MP_thin [OF OrdInd_lemma step])
apply (auto simp: j)
done

lemma OrdIndH:
assumes "atom (j::name) ♯ (i,A)"
and "H ⊢ All i (OrdP (Var i) IMP (All2 j (Var i) (A(i::= Var j)) IMP A))"
shows "insert (OrdP (Var i)) H ⊢ A"
by (metis assms thin1 Assume OrdInd)

section ‹Linearity of Ordinals›

lemma OrdP_linear_lemma:
assumes j: "atom j ♯ i"
shows "{ OrdP (Var i) } ⊢ All j (OrdP (Var j) IMP (Var i IN Var j OR Var i EQ Var j OR Var j IN Var i))"
(is "_ ⊢ ?scheme")
proof -
obtain k::name and l::name and m::name
where k: "atom k ♯ (i,j)" and l: "atom l ♯ (i,j,k)" and m: "atom m ♯ (i,j)"
by (metis obtain_fresh)
show ?thesis
proof (rule OrdIndH [where i=i and j=k])
show "atom k ♯ (i, ?scheme)"
using k  by (force simp add: fresh_Pair)
next
show "{} ⊢ All i (OrdP (Var i) IMP (All2 k (Var i) (?scheme(i::= Var k)) IMP ?scheme))"
using j k
apply simp
apply (rule All_I Imp_I)+
defer 1
apply auto [2]
apply (rule OrdIndH [where i=j and j=l]) using l
― ‹nested induction›
apply simp
apply (rule All_I Imp_I)+
prefer 2  apply force
apply (rule Disj_3I)
apply (rule Equality_I)
― ‹Now the opposite inclusion, @{term"Var j SUBS Var i"}›
apply (rule Subset_I [where i=m])
apply (rule All2_E [THEN rotate4]) using l m
apply auto
apply (blast intro: ContraProve [THEN rotate3] OrdP_Trans)
apply (blast intro: ContraProve [THEN rotate3] Mem_cong [OF Hyp Refl, THEN Iff_MP2_same])
― ‹Now the opposite inclusion, @{term"Var i SUBS Var j"}›
apply (rule Subset_I [where i=m])
apply (rule All2_E [THEN rotate6], auto)
apply (rule All_E [where x = "Var j"], auto)
apply (blast intro: ContraProve [THEN rotate4] Mem_cong [OF Hyp Refl, THEN Iff_MP_same])
apply (blast intro: ContraProve [THEN rotate4] OrdP_Trans)
done
qed
qed

lemma OrdP_linear_imp: "{} ⊢ OrdP x IMP OrdP y IMP x IN y OR x EQ y OR y IN x"
proof -
obtain i::name and j::name
where atoms: "atom i ♯ (x,y)" "atom j ♯ (x,y,i)"
by (metis obtain_fresh)
have "{ OrdP (Var i) } ⊢ (OrdP (Var j) IMP (Var i IN Var j OR Var i EQ Var j OR Var j IN Var i))(j::=y)"
using atoms  by (metis All_D OrdP_linear_lemma fresh_Pair)
hence "{} ⊢ OrdP (Var i) IMP OrdP y IMP (Var i IN y OR Var i EQ y OR y IN Var i)"
using atoms by auto
hence "{} ⊢ (OrdP (Var i) IMP OrdP y IMP (Var i IN y OR Var i EQ y OR y IN Var i))(i::=x)"
by (metis Subst empty_iff)
thus ?thesis
using atoms by auto
qed

lemma OrdP_linear:
assumes "H ⊢ OrdP x" "H ⊢ OrdP y"
"insert (x IN y) H ⊢ A" "insert (x EQ y) H ⊢ A" "insert (y IN x) H ⊢ A"
shows "H ⊢ A"
proof -
have "{ OrdP x, OrdP y } ⊢ x IN y OR x EQ y OR y IN x"
by (metis OrdP_linear_imp Imp_Imp_commute anti_deduction)
thus ?thesis
using assms  by (metis cut2 Disj_E cut_same)
qed

lemma Zero_In_SUCC: "{OrdP k} ⊢ Zero IN SUCC k"
by (rule OrdP_linear [OF OrdP_Zero OrdP_SUCC_I]) (force simp: SUCC_def)+

section ‹The predicate ‹OrdNotEqP››

nominal_function OrdNotEqP :: "tm ⇒ tm ⇒ fm"  (infixr "NEQ" 150)
where "OrdNotEqP x y = OrdP x AND OrdP y AND (x IN y OR y IN x)"
by (auto simp: eqvt_def OrdNotEqP_graph_aux_def)

nominal_termination (eqvt)
by lexicographic_order

lemma OrdNotEqP_fresh_iff [simp]: "a ♯ OrdNotEqP x y ⟷ a ♯ x ∧ a ♯ y"
by auto

lemma eval_fm_OrdNotEqP [simp]: "eval_fm e (OrdNotEqP x y) ⟷ Ord ⟦x⟧e ∧ Ord ⟦y⟧e ∧ ⟦x⟧e ≠ ⟦y⟧e"
by (auto simp: hmem_not_refl) (metis Ord_linear)

lemma OrdNotEqP_subst [simp]: "(OrdNotEqP x y)(i::=t) = OrdNotEqP (subst i t x) (subst i t y)"
by simp

lemma OrdNotEqP_cong: "H ⊢ x EQ x' ⟹ H ⊢ y EQ y' ⟹ H ⊢ OrdNotEqP x y IFF OrdNotEqP x' y'"
by (rule P2_cong) auto

lemma OrdNotEqP_self_contra: "{x NEQ x} ⊢ Fls"
by auto

lemma OrdNotEqP_OrdP_E: "insert (OrdP x) (insert (OrdP y) H) ⊢ A ⟹ insert (x NEQ y) H ⊢ A"
by (auto intro: thin1 rotate2)

lemma OrdNotEqP_I: "insert (x EQ y) H ⊢ Fls ⟹ H ⊢ OrdP x ⟹ H ⊢ OrdP y ⟹ H ⊢ x NEQ y"
by (rule OrdP_linear [of _ x y]) (auto intro: ExFalso thin1 Disj_I1 Disj_I2)

declare OrdNotEqP.simps [simp del]

lemma OrdNotEqP_imp_Neg_Eq: "{x NEQ y} ⊢ Neg (x EQ y)"
by (blast intro: OrdNotEqP_cong [THEN Iff_MP2_same]  OrdNotEqP_self_contra [of x, THEN cut1])

lemma OrdNotEqP_E: "H ⊢ x EQ y ⟹ insert (x NEQ y) H ⊢ A"
by (metis ContraProve OrdNotEqP_imp_Neg_Eq rcut1)

section ‹Predecessor of an Ordinal›

lemma OrdP_set_max_lemma:
assumes j: "atom (j::name) ♯ i" and k: "atom (k::name) ♯ (i,j)"
shows "{} ⊢ (Neg (Var i EQ Zero) AND (All2 j (Var i) (OrdP (Var j)))) IMP
(Ex j (Var j IN Var i AND (All2 k (Var i) (Var k SUBS Var j))))"
proof -
obtain l::name where l: "atom l ♯ (i,j,k)"
by (metis obtain_fresh)
show ?thesis
apply (rule Ind [of l i]) using j k l
apply simp_all
apply (metis Conj_E Refl Swap Imp_I)
apply (rule All_I Imp_I)+
apply simp_all
apply clarify
apply (rule thin1)
apply (rule thin1 [THEN rotate2])
apply (rule Disj_EH)
apply (rule Neg_Conj_E)
apply (auto simp: All2_Eats_E1)
apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2)
apply (metis Assume Eq_Zero_D rotate3)
apply (metis Assume EQ_imp_SUBS Neg_D thin1)
apply (rule Cases [where A = "Var j IN Var l"])
apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2)
apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2 ContraProve)
apply (rule Ex_I [where x="Var k"], auto)
apply (metis Assume Subset_trans OrdP_Mem_imp_Subset thin1)
apply (rule Ex_I [where x="Var l"], auto intro: Mem_Eats_I2 ContraProve)
apply (metis ContraProve EQ_imp_SUBS rotate3)
― ‹final case›
apply (rule All2_Eats_E [THEN rotate4], simp_all)
apply (rule Ex_I [where x="Var j"], auto intro: Mem_Eats_I1)
apply (rule All2_E [where x = "Var k", THEN rotate3], auto)
apply (rule Ex_I [where x="Var k"], simp)
apply (metis Assume NegNeg_I Neg_Disj_I rotate3)
apply (rule cut_same [where A = "OrdP (Var j)"])
apply (rule All2_E [where x = "Var j", THEN rotate3], auto)
apply (rule cut_same [where A = "Var l EQ Var j OR Var l IN Var j"])
apply (rule OrdP_linear [of _ "Var l" "Var j"], auto intro: Disj_CI)
apply (metis Assume ContraProve rotate7)
apply (metis ContraProve [THEN rotate4] EQ_imp_SUBS Subset_trans rotate3)
apply (blast intro: ContraProve [THEN rotate4] OrdP_Mem_imp_Subset Iff_MP2_same [OF Mem_cong])
done
qed

lemma OrdP_max_imp:
assumes j: "atom j ♯ (x)" and k: "atom k ♯ (x,j)"
shows  "{ OrdP x, Neg (x EQ Zero) } ⊢ Ex j (Var j IN x AND (All2 k x (Var k SUBS Var j)))"
proof -
obtain i::name where i: "atom i ♯ (x,j,k)"
by (metis obtain_fresh)
have "{} ⊢ ((Neg (Var i EQ Zero) AND (All2 j (Var i) (OrdP (Var j)))) IMP
(Ex j (Var j IN Var i AND (All2 k (Var i) (Var k SUBS Var j)))))(i::=x)"
apply (rule Subst [OF OrdP_set_max_lemma])
using i k apply auto
done
hence "{ Neg (x EQ Zero) AND (All2 j x (OrdP (Var j))) }
⊢ Ex j (Var j IN x AND (All2 k x (Var k SUBS Var j)))"
using i j k by simp (metis anti_deduction)
hence "{ All2 j x (OrdP (Var j)), Neg (x EQ Zero) }
⊢ Ex j (Var j IN x AND (All2 k x (Var k SUBS Var j)))"
by (rule cut1) (metis Assume Conj_I thin1)
moreover have "{ OrdP x } ⊢ All2 j x (OrdP (Var j))" using j
by auto (metis Assume Ord_IN_Ord thin1)
ultimately show ?thesis
by (metis rcut1)
qed

declare OrdP.simps [simp del]

section ‹Case Analysis and Zero/SUCC Induction›

lemma OrdP_cases_lemma:
assumes p: "atom p ♯ x"
shows  "{ OrdP x, Neg (x EQ Zero) } ⊢ Ex p (OrdP (Var p) AND x EQ SUCC (Var p))"
proof -
obtain j::name and k::name where j: "atom j ♯ (x,p)" and k: "atom k ♯ (x,j,p)"
by (metis obtain_fresh)
show ?thesis
apply (rule cut_same [OF OrdP_max_imp [of j x k]])
using p j k apply auto
apply (rule Ex_I [where x="Var j"], auto)
apply (metis Assume Ord_IN_Ord thin1)
apply (rule cut_same [where A = "OrdP (SUCC (Var j))"])
apply (metis Assume Ord_IN_Ord0 OrdP_SUCC_I rotate2 thin1)
apply (rule OrdP_linear [where x = x, OF _ Assume], auto intro!: Mem_SUCC_EH)
apply (metis Mem_not_sym rotate3)
apply (rule Mem_non_refl, blast intro: Mem_cong [OF Assume Refl, THEN Iff_MP2_same])
apply (force intro: thin1 All2_E [where x = "SUCC (Var j)", THEN rotate4])
done
qed

lemma OrdP_cases_disj:
assumes p: "atom p ♯ x"
shows  "insert (OrdP x) H ⊢ x EQ Zero OR Ex p (OrdP (Var p) AND x EQ SUCC (Var p))"
by (metis Disj_CI Assume cut2 [OF OrdP_cases_lemma [OF p]] Swap thin1)

lemma OrdP_cases_E:
"⟦insert (x EQ Zero) H ⊢ A;
insert (x EQ SUCC (Var k)) (insert (OrdP (Var k)) H) ⊢ A;
atom k ♯ (x,A);   ∀C ∈ H. atom k ♯ C⟧
⟹ insert (OrdP x) H ⊢ A"
by (rule cut_same [OF OrdP_cases_disj [of k]]) (auto simp: insert_commute intro: thin1)

lemma OrdInd2_lemma:
"{ OrdP (Var i), A(i::= Zero), (All i (OrdP (Var i) IMP A IMP (A(i::= SUCC (Var i))))) } ⊢ A"
proof -
obtain j::name and k::name  where atoms: "atom j ♯ (i,A)" "atom k ♯ (i,j,A)"
by (metis obtain_fresh)
show ?thesis
apply (rule OrdIndH [where i=i and j=j])
using atoms  apply auto
apply (rule OrdP_cases_E [where k=k, THEN rotate3])
apply (rule ContraProve [THEN rotate2]) using Var_Eq_imp_subst_Iff
apply (metis Assume AssumeH(3) Iff_MP_same)
apply (rule Ex_I [where x="Var k"], simp)
apply (rule Neg_Imp_I, blast)
apply (rule cut_same [where A = "A(i::=Var k)"])
apply (rule All2_E [where x = "Var k", THEN rotate5])
apply (auto intro: Mem_SUCC_I2 Mem_cong [OF Refl, THEN Iff_MP2_same])
apply (rule ContraProve [THEN rotate5])
by (metis Assume Iff_MP_left' Var_Eq_subst_Iff thin1)
qed

lemma OrdInd2:
assumes "H ⊢ OrdP (Var i)"
and "H ⊢ A(i::= Zero)"
and "H ⊢ All i (OrdP (Var i) IMP A IMP (A(i::= SUCC (Var i))))"
shows "H ⊢ A"
by (metis cut3 [OF OrdInd2_lemma] assms)

lemma OrdInd2H:
assumes "H ⊢ A(i::= Zero)"
and "H ⊢ All i (OrdP (Var i) IMP A IMP (A(i::= SUCC (Var i))))"
shows "insert (OrdP (Var i)) H ⊢ A"
by (metis assms thin1 Assume OrdInd2)

section ‹The predicate ‹HFun_Sigma››

text‹To characterise the concept of a function using only bounded universal quantifiers.›

text‹See the note after the proof of Lemma 2.3.›

definition hfun_sigma where
"hfun_sigma r ≡ ∀z ❙∈ r. ∀z' ❙∈ r. ∃x y x' y'. z = ⟨x,y⟩ ∧ z' = ⟨x',y'⟩ ∧ (x=x' ⟶ y=y')"

definition hfun_sigma_ord where
"hfun_sigma_ord r ≡ ∀z ❙∈ r. ∀z' ❙∈ r. ∃x y x' y'. z = ⟨x,y⟩ ∧ z' = ⟨x',y'⟩ ∧ Ord x ∧ Ord x' ∧ (x=x' ⟶ y=y')"

nominal_function HFun_Sigma :: "tm ⇒ fm"
where "⟦atom z ♯ (r,z',x,y,x',y'); atom z' ♯ (r,x,y,x',y');
atom x ♯ (r,y,x',y'); atom y ♯ (r,x',y'); atom x' ♯ (r,y'); atom y' ♯ (r) ⟧ ⟹
HFun_Sigma r =
All2 z r (All2 z' r (Ex x (Ex y (Ex x' (Ex y'
(Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y')
AND OrdP (Var x