Theory Refute_Nits

theory Refute_Nits
imports Main
(*  Title:      HOL/Nitpick_Examples/Refute_Nits.thy
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2009-2011

Refute examples adapted to Nitpick.
*)

header {* Refute Examples Adapted to Nitpick *}

theory Refute_Nits
imports Main
begin

nitpick_params [verbose, card = 1-6, max_potential = 0,
                sat_solver = MiniSat_JNI, max_threads = 1, timeout = 240]

lemma "P ∧ Q"
apply (rule conjI)
nitpick [expect = genuine] 1
nitpick [expect = genuine] 2
nitpick [expect = genuine]
nitpick [card = 5, expect = genuine]
nitpick [sat_solver = SAT4J, expect = genuine] 2
oops

subsection {* Examples and Test Cases *}

subsubsection {* Propositional logic *}

lemma "True"
nitpick [expect = none]
apply auto
done

lemma "False"
nitpick [expect = genuine]
oops

lemma "P"
nitpick [expect = genuine]
oops

lemma "¬ P"
nitpick [expect = genuine]
oops

lemma "P ∧ Q"
nitpick [expect = genuine]
oops

lemma "P ∨ Q"
nitpick [expect = genuine]
oops

lemma "P --> Q"
nitpick [expect = genuine]
oops

lemma "(P::bool) = Q"
nitpick [expect = genuine]
oops

lemma "(P ∨ Q) --> (P ∧ Q)"
nitpick [expect = genuine]
oops

subsubsection {* Predicate logic *}

lemma "P x y z"
nitpick [expect = genuine]
oops

lemma "P x y --> P y x"
nitpick [expect = genuine]
oops

lemma "P (f (f x)) --> P x --> P (f x)"
nitpick [expect = genuine]
oops

subsubsection {* Equality *}

lemma "P = True"
nitpick [expect = genuine]
oops

lemma "P = False"
nitpick [expect = genuine]
oops

lemma "x = y"
nitpick [expect = genuine]
oops

lemma "f x = g x"
nitpick [expect = genuine]
oops

lemma "(f::'a=>'b) = g"
nitpick [expect = genuine]
oops

lemma "(f::('d=>'d)=>('c=>'d)) = g"
nitpick [expect = genuine]
oops

lemma "distinct [a, b]"
nitpick [expect = genuine]
apply simp
nitpick [expect = genuine]
oops

subsubsection {* First-Order Logic *}

lemma "∃x. P x"
nitpick [expect = genuine]
oops

lemma "∀x. P x"
nitpick [expect = genuine]
oops

lemma "∃!x. P x"
nitpick [expect = genuine]
oops

lemma "Ex P"
nitpick [expect = genuine]
oops

lemma "All P"
nitpick [expect = genuine]
oops

lemma "Ex1 P"
nitpick [expect = genuine]
oops

lemma "(∃x. P x) --> (∀x. P x)"
nitpick [expect = genuine]
oops

lemma "(∀x. ∃y. P x y) --> (∃y. ∀x. P x y)"
nitpick [expect = genuine]
oops

lemma "(∃x. P x) --> (∃!x. P x)"
nitpick [expect = genuine]
oops

text {* A true statement (also testing names of free and bound variables being identical) *}

lemma "(∀x y. P x y --> P y x) --> (∀x. P x y) --> P y x"
nitpick [expect = none]
apply fast
done

text {* "A type has at most 4 elements." *}

lemma "¬ distinct [a, b, c, d, e]"
nitpick [expect = genuine]
apply simp
nitpick [expect = genuine]
oops

lemma "distinct [a, b, c, d]"
nitpick [expect = genuine]
apply simp
nitpick [expect = genuine]
oops

text {* "Every reflexive and symmetric relation is transitive." *}

lemma "[|∀x. P x x; ∀x y. P x y --> P y x|] ==> P x y --> P y z --> P x z"
nitpick [expect = genuine]
oops

text {* The ``Drinker's theorem'' *}

lemma "∃x. f x = g x --> f = g"
nitpick [expect = none]
apply (auto simp add: ext)
done

text {* And an incorrect version of it *}

lemma "(∃x. f x = g x) --> f = g"
nitpick [expect = genuine]
oops

text {* "Every function has a fixed point." *}

lemma "∃x. f x = x"
nitpick [expect = genuine]
oops

text {* "Function composition is commutative." *}

lemma "f (g x) = g (f x)"
nitpick [expect = genuine]
oops

text {* "Two functions that are equivalent wrt.\ the same predicate 'P' are equal." *}

lemma "((P::('a=>'b)=>bool) f = P g) --> (f x = g x)"
nitpick [expect = genuine]
oops

subsubsection {* Higher-Order Logic *}

lemma "∃P. P"
nitpick [expect = none]
apply auto
done

lemma "∀P. P"
nitpick [expect = genuine]
oops

lemma "∃!P. P"
nitpick [expect = none]
apply auto
done

lemma "∃!P. P x"
nitpick [expect = genuine]
oops

lemma "P Q ∨ Q x"
nitpick [expect = genuine]
oops

lemma "x ≠ All"
nitpick [expect = genuine]
oops

lemma "x ≠ Ex"
nitpick [expect = genuine]
oops

lemma "x ≠ Ex1"
nitpick [expect = genuine]
oops

text {* ``The transitive closure of an arbitrary relation is non-empty.'' *}

definition "trans" :: "('a => 'a => bool) => bool" where
"trans P ≡ (ALL x y z. P x y --> P y z --> P x z)"

definition
"subset" :: "('a => 'a => bool) => ('a => 'a => bool) => bool" where
"subset P Q ≡ (ALL x y. P x y --> Q x y)"

definition
"trans_closure" :: "('a => 'a => bool) => ('a => 'a => bool) => bool" where
"trans_closure P Q ≡ (subset Q P) ∧ (trans P) ∧ (ALL R. subset Q R --> trans R --> subset P R)"

lemma "trans_closure T P --> (∃x y. T x y)"
nitpick [expect = genuine]
oops

text {* ``The union of transitive closures is equal to the transitive closure of unions.'' *}

lemma "(∀x y. (P x y ∨ R x y) --> T x y) --> trans T --> (∀Q. (∀x y. (P x y ∨ R x y) --> Q x y) --> trans Q --> subset T Q)
 --> trans_closure TP P
 --> trans_closure TR R
 --> (T x y = (TP x y ∨ TR x y))"
nitpick [expect = genuine]
oops

text {* ``Every surjective function is invertible.'' *}

lemma "(∀y. ∃x. y = f x) --> (∃g. ∀x. g (f x) = x)"
nitpick [expect = genuine]
oops

text {* ``Every invertible function is surjective.'' *}

lemma "(∃g. ∀x. g (f x) = x) --> (∀y. ∃x. y = f x)"
nitpick [expect = genuine]
oops

text {* ``Every point is a fixed point of some function.'' *}

lemma "∃f. f x = x"
nitpick [card = 1-7, expect = none]
apply (rule_tac x = "λx. x" in exI)
apply simp
done

text {* Axiom of Choice: first an incorrect version *}

lemma "(∀x. ∃y. P x y) --> (∃!f. ∀x. P x (f x))"
nitpick [expect = genuine]
oops

text {* And now two correct ones *}

lemma "(∀x. ∃y. P x y) --> (∃f. ∀x. P x (f x))"
nitpick [card = 1-4, expect = none]
apply (simp add: choice)
done

lemma "(∀x. ∃!y. P x y) --> (∃!f. ∀x. P x (f x))"
nitpick [card = 1-3, expect = none]
apply auto
 apply (simp add: ex1_implies_ex choice)
apply (fast intro: ext)
done

subsubsection {* Metalogic *}

lemma "!!x. P x"
nitpick [expect = genuine]
oops

lemma "f x ≡ g x"
nitpick [expect = genuine]
oops

lemma "P ==> Q"
nitpick [expect = genuine]
oops

lemma "[|P; Q; R|] ==> S"
nitpick [expect = genuine]
oops

lemma "(x ≡ Pure.all) ==> False"
nitpick [expect = genuine]
oops

lemma "(x ≡ (op ≡)) ==> False"
nitpick [expect = genuine]
oops

lemma "(x ≡ (op ==>)) ==> False"
nitpick [expect = genuine]
oops

subsubsection {* Schematic Variables *}

schematic_lemma "?P"
nitpick [expect = none]
apply auto
done

schematic_lemma "x = ?y"
nitpick [expect = none]
apply auto
done

subsubsection {* Abstractions *}

lemma "(λx. x) = (λx. y)"
nitpick [expect = genuine]
oops

lemma "(λf. f x) = (λf. True)"
nitpick [expect = genuine]
oops

lemma "(λx. x) = (λy. y)"
nitpick [expect = none]
apply simp
done

subsubsection {* Sets *}

lemma "P (A::'a set)"
nitpick [expect = genuine]
oops

lemma "P (A::'a set set)"
nitpick [expect = genuine]
oops

lemma "{x. P x} = {y. P y}"
nitpick [expect = none]
apply simp
done

lemma "x ∈ {x. P x}"
nitpick [expect = genuine]
oops

lemma "P (op ∈)"
nitpick [expect = genuine]
oops

lemma "P (op ∈ x)"
nitpick [expect = genuine]
oops

lemma "P Collect"
nitpick [expect = genuine]
oops

lemma "A Un B = A Int B"
nitpick [expect = genuine]
oops

lemma "(A Int B) Un C = (A Un C) Int B"
nitpick [expect = genuine]
oops

lemma "Ball A P --> Bex A P"
nitpick [expect = genuine]
oops

subsubsection {* @{const undefined} *}

lemma "undefined"
nitpick [expect = genuine]
oops

lemma "P undefined"
nitpick [expect = genuine]
oops

lemma "undefined x"
nitpick [expect = genuine]
oops

lemma "undefined undefined"
nitpick [expect = genuine]
oops

subsubsection {* @{const The} *}

lemma "The P"
nitpick [expect = genuine]
oops

lemma "P The"
nitpick [expect = genuine]
oops

lemma "P (The P)"
nitpick [expect = genuine]
oops

lemma "(THE x. x=y) = z"
nitpick [expect = genuine]
oops

lemma "Ex P --> P (The P)"
nitpick [expect = genuine]
oops

subsubsection {* @{const Eps} *}

lemma "Eps P"
nitpick [expect = genuine]
oops

lemma "P Eps"
nitpick [expect = genuine]
oops

lemma "P (Eps P)"
nitpick [expect = genuine]
oops

lemma "(SOME x. x=y) = z"
nitpick [expect = genuine]
oops

lemma "Ex P --> P (Eps P)"
nitpick [expect = none]
apply (auto simp add: someI)
done

subsubsection {* Operations on Natural Numbers *}

lemma "(x::nat) + y = 0"
nitpick [expect = genuine]
oops

lemma "(x::nat) = x + x"
nitpick [expect = genuine]
oops

lemma "(x::nat) - y + y = x"
nitpick [expect = genuine]
oops

lemma "(x::nat) = x * x"
nitpick [expect = genuine]
oops

lemma "(x::nat) < x + y"
nitpick [card = 1, expect = genuine]
oops

text {* × *}

lemma "P (x::'a×'b)"
nitpick [expect = genuine]
oops

lemma "∀x::'a×'b. P x"
nitpick [expect = genuine]
oops

lemma "P (x, y)"
nitpick [expect = genuine]
oops

lemma "P (fst x)"
nitpick [expect = genuine]
oops

lemma "P (snd x)"
nitpick [expect = genuine]
oops

lemma "P Pair"
nitpick [expect = genuine]
oops

lemma "P (case x of Pair a b => f a b)"
nitpick [expect = genuine]
oops

subsubsection {* Subtypes (typedef), typedecl *}

text {* A completely unspecified non-empty subset of @{typ "'a"}: *}

definition "myTdef = insert (undefined::'a) (undefined::'a set)"

typedef 'a myTdef = "myTdef :: 'a set"
unfolding myTdef_def by auto

lemma "(x::'a myTdef) = y"
nitpick [expect = genuine]
oops

typedecl myTdecl

definition "T_bij = {(f::'a=>'a). ∀y. ∃!x. f x = y}"

typedef 'a T_bij = "T_bij :: ('a => 'a) set"
unfolding T_bij_def by auto

lemma "P (f::(myTdecl myTdef) T_bij)"
nitpick [expect = genuine]
oops

subsubsection {* Inductive Datatypes *}

text {* unit *}

lemma "P (x::unit)"
nitpick [expect = genuine]
oops

lemma "∀x::unit. P x"
nitpick [expect = genuine]
oops

lemma "P ()"
nitpick [expect = genuine]
oops

lemma "P (case x of () => u)"
nitpick [expect = genuine]
oops

text {* option *}

lemma "P (x::'a option)"
nitpick [expect = genuine]
oops

lemma "∀x::'a option. P x"
nitpick [expect = genuine]
oops

lemma "P None"
nitpick [expect = genuine]
oops

lemma "P (Some x)"
nitpick [expect = genuine]
oops

lemma "P (case x of None => n | Some u => s u)"
nitpick [expect = genuine]
oops

text {* + *}

lemma "P (x::'a+'b)"
nitpick [expect = genuine]
oops

lemma "∀x::'a+'b. P x"
nitpick [expect = genuine]
oops

lemma "P (Inl x)"
nitpick [expect = genuine]
oops

lemma "P (Inr x)"
nitpick [expect = genuine]
oops

lemma "P Inl"
nitpick [expect = genuine]
oops

lemma "P (case x of Inl a => l a | Inr b => r b)"
nitpick [expect = genuine]
oops

text {* Non-recursive datatypes *}

datatype T1 = A | B

lemma "P (x::T1)"
nitpick [expect = genuine]
oops

lemma "∀x::T1. P x"
nitpick [expect = genuine]
oops

lemma "P A"
nitpick [expect = genuine]
oops

lemma "P B"
nitpick [expect = genuine]
oops

lemma "rec_T1 a b A = a"
nitpick [expect = none]
apply simp
done

lemma "rec_T1 a b B = b"
nitpick [expect = none]
apply simp
done

lemma "P (rec_T1 a b x)"
nitpick [expect = genuine]
oops

lemma "P (case x of A => a | B => b)"
nitpick [expect = genuine]
oops

datatype 'a T2 = C T1 | D 'a

lemma "P (x::'a T2)"
nitpick [expect = genuine]
oops

lemma "∀x::'a T2. P x"
nitpick [expect = genuine]
oops

lemma "P D"
nitpick [expect = genuine]
oops

lemma "rec_T2 c d (C x) = c x"
nitpick [expect = none]
apply simp
done

lemma "rec_T2 c d (D x) = d x"
nitpick [expect = none]
apply simp
done

lemma "P (rec_T2 c d x)"
nitpick [expect = genuine]
oops

lemma "P (case x of C u => c u | D v => d v)"
nitpick [expect = genuine]
oops

datatype ('a, 'b) T3 = E "'a => 'b"

lemma "P (x::('a, 'b) T3)"
nitpick [expect = genuine]
oops

lemma "∀x::('a, 'b) T3. P x"
nitpick [expect = genuine]
oops

lemma "P E"
nitpick [expect = genuine]
oops

lemma "rec_T3 e (E x) = e x"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "P (rec_T3 e x)"
nitpick [expect = genuine]
oops

lemma "P (case x of E f => e f)"
nitpick [expect = genuine]
oops

text {* Recursive datatypes *}

text {* nat *}

lemma "P (x::nat)"
nitpick [expect = genuine]
oops

lemma "∀x::nat. P x"
nitpick [expect = genuine]
oops

lemma "P (Suc 0)"
nitpick [expect = genuine]
oops

lemma "P Suc"
nitpick [card = 1-7, expect = none]
oops

lemma "rec_nat zero suc 0 = zero"
nitpick [expect = none]
apply simp
done

lemma "rec_nat zero suc (Suc x) = suc x (rec_nat zero suc x)"
nitpick [expect = none]
apply simp
done

lemma "P (rec_nat zero suc x)"
nitpick [expect = genuine]
oops

lemma "P (case x of 0 => zero | Suc n => suc n)"
nitpick [expect = genuine]
oops

text {* 'a list *}

lemma "P (xs::'a list)"
nitpick [expect = genuine]
oops

lemma "∀xs::'a list. P xs"
nitpick [expect = genuine]
oops

lemma "P [x, y]"
nitpick [expect = genuine]
oops

lemma "rec_list nil cons [] = nil"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "rec_list nil cons (x#xs) = cons x xs (rec_list nil cons xs)"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "P (rec_list nil cons xs)"
nitpick [expect = genuine]
oops

lemma "P (case x of Nil => nil | Cons a b => cons a b)"
nitpick [expect = genuine]
oops

lemma "(xs::'a list) = ys"
nitpick [expect = genuine]
oops

lemma "a # xs = b # xs"
nitpick [expect = genuine]
oops

datatype BitList = BitListNil | Bit0 BitList | Bit1 BitList

lemma "P (x::BitList)"
nitpick [expect = genuine]
oops

lemma "∀x::BitList. P x"
nitpick [expect = genuine]
oops

lemma "P (Bit0 (Bit1 BitListNil))"
nitpick [expect = genuine]
oops

lemma "rec_BitList nil bit0 bit1 BitListNil = nil"
nitpick [expect = none]
apply simp
done

lemma "rec_BitList nil bit0 bit1 (Bit0 xs) = bit0 xs (rec_BitList nil bit0 bit1 xs)"
nitpick [expect = none]
apply simp
done

lemma "rec_BitList nil bit0 bit1 (Bit1 xs) = bit1 xs (rec_BitList nil bit0 bit1 xs)"
nitpick [expect = none]
apply simp
done

lemma "P (rec_BitList nil bit0 bit1 x)"
nitpick [expect = genuine]
oops

datatype 'a BinTree = Leaf 'a | Node "'a BinTree" "'a BinTree"

lemma "P (x::'a BinTree)"
nitpick [expect = genuine]
oops

lemma "∀x::'a BinTree. P x"
nitpick [expect = genuine]
oops

lemma "P (Node (Leaf x) (Leaf y))"
nitpick [expect = genuine]
oops

lemma "rec_BinTree l n (Leaf x) = l x"
nitpick [expect = none]
apply simp
done

lemma "rec_BinTree l n (Node x y) = n x y (rec_BinTree l n x) (rec_BinTree l n y)"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "P (rec_BinTree l n x)"
nitpick [expect = genuine]
oops

lemma "P (case x of Leaf a => l a | Node a b => n a b)"
nitpick [expect = genuine]
oops

text {* Mutually recursive datatypes *}

datatype 'a aexp = Number 'a | ITE "'a bexp" "'a aexp" "'a aexp"
 and 'a bexp = Equal "'a aexp" "'a aexp"

lemma "P (x::'a aexp)"
nitpick [expect = genuine]
oops

lemma "∀x::'a aexp. P x"
nitpick [expect = genuine]
oops

lemma "P (ITE (Equal (Number x) (Number y)) (Number x) (Number y))"
nitpick [expect = genuine]
oops

lemma "P (x::'a bexp)"
nitpick [expect = genuine]
oops

lemma "∀x::'a bexp. P x"
nitpick [expect = genuine]
oops

lemma "rec_aexp_bexp_1 number ite equal (Number x) = number x"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_aexp_bexp_1 number ite equal (ITE x y z) = ite x y z (rec_aexp_bexp_2 number ite equal x) (rec_aexp_bexp_1 number ite equal y) (rec_aexp_bexp_1 number ite equal z)"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "P (rec_aexp_bexp_1 number ite equal x)"
nitpick [expect = genuine]
oops

lemma "P (case x of Number a => number a | ITE b a1 a2 => ite b a1 a2)"
nitpick [expect = genuine]
oops

lemma "rec_aexp_bexp_2 number ite equal (Equal x y) = equal x y (rec_aexp_bexp_1 number ite equal x) (rec_aexp_bexp_1 number ite equal y)"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "P (rec_aexp_bexp_2 number ite equal x)"
nitpick [expect = genuine]
oops

lemma "P (case x of Equal a1 a2 => equal a1 a2)"
nitpick [expect = genuine]
oops

datatype X = A | B X | C Y
     and Y = D X | E Y | F

lemma "P (x::X)"
nitpick [expect = genuine]
oops

lemma "P (y::Y)"
nitpick [expect = genuine]
oops

lemma "P (B (B A))"
nitpick [expect = genuine]
oops

lemma "P (B (C F))"
nitpick [expect = genuine]
oops

lemma "P (C (D A))"
nitpick [expect = genuine]
oops

lemma "P (C (E F))"
nitpick [expect = genuine]
oops

lemma "P (D (B A))"
nitpick [expect = genuine]
oops

lemma "P (D (C F))"
nitpick [expect = genuine]
oops

lemma "P (E (D A))"
nitpick [expect = genuine]
oops

lemma "P (E (E F))"
nitpick [expect = genuine]
oops

lemma "P (C (D (C F)))"
nitpick [expect = genuine]
oops

lemma "rec_X_Y_1 a b c d e f A = a"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "rec_X_Y_1 a b c d e f (B x) = b x (rec_X_Y_1 a b c d e f x)"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "rec_X_Y_1 a b c d e f (C y) = c y (rec_X_Y_2 a b c d e f y)"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "rec_X_Y_2 a b c d e f (D x) = d x (rec_X_Y_1 a b c d e f x)"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "rec_X_Y_2 a b c d e f (E y) = e y (rec_X_Y_2 a b c d e f y)"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "rec_X_Y_2 a b c d e f F = f"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "P (rec_X_Y_1 a b c d e f x)"
nitpick [expect = genuine]
oops

lemma "P (rec_X_Y_2 a b c d e f y)"
nitpick [expect = genuine]
oops

text {* Other datatype examples *}

text {* Indirect recursion is implemented via mutual recursion. *}

datatype XOpt = CX "XOpt option" | DX "bool => XOpt option"

lemma "P (x::XOpt)"
nitpick [expect = genuine]
oops

lemma "P (CX None)"
nitpick [expect = genuine]
oops

lemma "P (CX (Some (CX None)))"
nitpick [expect = genuine]
oops

lemma "rec_XOpt_1 cx dx n1 s1 n2 s2 (CX x) = cx x (rec_XOpt_2 cx dx n1 s1 n2 s2 x)"
nitpick [card = 1-5, expect = none]
apply simp
done

lemma "rec_XOpt_1 cx dx n1 s1 n2 s2 (DX x) = dx x (λb. rec_XOpt_3 cx dx n1 s1 n2 s2 (x b))"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_XOpt_2 cx dx n1 s1 n2 s2 None = n1"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "rec_XOpt_2 cx dx n1 s1 n2 s2 (Some x) = s1 x (rec_XOpt_1 cx dx n1 s1 n2 s2 x)"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "rec_XOpt_3 cx dx n1 s1 n2 s2 None = n2"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "rec_XOpt_3 cx dx n1 s1 n2 s2 (Some x) = s2 x (rec_XOpt_1 cx dx n1 s1 n2 s2 x)"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "P (rec_XOpt_1 cx dx n1 s1 n2 s2 x)"
nitpick [expect = genuine]
oops

lemma "P (rec_XOpt_2 cx dx n1 s1 n2 s2 x)"
nitpick [expect = genuine]
oops

lemma "P (rec_XOpt_3 cx dx n1 s1 n2 s2 x)"
nitpick [expect = genuine]
oops

datatype 'a YOpt = CY "('a => 'a YOpt) option"

lemma "P (x::'a YOpt)"
nitpick [expect = genuine]
oops

lemma "P (CY None)"
nitpick [expect = genuine]
oops

lemma "P (CY (Some (λa. CY None)))"
nitpick [expect = genuine]
oops

lemma "rec_YOpt_1 cy n s (CY x) = cy x (rec_YOpt_2 cy n s x)"
nitpick [card = 1-2, expect = none]
apply simp
done

lemma "rec_YOpt_2 cy n s None = n"
nitpick [card = 1-2, expect = none]
apply simp
done

lemma "rec_YOpt_2 cy n s (Some x) = s x (λa. rec_YOpt_1 cy n s (x a))"
nitpick [card = 1-2, expect = none]
apply simp
done

lemma "P (rec_YOpt_1 cy n s x)"
nitpick [expect = genuine]
oops

lemma "P (rec_YOpt_2 cy n s x)"
nitpick [expect = genuine]
oops

datatype Trie = TR "Trie list"

lemma "P (x::Trie)"
nitpick [expect = genuine]
oops

lemma "∀x::Trie. P x"
nitpick [expect = genuine]
oops

lemma "P (TR [TR []])"
nitpick [expect = genuine]
oops

lemma "rec_Trie_1 tr nil cons (TR x) = tr x (rec_Trie_2 tr nil cons x)"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "rec_Trie_2 tr nil cons [] = nil"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "rec_Trie_2 tr nil cons (x#xs) = cons x xs (rec_Trie_1 tr nil cons x) (rec_Trie_2 tr nil cons xs)"
nitpick [card = 1-4, expect = none]
apply simp
done

lemma "P (rec_Trie_1 tr nil cons x)"
nitpick [card = 1, expect = genuine]
oops

lemma "P (rec_Trie_2 tr nil cons x)"
nitpick [card = 1, expect = genuine]
oops

datatype InfTree = Leaf | Node "nat => InfTree"

lemma "P (x::InfTree)"
nitpick [expect = genuine]
oops

lemma "∀x::InfTree. P x"
nitpick [expect = genuine]
oops

lemma "P (Node (λn. Leaf))"
nitpick [expect = genuine]
oops

lemma "rec_InfTree leaf node Leaf = leaf"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_InfTree leaf node (Node x) = node x (λn. rec_InfTree leaf node (x n))"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "P (rec_InfTree leaf node x)"
nitpick [expect = genuine]
oops

datatype 'a lambda = Var 'a | App "'a lambda" "'a lambda" | Lam "'a => 'a lambda"

lemma "P (x::'a lambda)"
nitpick [expect = genuine]
oops

lemma "∀x::'a lambda. P x"
nitpick [expect = genuine]
oops

lemma "P (Lam (λa. Var a))"
nitpick [card = 1-5, expect = none]
nitpick [card 'a = 4, card "'a lambda" = 5, expect = genuine]
oops

lemma "rec_lambda var app lam (Var x) = var x"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_lambda var app lam (App x y) = app x y (rec_lambda var app lam x) (rec_lambda var app lam y)"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_lambda var app lam (Lam x) = lam x (λa. rec_lambda var app lam (x a))"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "P (rec_lambda v a l x)"
nitpick [expect = genuine]
oops

text {* Taken from "Inductive datatypes in HOL", p. 8: *}

datatype ('a, 'b) T = C "'a => bool" | D "'b list"
datatype 'c U = E "('c, 'c U) T"

lemma "P (x::'c U)"
nitpick [expect = genuine]
oops

lemma "∀x::'c U. P x"
nitpick [expect = genuine]
oops

lemma "P (E (C (λa. True)))"
nitpick [expect = genuine]
oops

lemma "rec_U_1 e c d nil cons (E x) = e x (rec_U_2 e c d nil cons x)"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_U_2 e c d nil cons (C x) = c x"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_U_2 e c d nil cons (D x) = d x (rec_U_3 e c d nil cons x)"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_U_3 e c d nil cons [] = nil"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "rec_U_3 e c d nil cons (x#xs) = cons x xs (rec_U_1 e c d nil cons x) (rec_U_3 e c d nil cons xs)"
nitpick [card = 1-3, expect = none]
apply simp
done

lemma "P (rec_U_1 e c d nil cons x)"
nitpick [expect = genuine]
oops

lemma "P (rec_U_2 e c d nil cons x)"
nitpick [card = 1, expect = genuine]
oops

lemma "P (rec_U_3 e c d nil cons x)"
nitpick [card = 1, expect = genuine]
oops

subsubsection {* Records *}

record ('a, 'b) point =
  xpos :: 'a
  ypos :: 'b

lemma "(x::('a, 'b) point) = y"
nitpick [expect = genuine]
oops

record ('a, 'b, 'c) extpoint = "('a, 'b) point" +
  ext :: 'c

lemma "(x::('a, 'b, 'c) extpoint) = y"
nitpick [expect = genuine]
oops

subsubsection {* Inductively Defined Sets *}

inductive_set undefinedSet :: "'a set" where
"undefined ∈ undefinedSet"

lemma "x ∈ undefinedSet"
nitpick [expect = genuine]
oops

inductive_set evenCard :: "'a set set"
where
"{} ∈ evenCard" |
"[|S ∈ evenCard; x ∉ S; y ∉ S; x ≠ y|] ==> S ∪ {x, y} ∈ evenCard"

lemma "S ∈ evenCard"
nitpick [expect = genuine]
oops

inductive_set
even :: "nat set"
and odd :: "nat set"
where
"0 ∈ even" |
"n ∈ even ==> Suc n ∈ odd" |
"n ∈ odd ==> Suc n ∈ even"

lemma "n ∈ odd"
nitpick [expect = genuine]
oops

consts f :: "'a => 'a"

inductive_set a_even :: "'a set" and a_odd :: "'a set" where
"undefined ∈ a_even" |
"x ∈ a_even ==> f x ∈ a_odd" |
"x ∈ a_odd ==> f x ∈ a_even"

lemma "x ∈ a_odd"
nitpick [expect = genuine]
oops

subsubsection {* Examples Involving Special Functions *}

lemma "card x = 0"
nitpick [expect = genuine]
oops

lemma "finite x"
nitpick [expect = none]
oops

lemma "xs @ [] = ys @ []"
nitpick [expect = genuine]
oops

lemma "xs @ ys = ys @ xs"
nitpick [expect = genuine]
oops

lemma "f (lfp f) = lfp f"
nitpick [card = 2, expect = genuine]
oops

lemma "f (gfp f) = gfp f"
nitpick [card = 2, expect = genuine]
oops

lemma "lfp f = gfp f"
nitpick [card = 2, expect = genuine]
oops

subsubsection {* Axiomatic Type Classes and Overloading *}

text {* A type class without axioms: *}

class classA

lemma "P (x::'a::classA)"
nitpick [expect = genuine]
oops

text {* An axiom with a type variable (denoting types which have at least two elements): *}

class classC =
  assumes classC_ax: "∃x y. x ≠ y"

lemma "P (x::'a::classC)"
nitpick [expect = genuine]
oops

lemma "∃x y. (x::'a::classC) ≠ y"
nitpick [expect = none]
sorry

text {* A type class for which a constant is defined: *}

class classD =
  fixes classD_const :: "'a => 'a"
  assumes classD_ax: "classD_const (classD_const x) = classD_const x"

lemma "P (x::'a::classD)"
nitpick [expect = genuine]
oops

text {* A type class with multiple superclasses: *}

class classE = classC + classD

lemma "P (x::'a::classE)"
nitpick [expect = genuine]
oops

text {* OFCLASS: *}

lemma "OFCLASS('a::type, type_class)"
nitpick [expect = none]
apply intro_classes
done

lemma "OFCLASS('a::classC, type_class)"
nitpick [expect = none]
apply intro_classes
done

lemma "OFCLASS('a::type, classC_class)"
nitpick [expect = genuine]
oops

text {* Overloading: *}

consts inverse :: "'a => 'a"

defs (overloaded)
inverse_bool: "inverse (b::bool) ≡ ¬ b"
inverse_set: "inverse (S::'a set) ≡ -S"
inverse_pair: "inverse p ≡ (inverse (fst p), inverse (snd p))"

lemma "inverse b"
nitpick [expect = genuine]
oops

lemma "P (inverse (S::'a set))"
nitpick [expect = genuine]
oops

lemma "P (inverse (p::'a×'b))"
nitpick [expect = genuine]
oops

end