(* Title: HOL/ex/Refute_Examples.thy

Author: Tjark Weber

Copyright 2003-2007

See HOL/Refute.thy for help.

*)

header {* Examples for the 'refute' command *}

theory Refute_Examples

imports "~~/src/HOL/Library/Refute"

begin

refute_params [satsolver = "dpll"]

lemma "P ∧ Q"

apply (rule conjI)

refute [expect = genuine] 1 -- {* refutes @{term "P"} *}

refute [expect = genuine] 2 -- {* refutes @{term "Q"} *}

refute [expect = genuine] -- {* equivalent to 'refute 1' *}

-- {* here 'refute 3' would cause an exception, since we only have 2 subgoals *}

refute [maxsize = 5, expect = genuine] -- {* we can override parameters ... *}

refute [satsolver = "dpll", expect = genuine] 2

-- {* ... and specify a subgoal at the same time *}

oops

(*****************************************************************************)

subsection {* Examples and Test Cases *}

subsubsection {* Propositional logic *}

lemma "True"

refute [expect = none]

by auto

lemma "False"

refute [expect = genuine]

oops

lemma "P"

refute [expect = genuine]

oops

lemma "~ P"

refute [expect = genuine]

oops

lemma "P & Q"

refute [expect = genuine]

oops

lemma "P | Q"

refute [expect = genuine]

oops

lemma "P --> Q"

refute [expect = genuine]

oops

lemma "(P::bool) = Q"

refute [expect = genuine]

oops

lemma "(P | Q) --> (P & Q)"

refute [expect = genuine]

oops

(*****************************************************************************)

subsubsection {* Predicate logic *}

lemma "P x y z"

refute [expect = genuine]

oops

lemma "P x y --> P y x"

refute [expect = genuine]

oops

lemma "P (f (f x)) --> P x --> P (f x)"

refute [expect = genuine]

oops

(*****************************************************************************)

subsubsection {* Equality *}

lemma "P = True"

refute [expect = genuine]

oops

lemma "P = False"

refute [expect = genuine]

oops

lemma "x = y"

refute [expect = genuine]

oops

lemma "f x = g x"

refute [expect = genuine]

oops

lemma "(f::'a=>'b) = g"

refute [expect = genuine]

oops

lemma "(f::('d=>'d)=>('c=>'d)) = g"

refute [expect = genuine]

oops

lemma "distinct [a, b]"

(* refute *)

apply simp

refute [expect = genuine]

oops

(*****************************************************************************)

subsubsection {* First-Order Logic *}

lemma "∃x. P x"

refute [expect = genuine]

oops

lemma "∀x. P x"

refute [expect = genuine]

oops

lemma "EX! x. P x"

refute [expect = genuine]

oops

lemma "Ex P"

refute [expect = genuine]

oops

lemma "All P"

refute [expect = genuine]

oops

lemma "Ex1 P"

refute [expect = genuine]

oops

lemma "(∃x. P x) --> (∀x. P x)"

refute [expect = genuine]

oops

lemma "(∀x. ∃y. P x y) --> (∃y. ∀x. P x y)"

refute [expect = genuine]

oops

lemma "(∃x. P x) --> (EX! x. P x)"

refute [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"

refute [maxsize = 4, expect = none]

by fast

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

lemma "a=b | a=c | a=d | a=e | b=c | b=d | b=e | c=d | c=e | d=e"

refute [expect = genuine]

oops

lemma "∀a b c d e. a=b | a=c | a=d | a=e | b=c | b=d | b=e | c=d | c=e | d=e"

refute [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"

refute [expect = genuine]

oops

text {* The "Drinker's theorem" ... *}

lemma "∃x. f x = g x --> f = g"

refute [maxsize = 4, expect = none]

by (auto simp add: ext)

text {* ... and an incorrect version of it *}

lemma "(∃x. f x = g x) --> f = g"

refute [expect = genuine]

oops

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

lemma "∃x. f x = x"

refute [expect = genuine]

oops

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

lemma "f (g x) = g (f x)"

refute [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)"

refute [expect = genuine]

oops

(*****************************************************************************)

subsubsection {* Higher-Order Logic *}

lemma "∃P. P"

refute [expect = none]

by auto

lemma "∀P. P"

refute [expect = genuine]

oops

lemma "EX! P. P"

refute [expect = none]

by auto

lemma "EX! P. P x"

refute [expect = genuine]

oops

lemma "P Q | Q x"

refute [expect = genuine]

oops

lemma "x ≠ All"

refute [expect = genuine]

oops

lemma "x ≠ Ex"

refute [expect = genuine]

oops

lemma "x ≠ Ex1"

refute [expect = genuine]

oops

text {* "The transitive closure 'T' of an arbitrary relation 'P' 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)"

refute [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))"

refute [expect = genuine]

oops

text {* "Every surjective function is invertible." *}

lemma "(∀y. ∃x. y = f x) --> (∃g. ∀x. g (f x) = x)"

refute [expect = genuine]

oops

text {* "Every invertible function is surjective." *}

lemma "(∃g. ∀x. g (f x) = x) --> (∀y. ∃x. y = f x)"

refute [expect = genuine]

oops

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

lemma "∃f. f x = x"

refute [maxsize = 4, expect = none]

apply (rule_tac x="λx. x" in exI)

by simp

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

lemma "(∀x. ∃y. P x y) --> (EX!f. ∀x. P x (f x))"

refute [expect = genuine]

oops

text {* ... and now two correct ones *}

lemma "(∀x. ∃y. P x y) --> (∃f. ∀x. P x (f x))"

refute [maxsize = 4, expect = none]

by (simp add: choice)

lemma "(∀x. EX!y. P x y) --> (EX!f. ∀x. P x (f x))"

refute [maxsize = 2, expect = none]

apply auto

apply (simp add: ex1_implies_ex choice)

by (fast intro: ext)

(*****************************************************************************)

subsubsection {* Meta-logic *}

lemma "!!x. P x"

refute [expect = genuine]

oops

lemma "f x == g x"

refute [expect = genuine]

oops

lemma "P ==> Q"

refute [expect = genuine]

oops

lemma "[| P; Q; R |] ==> S"

refute [expect = genuine]

oops

lemma "(x == all) ==> False"

refute [expect = genuine]

oops

lemma "(x == (op ==)) ==> False"

refute [expect = genuine]

oops

lemma "(x == (op ==>)) ==> False"

refute [expect = genuine]

oops

(*****************************************************************************)

subsubsection {* Schematic variables *}

schematic_lemma "?P"

refute [expect = none]

by auto

schematic_lemma "x = ?y"

refute [expect = none]

by auto

(******************************************************************************)

subsubsection {* Abstractions *}

lemma "(λx. x) = (λx. y)"

refute [expect = genuine]

oops

lemma "(λf. f x) = (λf. True)"

refute [expect = genuine]

oops

lemma "(λx. x) = (λy. y)"

refute

by simp

(*****************************************************************************)

subsubsection {* Sets *}

lemma "P (A::'a set)"

refute

oops

lemma "P (A::'a set set)"

refute

oops

lemma "{x. P x} = {y. P y}"

refute

by simp

lemma "x : {x. P x}"

refute

oops

lemma "P op:"

refute

oops

lemma "P (op: x)"

refute

oops

lemma "P Collect"

refute

oops

lemma "A Un B = A Int B"

refute

oops

lemma "(A Int B) Un C = (A Un C) Int B"

refute

oops

lemma "Ball A P --> Bex A P"

refute

oops

(*****************************************************************************)

subsubsection {* undefined *}

lemma "undefined"

refute [expect = genuine]

oops

lemma "P undefined"

refute [expect = genuine]

oops

lemma "undefined x"

refute [expect = genuine]

oops

lemma "undefined undefined"

refute [expect = genuine]

oops

(*****************************************************************************)

subsubsection {* The *}

lemma "The P"

refute [expect = genuine]

oops

lemma "P The"

refute [expect = genuine]

oops

lemma "P (The P)"

refute [expect = genuine]

oops

lemma "(THE x. x=y) = z"

refute [expect = genuine]

oops

lemma "Ex P --> P (The P)"

refute [expect = genuine]

oops

(*****************************************************************************)

subsubsection {* Eps *}

lemma "Eps P"

refute [expect = genuine]

oops

lemma "P Eps"

refute [expect = genuine]

oops

lemma "P (Eps P)"

refute [expect = genuine]

oops

lemma "(SOME x. x=y) = z"

refute [expect = genuine]

oops

lemma "Ex P --> P (Eps P)"

refute [maxsize = 3, expect = none]

by (auto simp add: someI)

(*****************************************************************************)

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"

refute

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)"

refute

oops

(*****************************************************************************)

subsubsection {* Inductive datatypes *}

text {* With @{text quick_and_dirty} set, the datatype package does

not generate certain axioms for recursion operators. Without these

axioms, Refute may find spurious countermodels. *}

text {* unit *}

lemma "P (x::unit)"

refute [expect = genuine]

oops

lemma "∀x::unit. P x"

refute [expect = genuine]

oops

lemma "P ()"

refute [expect = genuine]

oops

lemma "unit_rec u x = u"

refute [expect = none]

by simp

lemma "P (unit_rec u x)"

refute [expect = genuine]

oops

lemma "P (case x of () => u)"

refute [expect = genuine]

oops

text {* option *}

lemma "P (x::'a option)"

refute [expect = genuine]

oops

lemma "∀x::'a option. P x"

refute [expect = genuine]

oops

lemma "P None"

refute [expect = genuine]

oops

lemma "P (Some x)"

refute [expect = genuine]

oops

lemma "option_rec n s None = n"

refute [expect = none]

by simp

lemma "option_rec n s (Some x) = s x"

refute [maxsize = 4, expect = none]

by simp

lemma "P (option_rec n s x)"

refute [expect = genuine]

oops

lemma "P (case x of None => n | Some u => s u)"

refute [expect = genuine]

oops

text {* * *}

lemma "P (x::'a*'b)"

refute [expect = genuine]

oops

lemma "∀x::'a*'b. P x"

refute [expect = genuine]

oops

lemma "P (x, y)"

refute [expect = genuine]

oops

lemma "P (fst x)"

refute [expect = genuine]

oops

lemma "P (snd x)"

refute [expect = genuine]

oops

lemma "P Pair"

refute [expect = genuine]

oops

lemma "prod_rec p (a, b) = p a b"

refute [maxsize = 2, expect = none]

by simp

lemma "P (prod_rec p x)"

refute [expect = genuine]

oops

lemma "P (case x of Pair a b => p a b)"

refute [expect = genuine]

oops

text {* + *}

lemma "P (x::'a+'b)"

refute [expect = genuine]

oops

lemma "∀x::'a+'b. P x"

refute [expect = genuine]

oops

lemma "P (Inl x)"

refute [expect = genuine]

oops

lemma "P (Inr x)"

refute [expect = genuine]

oops

lemma "P Inl"

refute [expect = genuine]

oops

lemma "sum_rec l r (Inl x) = l x"

refute [maxsize = 3, expect = none]

by simp

lemma "sum_rec l r (Inr x) = r x"

refute [maxsize = 3, expect = none]

by simp

lemma "P (sum_rec l r x)"

refute [expect = genuine]

oops

lemma "P (case x of Inl a => l a | Inr b => r b)"

refute [expect = genuine]

oops

text {* Non-recursive datatypes *}

datatype T1 = A | B

lemma "P (x::T1)"

refute [expect = genuine]

oops

lemma "∀x::T1. P x"

refute [expect = genuine]

oops

lemma "P A"

refute [expect = genuine]

oops

lemma "P B"

refute [expect = genuine]

oops

lemma "T1_rec a b A = a"

refute [expect = none]

by simp

lemma "T1_rec a b B = b"

refute [expect = none]

by simp

lemma "P (T1_rec a b x)"

refute [expect = genuine]

oops

lemma "P (case x of A => a | B => b)"

refute [expect = genuine]

oops

datatype 'a T2 = C T1 | D 'a

lemma "P (x::'a T2)"

refute [expect = genuine]

oops

lemma "∀x::'a T2. P x"

refute [expect = genuine]

oops

lemma "P D"

refute [expect = genuine]

oops

lemma "T2_rec c d (C x) = c x"

refute [maxsize = 4, expect = none]

by simp

lemma "T2_rec c d (D x) = d x"

refute [maxsize = 4, expect = none]

by simp

lemma "P (T2_rec c d x)"

refute [expect = genuine]

oops

lemma "P (case x of C u => c u | D v => d v)"

refute [expect = genuine]

oops

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

lemma "P (x::('a,'b) T3)"

refute [expect = genuine]

oops

lemma "∀x::('a,'b) T3. P x"

refute [expect = genuine]

oops

lemma "P E"

refute [expect = genuine]

oops

lemma "T3_rec e (E x) = e x"

refute [maxsize = 2, expect = none]

by simp

lemma "P (T3_rec e x)"

refute [expect = genuine]

oops

lemma "P (case x of E f => e f)"

refute [expect = genuine]

oops

text {* Recursive datatypes *}

text {* nat *}

lemma "P (x::nat)"

refute [expect = potential]

oops

lemma "∀x::nat. P x"

refute [expect = potential]

oops

lemma "P (Suc 0)"

refute [expect = potential]

oops

lemma "P Suc"

refute [maxsize = 3, expect = none]

-- {* @{term Suc} is a partial function (regardless of the size

of the model), hence @{term "P Suc"} is undefined and no

model will be found *}

oops

lemma "nat_rec zero suc 0 = zero"

refute [expect = none]

by simp

lemma "nat_rec zero suc (Suc x) = suc x (nat_rec zero suc x)"

refute [maxsize = 2, expect = none]

by simp

lemma "P (nat_rec zero suc x)"

refute [expect = potential]

oops

lemma "P (case x of 0 => zero | Suc n => suc n)"

refute [expect = potential]

oops

text {* 'a list *}

lemma "P (xs::'a list)"

refute [expect = potential]

oops

lemma "∀xs::'a list. P xs"

refute [expect = potential]

oops

lemma "P [x, y]"

refute [expect = potential]

oops

lemma "list_rec nil cons [] = nil"

refute [maxsize = 3, expect = none]

by simp

lemma "list_rec nil cons (x#xs) = cons x xs (list_rec nil cons xs)"

refute [maxsize = 2, expect = none]

by simp

lemma "P (list_rec nil cons xs)"

refute [expect = potential]

oops

lemma "P (case x of Nil => nil | Cons a b => cons a b)"

refute [expect = potential]

oops

lemma "(xs::'a list) = ys"

refute [expect = potential]

oops

lemma "a # xs = b # xs"

refute [expect = potential]

oops

datatype BitList = BitListNil | Bit0 BitList | Bit1 BitList

lemma "P (x::BitList)"

refute [expect = potential]

oops

lemma "∀x::BitList. P x"

refute [expect = potential]

oops

lemma "P (Bit0 (Bit1 BitListNil))"

refute [expect = potential]

oops

lemma "BitList_rec nil bit0 bit1 BitListNil = nil"

refute [maxsize = 4, expect = none]

by simp

lemma "BitList_rec nil bit0 bit1 (Bit0 xs) = bit0 xs (BitList_rec nil bit0 bit1 xs)"

refute [maxsize = 2, expect = none]

by simp

lemma "BitList_rec nil bit0 bit1 (Bit1 xs) = bit1 xs (BitList_rec nil bit0 bit1 xs)"

refute [maxsize = 2, expect = none]

by simp

lemma "P (BitList_rec nil bit0 bit1 x)"

refute [expect = potential]

oops

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

lemma "P (x::'a BinTree)"

refute [expect = potential]

oops

lemma "∀x::'a BinTree. P x"

refute [expect = potential]

oops

lemma "P (Node (Leaf x) (Leaf y))"

refute [expect = potential]

oops

lemma "BinTree_rec l n (Leaf x) = l x"

refute [maxsize = 1, expect = none]

(* The "maxsize = 1" tests are a bit pointless: for some formulae

below, refute will find no countermodel simply because this

size makes involved terms undefined. Unfortunately, any

larger size already takes too long. *)

by simp

lemma "BinTree_rec l n (Node x y) = n x y (BinTree_rec l n x) (BinTree_rec l n y)"

refute [maxsize = 1, expect = none]

by simp

lemma "P (BinTree_rec l n x)"

refute [expect = potential]

oops

lemma "P (case x of Leaf a => l a | Node a b => n a b)"

refute [expect = potential]

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)"

refute [expect = potential]

oops

lemma "∀x::'a aexp. P x"

refute [expect = potential]

oops

lemma "P (ITE (Equal (Number x) (Number y)) (Number x) (Number y))"

refute [expect = potential]

oops

lemma "P (x::'a bexp)"

refute [expect = potential]

oops

lemma "∀x::'a bexp. P x"

refute [expect = potential]

oops

lemma "aexp_bexp_rec_1 number ite equal (Number x) = number x"

refute [maxsize = 1, expect = none]

by simp

lemma "aexp_bexp_rec_1 number ite equal (ITE x y z) = ite x y z (aexp_bexp_rec_2 number ite equal x) (aexp_bexp_rec_1 number ite equal y) (aexp_bexp_rec_1 number ite equal z)"

refute [maxsize = 1, expect = none]

by simp

lemma "P (aexp_bexp_rec_1 number ite equal x)"

refute [expect = potential]

oops

lemma "P (case x of Number a => number a | ITE b a1 a2 => ite b a1 a2)"

refute [expect = potential]

oops

lemma "aexp_bexp_rec_2 number ite equal (Equal x y) = equal x y (aexp_bexp_rec_1 number ite equal x) (aexp_bexp_rec_1 number ite equal y)"

refute [maxsize = 1, expect = none]

by simp

lemma "P (aexp_bexp_rec_2 number ite equal x)"

refute [expect = potential]

oops

lemma "P (case x of Equal a1 a2 => equal a1 a2)"

refute [expect = potential]

oops

datatype X = A | B X | C Y

and Y = D X | E Y | F

lemma "P (x::X)"

refute [expect = potential]

oops

lemma "P (y::Y)"

refute [expect = potential]

oops

lemma "P (B (B A))"

refute [expect = potential]

oops

lemma "P (B (C F))"

refute [expect = potential]

oops

lemma "P (C (D A))"

refute [expect = potential]

oops

lemma "P (C (E F))"

refute [expect = potential]

oops

lemma "P (D (B A))"

refute [expect = potential]

oops

lemma "P (D (C F))"

refute [expect = potential]

oops

lemma "P (E (D A))"

refute [expect = potential]

oops

lemma "P (E (E F))"

refute [expect = potential]

oops

lemma "P (C (D (C F)))"

refute [expect = potential]

oops

lemma "X_Y_rec_1 a b c d e f A = a"

refute [maxsize = 3, expect = none]

by simp

lemma "X_Y_rec_1 a b c d e f (B x) = b x (X_Y_rec_1 a b c d e f x)"

refute [maxsize = 1, expect = none]

by simp

lemma "X_Y_rec_1 a b c d e f (C y) = c y (X_Y_rec_2 a b c d e f y)"

refute [maxsize = 1, expect = none]

by simp

lemma "X_Y_rec_2 a b c d e f (D x) = d x (X_Y_rec_1 a b c d e f x)"

refute [maxsize = 1, expect = none]

by simp

lemma "X_Y_rec_2 a b c d e f (E y) = e y (X_Y_rec_2 a b c d e f y)"

refute [maxsize = 1, expect = none]

by simp

lemma "X_Y_rec_2 a b c d e f F = f"

refute [maxsize = 3, expect = none]

by simp

lemma "P (X_Y_rec_1 a b c d e f x)"

refute [expect = potential]

oops

lemma "P (X_Y_rec_2 a b c d e f y)"

refute [expect = potential]

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)"

refute [expect = potential]

oops

lemma "P (CX None)"

refute [expect = potential]

oops

lemma "P (CX (Some (CX None)))"

refute [expect = potential]

oops

lemma "XOpt_rec_1 cx dx n1 s1 n2 s2 (CX x) = cx x (XOpt_rec_2 cx dx n1 s1 n2 s2 x)"

refute [maxsize = 1, expect = none]

by simp

lemma "XOpt_rec_1 cx dx n1 s1 n2 s2 (DX x) = dx x (λb. XOpt_rec_3 cx dx n1 s1 n2 s2 (x b))"

refute [maxsize = 1, expect = none]

by simp

lemma "XOpt_rec_2 cx dx n1 s1 n2 s2 None = n1"

refute [maxsize = 2, expect = none]

by simp

lemma "XOpt_rec_2 cx dx n1 s1 n2 s2 (Some x) = s1 x (XOpt_rec_1 cx dx n1 s1 n2 s2 x)"

refute [maxsize = 1, expect = none]

by simp

lemma "XOpt_rec_3 cx dx n1 s1 n2 s2 None = n2"

refute [maxsize = 2, expect = none]

by simp

lemma "XOpt_rec_3 cx dx n1 s1 n2 s2 (Some x) = s2 x (XOpt_rec_1 cx dx n1 s1 n2 s2 x)"

refute [maxsize = 1, expect = none]

by simp

lemma "P (XOpt_rec_1 cx dx n1 s1 n2 s2 x)"

refute [expect = potential]

oops

lemma "P (XOpt_rec_2 cx dx n1 s1 n2 s2 x)"

refute [expect = potential]

oops

lemma "P (XOpt_rec_3 cx dx n1 s1 n2 s2 x)"

refute [expect = potential]

oops

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

lemma "P (x::'a YOpt)"

refute [expect = potential]

oops

lemma "P (CY None)"

refute [expect = potential]

oops

lemma "P (CY (Some (λa. CY None)))"

refute [expect = potential]

oops

lemma "YOpt_rec_1 cy n s (CY x) = cy x (YOpt_rec_2 cy n s x)"

refute [maxsize = 1, expect = none]

by simp

lemma "YOpt_rec_2 cy n s None = n"

refute [maxsize = 2, expect = none]

by simp

lemma "YOpt_rec_2 cy n s (Some x) = s x (λa. YOpt_rec_1 cy n s (x a))"

refute [maxsize = 1, expect = none]

by simp

lemma "P (YOpt_rec_1 cy n s x)"

refute [expect = potential]

oops

lemma "P (YOpt_rec_2 cy n s x)"

refute [expect = potential]

oops

datatype Trie = TR "Trie list"

lemma "P (x::Trie)"

refute [expect = potential]

oops

lemma "∀x::Trie. P x"

refute [expect = potential]

oops

lemma "P (TR [TR []])"

refute [expect = potential]

oops

lemma "Trie_rec_1 tr nil cons (TR x) = tr x (Trie_rec_2 tr nil cons x)"

refute [maxsize = 1, expect = none]

by simp

lemma "Trie_rec_2 tr nil cons [] = nil"

refute [maxsize = 3, expect = none]

by simp

lemma "Trie_rec_2 tr nil cons (x#xs) = cons x xs (Trie_rec_1 tr nil cons x) (Trie_rec_2 tr nil cons xs)"

refute [maxsize = 1, expect = none]

by simp

lemma "P (Trie_rec_1 tr nil cons x)"

refute [expect = potential]

oops

lemma "P (Trie_rec_2 tr nil cons x)"

refute [expect = potential]

oops

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

lemma "P (x::InfTree)"

refute [expect = potential]

oops

lemma "∀x::InfTree. P x"

refute [expect = potential]

oops

lemma "P (Node (λn. Leaf))"

refute [expect = potential]

oops

lemma "InfTree_rec leaf node Leaf = leaf"

refute [maxsize = 2, expect = none]

by simp

lemma "InfTree_rec leaf node (Node x) = node x (λn. InfTree_rec leaf node (x n))"

refute [maxsize = 1, expect = none]

by simp

lemma "P (InfTree_rec leaf node x)"

refute [expect = potential]

oops

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

lemma "P (x::'a lambda)"

refute [expect = potential]

oops

lemma "∀x::'a lambda. P x"

refute [expect = potential]

oops

lemma "P (Lam (λa. Var a))"

refute [expect = potential]

oops

lemma "lambda_rec var app lam (Var x) = var x"

refute [maxsize = 1, expect = none]

by simp

lemma "lambda_rec var app lam (App x y) = app x y (lambda_rec var app lam x) (lambda_rec var app lam y)"

refute [maxsize = 1, expect = none]

by simp

lemma "lambda_rec var app lam (Lam x) = lam x (λa. lambda_rec var app lam (x a))"

refute [maxsize = 1, expect = none]

by simp

lemma "P (lambda_rec v a l x)"

refute [expect = potential]

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)"

refute [expect = potential]

oops

lemma "∀x::'c U. P x"

refute [expect = potential]

oops

lemma "P (E (C (λa. True)))"

refute [expect = potential]

oops

lemma "U_rec_1 e c d nil cons (E x) = e x (U_rec_2 e c d nil cons x)"

refute [maxsize = 1, expect = none]

by simp

lemma "U_rec_2 e c d nil cons (C x) = c x"

refute [maxsize = 1, expect = none]

by simp

lemma "U_rec_2 e c d nil cons (D x) = d x (U_rec_3 e c d nil cons x)"

refute [maxsize = 1, expect = none]

by simp

lemma "U_rec_3 e c d nil cons [] = nil"

refute [maxsize = 2, expect = none]

by simp

lemma "U_rec_3 e c d nil cons (x#xs) = cons x xs (U_rec_1 e c d nil cons x) (U_rec_3 e c d nil cons xs)"

refute [maxsize = 1, expect = none]

by simp

lemma "P (U_rec_1 e c d nil cons x)"

refute [expect = potential]

oops

lemma "P (U_rec_2 e c d nil cons x)"

refute [expect = potential]

oops

lemma "P (U_rec_3 e c d nil cons x)"

refute [expect = potential]

oops

(*****************************************************************************)

subsubsection {* Records *}

(*TODO: make use of pair types, rather than typedef, for record types*)

record ('a, 'b) point =

xpos :: 'a

ypos :: 'b

lemma "(x::('a, 'b) point) = y"

refute

oops

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

ext :: 'c

lemma "(x::('a, 'b, 'c) extpoint) = y"

refute

oops

(*****************************************************************************)

subsubsection {* Inductively defined sets *}

inductive_set arbitrarySet :: "'a set"

where

"undefined : arbitrarySet"

lemma "x : arbitrarySet"

refute

oops

inductive_set evenCard :: "'a set set"

where

"{} : evenCard"

| "[| S : evenCard; x ∉ S; y ∉ S; x ≠ y |] ==> S ∪ {x, y} : evenCard"

lemma "S : evenCard"

refute

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"

(* refute *) (* TODO: there seems to be an issue here with undefined terms

because of the recursive datatype "nat" *)

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"

(* refute [expect = genuine] -- {* finds a model of size 2 *}

NO LONGER WORKS since "lfp"'s interpreter is disabled *)

oops

(*****************************************************************************)

subsubsection {* Examples involving special functions *}

lemma "card x = 0"

refute

oops

lemma "finite x"

refute -- {* no finite countermodel exists *}

oops

lemma "(x::nat) + y = 0"

refute [expect = potential]

oops

lemma "(x::nat) = x + x"

refute [expect = potential]

oops

lemma "(x::nat) - y + y = x"

refute [expect = potential]

oops

lemma "(x::nat) = x * x"

refute [expect = potential]

oops

lemma "(x::nat) < x + y"

refute [expect = potential]

oops

lemma "xs @ [] = ys @ []"

refute [expect = potential]

oops

lemma "xs @ ys = ys @ xs"

refute [expect = potential]

oops

lemma "f (lfp f) = lfp f"

refute

oops

lemma "f (gfp f) = gfp f"

refute

oops

lemma "lfp f = gfp f"

refute

oops

(*****************************************************************************)

subsubsection {* Type classes and overloading *}

text {* A type class without axioms: *}

class classA

lemma "P (x::'a::classA)"

refute [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)"

refute [expect = genuine]

oops

lemma "∃x y. (x::'a::classC) ≠ y"

(* refute [expect = none] FIXME *)

oops

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)"

refute [expect = genuine]

oops

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

class classE = classC + classD

lemma "P (x::'a::classE)"

refute [expect = genuine]

oops

text {* OFCLASS: *}

lemma "OFCLASS('a::type, type_class)"

refute [expect = none]

by intro_classes

lemma "OFCLASS('a::classC, type_class)"

refute [expect = none]

by intro_classes

lemma "OFCLASS('a::type, classC_class)"

refute [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"

refute [expect = genuine]

oops

lemma "P (inverse (S::'a set))"

refute [expect = genuine]

oops

lemma "P (inverse (p::'a×'b))"

refute [expect = genuine]

oops

text {* Structured proofs *}

lemma "x = y"

proof cases

assume "x = y"

show ?thesis

refute [expect = none]

refute [no_assms, expect = genuine]

refute [no_assms = false, expect = none]

oops

refute_params [satsolver = "auto"]

end