(* Title: HOL/Integ/Presburger.thy
ID: $Id$
Author: Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen
File containing necessary theorems for the proof
generation for Cooper Algorithm
*)
header {* Presburger Arithmetic: Cooper's Algorithm *}
theory Presburger
imports NatSimprocs
uses
("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML")
("reflected_presburger.ML") ("reflected_cooper.ML") ("presburger.ML")
begin
text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)"
apply (rule iffI)
apply (erule exE)
apply (rule_tac x = "l * x" in exI)
apply simp
apply (erule exE)
apply (erule conjE)
apply (erule dvdE)
apply (rule_tac x = k in exI)
apply simp
done
lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)"
apply(unfold dvd_def)
apply(rule iffI)
apply(clarsimp)
apply(rename_tac k)
apply(rule_tac x = "-k" in exI)
apply simp
apply(clarsimp)
apply(rename_tac k)
apply(rule_tac x = "-k" in exI)
apply simp
done
lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)"
apply(unfold dvd_def)
apply(rule iffI)
apply(clarsimp)
apply(rule_tac x = "-k" in exI)
apply simp
apply(clarsimp)
apply(rule_tac x = "-k" in exI)
apply simp
done
text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} less than some integer @{text z}.*}
theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
\<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
\<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
apply (erule exE)+
apply (rule_tac x = "min z1 z2" in exI)
apply simp
done
theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
\<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
\<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
apply (erule exE)+
apply (rule_tac x = "min z1 z2" in exI)
apply simp
done
text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} greather than some integer @{text z}.*}
theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
\<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
\<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))"
apply (erule exE)+
apply (rule_tac x = "max z1 z2" in exI)
apply simp
done
theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow>
\<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow>
\<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))"
apply (erule exE)+
apply (rule_tac x = "max z1 z2" in exI)
apply simp
done
text {*
\medskip Theorems for the combination of proofs of the modulo @{text
D} property for @{text "P plusinfinity"}
FIXME: This is THE SAME theorem as for the @{text minusinf} version,
but with @{text "+k.."} instead of @{text "-k.."} In the future
replace these both with only one. *}
theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
\<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
\<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))"
by simp
theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow>
\<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow>
\<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))"
by simp
text {*
This is one of the cases where the simplifed formula is prooved to
habe some property (in relation to @{text P_m}) but we need to prove
the property for the original formula (@{text P_m})
FIXME: This is exaclty the same thm as for @{text minusinf}. *}
lemma pinf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x)) ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
by blast
text {*
\medskip Theorems for the combination of proofs of the modulo @{text D}
property for @{text "P minusinfinity"} *}
theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
\<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
\<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))"
by simp
theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow>
\<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow>
\<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))"
by simp
text {*
This is one of the cases where the simplifed formula is prooved to
have some property (in relation to @{text P_m}) but we need to
prove the property for the original formula (@{text P_m}). *}
lemma minf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x)) ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) "
by blast
text {*
Theorem needed for proving at runtime divide properties using the
arithmetic tactic (which knows only about modulo = 0). *}
lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
by(simp add:dvd_def zmod_eq_0_iff)
text {*
\medskip Theorems used for the combination of proof for the
backwards direction of Cooper's Theorem. They rely exclusively on
Predicate calculus.*}
lemma not_ast_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P1(x) --> P1(x + d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<or> P2(x)) --> (P1(x + d) \<or> P2(x + d))) "
by blast
lemma not_ast_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a- j)) --> P1(x) --> P1(x + d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<and> P2(x)) --> (P1(x + d)
\<and> P2(x + d))) "
by blast
lemma not_ast_p_Q_elim: "
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d))
==> ( P = Q )
==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))"
by blast
text {*
\medskip Theorems used for the combination of proof for the
backwards direction of Cooper's Theorem. They rely exclusively on
Predicate calculus.*}
lemma not_bst_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<or> P2(x)) --> (P1(x - d)
\<or> P2(x-d))) "
by blast
lemma not_bst_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d))
==>
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<and> P2(x)) --> (P1(x - d)
\<and> P2(x-d))) "
by blast
lemma not_bst_p_Q_elim: "
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d))
==> ( P = Q )
==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))"
by blast
text {* \medskip This is the first direction of Cooper's Theorem. *}
lemma cooper_thm: "(R --> (EX x::int. P x)) ==> (Q -->(EX x::int. P x )) ==> ((R|Q) --> (EX x::int. P x )) "
by blast
text {*
\medskip The full Cooper's Theorem in its equivalence Form. Given
the premises it is trivial too, it relies exclusively on prediacte calculus.*}
lemma cooper_eq_thm: "(R --> (EX x::int. P x)) ==> (Q -->(EX x::int. P x )) ==> ((~Q)
--> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q "
by blast
text {*
\medskip Some of the atomic theorems generated each time the atom
does not depend on @{text x}, they are trivial.*}
lemma fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) "
by blast
lemma fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)"
by blast
lemma not_bst_p_fm: "ALL (x::int). Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> fm --> fm"
by blast
lemma fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) "
by blast
text {* The next two thms are the same as the @{text minusinf} version. *}
lemma fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)"
by blast
lemma not_ast_p_fm: "ALL (x::int). Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> fm --> fm"
by blast
text {* Theorems to be deleted from simpset when proving simplified formulaes. *}
lemma P_eqtrue: "(P=True) = P"
by iprover
lemma P_eqfalse: "(P=False) = (~P)"
by iprover
text {*
\medskip Theorems for the generation of the bachwards direction of
Cooper's Theorem.
These are the 6 interesting atomic cases which have to be proved relying on the
properties of B-set and the arithmetic and contradiction proofs. *}
lemma not_bst_p_lt: "0 < (d::int) ==>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ( 0 < -x + a) --> (0 < -(x - d) + a )"
by arith
lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 < (x) + a) --> ( 0 < (x - d) + a)"
apply clarsimp
apply(rule ccontr)
apply(drule_tac x = "x+a" in bspec)
apply(simp add:atLeastAtMost_iff)
apply(drule_tac x = "-a" in bspec)
apply assumption
apply(simp)
done
lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 = x + a) --> (0 = (x - d) + a )"
apply clarsimp
apply(subgoal_tac "x = -a")
prefer 2 apply arith
apply(drule_tac x = "1" in bspec)
apply(simp add:atLeastAtMost_iff)
apply(drule_tac x = "-a- 1" in bspec)
apply assumption
apply(simp)
done
lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(0 = x + a) --> ~(0 = (x - d) + a)"
apply clarsimp
apply(subgoal_tac "x = -a+d")
prefer 2 apply arith
apply(drule_tac x = "d" in bspec)
apply(simp add:atLeastAtMost_iff)
apply(drule_tac x = "-a" in bspec)
apply assumption
apply(simp)
done
lemma not_bst_p_dvd: "(d1::int) dvd d ==>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> d1 dvd (x + a) --> d1 dvd ((x - d) + a )"
apply(clarsimp simp add:dvd_def)
apply(rename_tac m)
apply(rule_tac x = "m - k" in exI)
apply(simp add:int_distrib)
done
lemma not_bst_p_ndvd: "(d1::int) dvd d ==>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(d1 dvd (x + a)) --> ~(d1 dvd ((x - d) + a ))"
apply(clarsimp simp add:dvd_def)
apply(rename_tac m)
apply(erule_tac x = "m + k" in allE)
apply(simp add:int_distrib)
done
text {*
\medskip Theorems for the generation of the bachwards direction of
Cooper's Theorem.
These are the 6 interesting atomic cases which have to be proved
relying on the properties of A-set ant the arithmetic and
contradiction proofs. *}
lemma not_ast_p_gt: "0 < (d::int) ==>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ( 0 < x + t) --> (0 < (x + d) + t )"
by arith
lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 < -x + t) --> ( 0 < -(x + d) + t)"
apply clarsimp
apply (rule ccontr)
apply (drule_tac x = "t-x" in bspec)
apply simp
apply (drule_tac x = "t" in bspec)
apply assumption
apply simp
done
lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 = x + t) --> (0 = (x + d) + t )"
apply clarsimp
apply (drule_tac x="1" in bspec)
apply simp
apply (drule_tac x="- t + 1" in bspec)
apply assumption
apply(subgoal_tac "x = -t")
prefer 2 apply arith
apply simp
done
lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(0 = x + t) --> ~(0 = (x + d) + t)"
apply clarsimp
apply (subgoal_tac "x = -t-d")
prefer 2 apply arith
apply (drule_tac x = "d" in bspec)
apply simp
apply (drule_tac x = "-t" in bspec)
apply assumption
apply simp
done
lemma not_ast_p_dvd: "(d1::int) dvd d ==>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> d1 dvd (x + t) --> d1 dvd ((x + d) + t )"
apply(clarsimp simp add:dvd_def)
apply(rename_tac m)
apply(rule_tac x = "m + k" in exI)
apply(simp add:int_distrib)
done
lemma not_ast_p_ndvd: "(d1::int) dvd d ==>
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(d1 dvd (x + t)) --> ~(d1 dvd ((x + d) + t ))"
apply(clarsimp simp add:dvd_def)
apply(rename_tac m)
apply(erule_tac x = "m - k" in allE)
apply(simp add:int_distrib)
done
text {*
\medskip These are the atomic cases for the proof generation for the
modulo @{text D} property for @{text "P plusinfinity"}
They are fully based on arithmetics. *}
lemma dvd_modd_pinf: "((d::int) dvd d1) ==>
(ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))"
apply(clarsimp simp add:dvd_def)
apply(rule iffI)
apply(clarsimp)
apply(rename_tac n m)
apply(rule_tac x = "m + n*k" in exI)
apply(simp add:int_distrib)
apply(clarsimp)
apply(rename_tac n m)
apply(rule_tac x = "m - n*k" in exI)
apply(simp add:int_distrib mult_ac)
done
lemma not_dvd_modd_pinf: "((d::int) dvd d1) ==>
(ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))"
apply(clarsimp simp add:dvd_def)
apply(rule iffI)
apply(clarsimp)
apply(rename_tac n m)
apply(erule_tac x = "m - n*k" in allE)
apply(simp add:int_distrib mult_ac)
apply(clarsimp)
apply(rename_tac n m)
apply(erule_tac x = "m + n*k" in allE)
apply(simp add:int_distrib mult_ac)
done
text {*
\medskip These are the atomic cases for the proof generation for the
equivalence of @{text P} and @{text "P plusinfinity"} for integers
@{text x} greater than some integer @{text z}.
They are fully based on arithmetics. *}
lemma eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )"
apply(rule_tac x = "-t" in exI)
apply simp
done
lemma neq_eq_pinf: "EX z::int. ALL x. z < x --> ((~( 0 = x +t )) = True )"
apply(rule_tac x = "-t" in exI)
apply simp
done
lemma le_eq_pinf: "EX z::int. ALL x. z < x --> ( 0 < x +t = True )"
apply(rule_tac x = "-t" in exI)
apply simp
done
lemma len_eq_pinf: "EX z::int. ALL x. z < x --> (0 < -x +t = False )"
apply(rule_tac x = "t" in exI)
apply simp
done
lemma dvd_eq_pinf: "EX z::int. ALL x. z < x --> ((d dvd (x + t)) = (d dvd (x + t))) "
by simp
lemma not_dvd_eq_pinf: "EX z::int. ALL x. z < x --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
by simp
text {*
\medskip These are the atomic cases for the proof generation for the
modulo @{text D} property for @{text "P minusinfinity"}.
They are fully based on arithmetics. *}
lemma dvd_modd_minf: "((d::int) dvd d1) ==>
(ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))"
apply(clarsimp simp add:dvd_def)
apply(rule iffI)
apply(clarsimp)
apply(rename_tac n m)
apply(rule_tac x = "m - n*k" in exI)
apply(simp add:int_distrib)
apply(clarsimp)
apply(rename_tac n m)
apply(rule_tac x = "m + n*k" in exI)
apply(simp add:int_distrib mult_ac)
done
lemma not_dvd_modd_minf: "((d::int) dvd d1) ==>
(ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))"
apply(clarsimp simp add:dvd_def)
apply(rule iffI)
apply(clarsimp)
apply(rename_tac n m)
apply(erule_tac x = "m + n*k" in allE)
apply(simp add:int_distrib mult_ac)
apply(clarsimp)
apply(rename_tac n m)
apply(erule_tac x = "m - n*k" in allE)
apply(simp add:int_distrib mult_ac)
done
text {*
\medskip These are the atomic cases for the proof generation for the
equivalence of @{text P} and @{text "P minusinfinity"} for integers
@{text x} less than some integer @{text z}.
They are fully based on arithmetics. *}
lemma eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )"
apply(rule_tac x = "-t" in exI)
apply simp
done
lemma neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )"
apply(rule_tac x = "-t" in exI)
apply simp
done
lemma le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t = False )"
apply(rule_tac x = "-t" in exI)
apply simp
done
lemma len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t = True )"
apply(rule_tac x = "t" in exI)
apply simp
done
lemma dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) "
by simp
lemma not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) "
by simp
text {*
\medskip This Theorem combines whithnesses about @{text "P
minusinfinity"} to show one component of the equivalence proof for
Cooper's Theorem.
FIXME: remove once they are part of the distribution. *}
theorem int_ge_induct[consumes 1,case_names base step]:
assumes ge: "k \<le> (i::int)" and
base: "P(k)" and
step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
shows "P i"
proof -
{ fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i"
proof (induct n)
case 0
hence "i = k" by arith
thus "P i" using base by simp
next
case (Suc n)
hence "n = nat((i - 1) - k)" by arith
moreover
have ki1: "k \<le> i - 1" using Suc.prems by arith
ultimately
have "P(i - 1)" by(rule Suc.hyps)
from step[OF ki1 this] show ?case by simp
qed
}
from this ge show ?thesis by fast
qed
theorem int_gr_induct[consumes 1,case_names base step]:
assumes gr: "k < (i::int)" and
base: "P(k+1)" and
step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
shows "P i"
apply(rule int_ge_induct[of "k + 1"])
using gr apply arith
apply(rule base)
apply(rule step)
apply simp+
done
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
apply(induct rule: int_gr_induct)
apply simp
apply (simp add:int_distrib)
done
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
apply(induct rule: int_gr_induct)
apply simp
apply (simp add:int_distrib)
done
lemma minusinfinity:
assumes "0 < d" and
P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and
ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
proof
assume eP1: "EX x. P1 x"
then obtain x where P1: "P1 x" ..
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
let ?w = "x - (abs(x-z)+1) * d"
show "EX x. P x"
proof
have w: "?w < z" by(rule decr_lemma)
have "P1 x = P1 ?w" using P1eqP1 by blast
also have "\<dots> = P(?w)" using w P1eqP by blast
finally show "P ?w" using P1 by blast
qed
qed
text {*
\medskip This Theorem combines whithnesses about @{text "P
minusinfinity"} to show one component of the equivalence proof for
Cooper's Theorem. *}
lemma plusinfinity:
assumes "0 < d" and
P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and
ePeqP1: "EX z::int. ALL x. z < x --> (P x = P1 x)"
shows "(EX x::int. P1 x) --> (EX x::int. P x)"
proof
assume eP1: "EX x. P1 x"
then obtain x where P1: "P1 x" ..
from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" ..
let ?w = "x + (abs(x-z)+1) * d"
show "EX x. P x"
proof
have w: "z < ?w" by(rule incr_lemma)
have "P1 x = P1 ?w" using P1eqP1 by blast
also have "\<dots> = P(?w)" using w P1eqP by blast
finally show "P ?w" using P1 by blast
qed
qed
text {*
\medskip Theorem for periodic function on discrete sets. *}
lemma minf_vee:
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
shows "(EX x. P x) = (EX j : {1..d}. P j)"
(is "?LHS = ?RHS")
proof
assume ?LHS
then obtain x where P: "P x" ..
have "x mod d = x - (x div d)*d"
by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
hence Pmod: "P x = P(x mod d)" using modd by simp
show ?RHS
proof (cases)
assume "x mod d = 0"
hence "P 0" using P Pmod by simp
moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
ultimately have "P d" by simp
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
ultimately show ?RHS ..
next
assume not0: "x mod d \<noteq> 0"
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
moreover have "x mod d : {1..d}"
proof -
have "0 \<le> x mod d" by(rule pos_mod_sign)
moreover have "x mod d < d" by(rule pos_mod_bound)
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
qed
ultimately show ?RHS ..
qed
next
assume ?RHS thus ?LHS by blast
qed
text {*
\medskip Theorem for periodic function on discrete sets. *}
lemma pinf_vee:
assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)"
shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)"
(is "?LHS = ?RHS")
proof
assume ?LHS
then obtain x where P: "P x" ..
have "x mod d = x + (-(x div d))*d"
by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
hence Pmod: "P x = P(x mod d)" using modd by (simp only:)
show ?RHS
proof (cases)
assume "x mod d = 0"
hence "P 0" using P Pmod by simp
moreover have "P 0 = P(0 + 1*d)" using modd by blast
ultimately have "P d" by simp
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
ultimately show ?RHS ..
next
assume not0: "x mod d \<noteq> 0"
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
moreover have "x mod d : {1..d}"
proof -
have "0 \<le> x mod d" by(rule pos_mod_sign)
moreover have "x mod d < d" by(rule pos_mod_bound)
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
qed
ultimately show ?RHS ..
qed
next
assume ?RHS thus ?LHS by blast
qed
lemma decr_mult_lemma:
assumes dpos: "(0::int) < d" and
minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and
knneg: "0 <= k"
shows "ALL x. P x \<longrightarrow> P(x - k*d)"
using knneg
proof (induct rule:int_ge_induct)
case base thus ?case by simp
next
case (step i)
show ?case
proof
fix x
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)"
using minus[THEN spec, of "x - i * d"]
by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast
qed
qed
lemma incr_mult_lemma:
assumes dpos: "(0::int) < d" and
plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and
knneg: "0 <= k"
shows "ALL x. P x \<longrightarrow> P(x + k*d)"
using knneg
proof (induct rule:int_ge_induct)
case base thus ?case by simp
next
case (step i)
show ?case
proof
fix x
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)"
using plus[THEN spec, of "x + i * d"]
by (simp add:int_distrib zadd_ac)
ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast
qed
qed
lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
apply(rule iffI)
prefer 2
apply(drule minusinfinity)
apply assumption+
apply(fastsimp)
apply clarsimp
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
apply(frule_tac x = x and z=z in decr_lemma)
apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
prefer 2
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
prefer 2 apply arith
apply fastsimp
apply(drule (1) minf_vee)
apply blast
apply(blast dest:decr_mult_lemma)
done
text {* Cooper Theorem, plus infinity version. *}
lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x))
==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D)
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D))))
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))"
apply(rule iffI)
prefer 2
apply(drule plusinfinity)
apply assumption+
apply(fastsimp)
apply clarsimp
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)")
apply(frule_tac x = x and z=z in incr_lemma)
apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)")
prefer 2
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
prefer 2 apply arith
apply fastsimp
apply(drule (1) pinf_vee)
apply blast
apply(blast dest:incr_mult_lemma)
done
text {*
\bigskip Theorems for the quantifier elminination Functions. *}
lemma qe_ex_conj: "(EX (x::int). A x) = R
==> (EX (x::int). P x) = (Q & (EX x::int. A x))
==> (EX (x::int). P x) = (Q & R)"
by blast
lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q)
==> (EX (x::int). P x) = Q"
by blast
lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)"
by blast
lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)"
by blast
lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)"
by blast
lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)"
by blast
lemma qe_Not: "P = Q ==> (~P) = (~Q)"
by blast
lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)"
by blast
text {* \bigskip Theorems for proving NNF *}
lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))"
by blast
lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))"
by blast
lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)"
by blast
lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))"
by blast
lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))"
by blast
lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))"
by blast
lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))"
by blast
lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))"
by blast
lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
by simp
lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))"
by iprover
lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))"
by iprover
lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j)))
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) "
by blast
lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j)))
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) "
by blast
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
apply(fastsimp)
done
text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *}
lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)"
shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q")
proof
assume ?P
thus ?Q
apply(simp add:dvd_def)
apply clarify
apply(rename_tac d)
apply(drule_tac f = "op * k" in arg_cong)
apply(simp only:int_distrib)
apply(rule_tac x = "d" in exI)
apply(simp only:mult_ac)
done
next
assume ?Q
then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def)
hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac)
hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
thus ?P by(simp add:dvd_def)
qed
lemma ac_lt_eq: assumes gr0: "0 < (k::int)"
shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q")
proof
assume P: ?P
show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac)
next
assume ?Q
hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac)
with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff)
thus ?P by(simp)
qed
lemma ac_eq_eq : assumes not0: "0 ~= (k::int)" shows "((m::int) = (c*n+t)) = (k*m =((k*c)*n+(k*t)) )" (is "?P = ?Q")
proof
assume ?P
thus ?Q
apply(drule_tac f = "op * k" in arg_cong)
apply(simp only:int_distrib)
done
next
assume ?Q
hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac)
hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"])
thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]])
qed
lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))"
proof -
have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith
also have "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac)
also have "0<(-1*c)*n + (-t+1) = (0 < (k*(-1*c)*n) + (k*(-t+1)))" by(rule ac_lt_eq[of _ 0,OF gr0,simplified])
also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac)
finally show ?thesis .
qed
lemma binminus_uminus_conv: "(a::int) - b = a + (-b)"
by arith
lemma linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)"
by simp
lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)"
by simp
lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)"
by simp
lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)"
by simp
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
by (simp split add: split_nat)
theorem zdiff_int_split: "P (int (x - y)) =
((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
apply (case_tac "y \<le> x")
apply (simp_all add: zdiff_int)
done
theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)"
by simp
theorem number_of2: "(0::int) <= Numeral0" by simp
theorem Suc_plus1: "Suc n = n + 1" by simp
text {*
\medskip Specific instances of congruence rules, to prevent
simplifier from looping. *}
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')"
by simp
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
by (simp cong: conj_cong)
(* Theorems used in presburger.ML for the computation simpset*)
(* FIXME: They are present in Float.thy, so may be Float.thy should be lightened.*)
lemma lift_bool: "x \<Longrightarrow> x=True"
by simp
lemma nlift_bool: "~x \<Longrightarrow> x=False"
by simp
lemma not_false_eq_true: "(~ False) = True" by simp
lemma not_true_eq_false: "(~ True) = False" by simp
lemma int_eq_number_of_eq:
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
by simp
lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
by (simp only: iszero_number_of_Pls)
lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
by simp
lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
by simp
lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
by simp
lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
by simp
lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
by simp
lemma int_neg_number_of_Min: "neg (-1::int)"
by simp
lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
by simp
lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
by simp
lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
by simp
lemma int_number_of_diff_sym:
"((number_of v)::int) - number_of w = number_of (v + (uminus w))"
by simp
lemma int_number_of_mult_sym:
"((number_of v)::int) * number_of w = number_of (v * w)"
by simp
lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
by simp
lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
by simp
lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
by simp
lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
by simp
lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
by simp
lemma int_pow_0: "(a::int)^(Numeral0) = 1"
by simp
lemma int_pow_1: "(a::int)^(Numeral1) = a"
by simp
lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
by simp
lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
by simp
lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
by simp
lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
by simp
lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
by simp
lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
proof -
have 1:"((-1)::nat) = 0"
by simp
show ?thesis by (simp add: 1)
qed
use "cooper_dec.ML"
use "reflected_presburger.ML"
use "reflected_cooper.ML"
oracle
presburger_oracle ("term") = ReflectedCooper.presburger_oracle
use "cooper_proof.ML"
use "qelim.ML"
use "presburger.ML"
setup "Presburger.setup"
text {* Code generator setup *}
text {*
Presburger arithmetic is necessary (at least convenient) to prove some
of the following code lemmas on integer numerals.
*}
lemma eq_number_of [code func]:
"((number_of k)\<Colon>int) = number_of l \<longleftrightarrow> k = l"
unfolding number_of_is_id ..
lemma eq_Pls_Pls:
"Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by rule+
lemma eq_Pls_Min:
"Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
unfolding Pls_def Min_def by auto
lemma eq_Pls_Bit0:
"Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
unfolding Pls_def Bit_def bit.cases by auto
lemma eq_Pls_Bit1:
"Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
unfolding Pls_def Bit_def bit.cases by arith
lemma eq_Min_Pls:
"Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
unfolding Pls_def Min_def by auto
lemma eq_Min_Min:
"Numeral.Min = Numeral.Min \<longleftrightarrow> True" by rule+
lemma eq_Min_Bit0:
"Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
unfolding Min_def Bit_def bit.cases by arith
lemma eq_Min_Bit1:
"Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
unfolding Min_def Bit_def bit.cases by auto
lemma eq_Bit0_Pls:
"Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
unfolding Pls_def Bit_def bit.cases by auto
lemma eq_Bit1_Pls:
"Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
unfolding Pls_def Bit_def bit.cases by arith
lemma eq_Bit0_Min:
"Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
unfolding Min_def Bit_def bit.cases by arith
lemma eq_Bit1_Min:
"(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
unfolding Min_def Bit_def bit.cases by auto
lemma eq_Bit_Bit:
"Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
v1 = v2 \<and> k1 = k2"
unfolding Bit_def
apply (cases v1)
apply (cases v2)
apply auto
apply arith
apply (cases v2)
apply auto
apply arith
apply (cases v2)
apply auto
done
lemmas eq_numeral_code [code func] =
eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
lemma less_eq_number_of [code func]:
"(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
unfolding number_of_is_id ..
lemma less_eq_Pls_Pls:
"Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
lemma less_eq_Pls_Min:
"Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
unfolding Pls_def Min_def by auto
lemma less_eq_Pls_Bit:
"Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
unfolding Pls_def Bit_def by (cases v) auto
lemma less_eq_Min_Pls:
"Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
unfolding Pls_def Min_def by auto
lemma less_eq_Min_Min:
"Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
lemma less_eq_Min_Bit0:
"Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
unfolding Min_def Bit_def by auto
lemma less_eq_Min_Bit1:
"Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
unfolding Min_def Bit_def by auto
lemma less_eq_Bit0_Pls:
"Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
unfolding Pls_def Bit_def by simp
lemma less_eq_Bit1_Pls:
"Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
unfolding Pls_def Bit_def by auto
lemma less_eq_Bit_Min:
"Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
unfolding Min_def Bit_def by (cases v) auto
lemma less_eq_Bit0_Bit:
"Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
unfolding Bit_def bit.cases by (cases v) auto
lemma less_eq_Bit_Bit1:
"Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
unfolding Bit_def bit.cases by (cases v) auto
lemma less_eq_Bit1_Bit0:
"Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
unfolding Bit_def by (auto split: bit.split)
lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
lemma less_eq_number_of [code func]:
"(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
unfolding number_of_is_id ..
lemma less_Pls_Pls:
"Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by auto
lemma less_Pls_Min:
"Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
unfolding Pls_def Min_def by auto
lemma less_Pls_Bit0:
"Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
unfolding Pls_def Bit_def by auto
lemma less_Pls_Bit1:
"Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
unfolding Pls_def Bit_def by auto
lemma less_Min_Pls:
"Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
unfolding Pls_def Min_def by auto
lemma less_Min_Min:
"Numeral.Min < Numeral.Min \<longleftrightarrow> False" by auto
lemma less_Min_Bit:
"Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
unfolding Min_def Bit_def by (auto split: bit.split)
lemma less_Bit_Pls:
"Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
unfolding Pls_def Bit_def by (auto split: bit.split)
lemma less_Bit0_Min:
"Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
unfolding Min_def Bit_def by auto
lemma less_Bit1_Min:
"Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
unfolding Min_def Bit_def by auto
lemma less_Bit_Bit0:
"Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
unfolding Bit_def by (auto split: bit.split)
lemma less_Bit1_Bit:
"Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
unfolding Bit_def by (auto split: bit.split)
lemma less_Bit0_Bit1:
"Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
unfolding Bit_def bit.cases by auto
lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0
less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls
less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
end