fact stringlengths 6 103k | type stringclasses 14 values | library stringclasses 13 values | imports listlengths 1 10 | filename stringclasses 157 values | symbolic_name stringlengths 1 54 | docstring stringclasses 185 values |
|---|---|---|---|---|---|---|
use_SCC_all_hypsM i Hi Hj :=
let rec aux x :=
match x with
| 0 => lia
| S ?y => destruct i; [use_SCC_hyp M Hi | aux y]
end in
match type of Hj with
| le _ ?Y => aux Y
end. | Ltac | DP | [
"From Stdlib Require Import Permutation Multiset Setoid PermutSetoid",
"From CoLoR Require Import SCCTopoOrdering AGraph ATrs RelUtil RelSub BoundNat"
] | DP/ASCCUnion.v | use_SCC_all_hyps | null |
SCC_namen1 n2 :=
match goal with
| |- WF (chain ?R) => set (n1 := dp R); set (n2 := int_red R #)
| |- WF (hd_red_mod ?E ?R) => set (n1 := R); set (n2 := red E #)
| |- WF (?X @ hd_red ?R) => set (n1 := R); set (n2 := X)
| |- WF (hd_red_Mod ?E ?R) => set (n1 := R); set (n2 := E)
end. | Ltac | DP | [
"From Stdlib Require Import Permutation Multiset Setoid PermutSetoid",
"From CoLoR Require Import SCCTopoOrdering AGraph ATrs RelUtil RelSub BoundNat"
] | DP/ASCCUnion.v | SCC_name | null |
filterp := hd_red_mod; apply WF_hd_red_mod_filter with (pi:=p). | Ltac | Filter | [
"From CoLoR Require Import ATrs VecFilter VecUtil LogicUtil RelUtil SN NatUtil",
"From Stdlib Require Import List"
] | Filter/AFilterBool.v | filter | tactics |
non_dup:=
match goal with
| |- non_dup (build_pi _ _) => rewrite <- bnon_dup_ok; check_eq
| |- non_dup ?pi => unfold pi; non_dup
end || fail 10 "duplicating arguments filter". | Ltac | Filter | [
"From CoLoR Require Import LogicUtil ATrs ListUtil NatUtil VecUtil SN BoolUtil"
] | Filter/AFilterPerm.v | non_dup | tactics |
permut:=
match goal with
| |- permut (build_pi _ _) => rewrite <- bpermut_ok; check_eq
| |- permut ?pi => unfold pi; permut
end || fail 10 "non-permutative arguments filter".
(*COQ: does not work... the recursive call on filter fails... | Ltac | Filter | [
"From CoLoR Require Import LogicUtil ATrs ListUtil NatUtil VecUtil SN BoolUtil"
] | Filter/AFilterPerm.v | permut | null |
filterp :=
match goal with
| |- WF (hd_red_Mod _ _) => hd_red_mod; filter p
| |- WF (hd_red_mod _ _) =>
apply WF_hd_red_mod_filter with (pi:=p); [non_dup | idtac]
end.*) | Ltac | Filter | [
"From CoLoR Require Import LogicUtil ATrs ListUtil NatUtil VecUtil SN BoolUtil"
] | Filter/AFilterPerm.v | filter | null |
filterp :=
hd_red_mod; apply WF_hd_red_mod_filter with (pi:=p); [non_dup | idtac]. | Ltac | Filter | [
"From CoLoR Require Import LogicUtil ATrs ListUtil NatUtil VecUtil SN BoolUtil"
] | Filter/AFilterPerm.v | filter | null |
prove_cc_succtac :=
apply filter_strong_cont_closed; [non_dup | permut | tac]. | Ltac | Filter | [
"From CoLoR Require Import LogicUtil ATrs ListUtil NatUtil VecUtil SN BoolUtil"
] | Filter/AFilterPerm.v | prove_cc_succ | null |
valid:= forall f k, raw_pi f = Some k -> k < arity f. | Definition | Filter | [
"From CoLoR Require Import ATrs VecUtil LogicUtil ListUtil SN ARelation RelUtil"
] | Filter/AProj.v | valid | null |
bvalid_symbf :=
match raw_pi f with
| Some k => bgt_nat (arity f) k
| None => true
end. | Definition | Filter | [
"From CoLoR Require Import ATrs VecUtil LogicUtil ListUtil SN ARelation RelUtil"
] | Filter/AProj.v | bvalid_symb | null |
bvalid:= forallb bvalid_symb Fs. | Definition | Filter | [
"From CoLoR Require Import ATrs VecUtil LogicUtil ListUtil SN ARelation RelUtil"
] | Filter/AProj.v | bvalid | null |
bvalid_ok: bvalid = true <-> valid.
Proof.
unfold bvalid, valid. apply forallb_ok_fintype. 2: hyp.
intro f. unfold bvalid_symb. destruct (raw_pi f).
rewrite bgt_nat_ok. intuition. inversion H1. subst. auto.
intuition. discr.
Qed. | Lemma | Filter | [
"From CoLoR Require Import ATrs VecUtil LogicUtil ListUtil SN ARelation RelUtil"
] | Filter/AProj.v | bvalid_ok | null |
projp := hd_red_mod; apply WF_hd_red_mod_proj with (pi:=p). | Ltac | Filter | [
"From CoLoR Require Import ATrs VecUtil LogicUtil ListUtil SN ARelation RelUtil"
] | Filter/AProj.v | proj | tactics |
validFs_ok :=
match goal with
| |- valid ?p => rewrite <- (@bvalid_ok _ p _ Fs_ok);
(check_eq || fail 10 "invalid projection")
end. | Ltac | Filter | [
"From CoLoR Require Import ATrs VecUtil LogicUtil ListUtil SN ARelation RelUtil"
] | Filter/AProj.v | valid | null |
matrixInt:= @matrixInt A matrix. | Definition | MatrixInt | [
"From Stdlib Require Import ZArith Lia",
"From CoLoR Require Import LogicUtil Matrix AMonAlg AArcticBasedInt VecUtil"
] | MatrixInt/AArcticBZInt.v | matrixInt | null |
mkMatrixInt:= @mkMatrixInt A matrix. | Definition | MatrixInt | [
"From Stdlib Require Import ZArith Lia",
"From CoLoR Require Import LogicUtil Matrix AMonAlg AArcticBasedInt VecUtil"
] | MatrixInt/AArcticBZInt.v | mkMatrixInt | null |
absolute_finiteFs_ok :=
apply (fin_absolute_finite _ _ Fs_ok);
(check_eq || fail 10 "invalid below-zero arctic interpretation"). | Ltac | MatrixInt | [
"From Stdlib Require Import ZArith Lia",
"From CoLoR Require Import LogicUtil Matrix AMonAlg AArcticBasedInt VecUtil"
] | MatrixInt/AArcticBZInt.v | absolute_finite | null |
matrixInt:= @matrixInt A matrix. | Definition | MatrixInt | [
"From CoLoR Require Import AArcticBasedInt Matrix OrdSemiRing VecUtil AMonAlg SN"
] | MatrixInt/AArcticInt.v | matrixInt | null |
mkMatrixInt:= @mkMatrixInt A matrix. | Definition | MatrixInt | [
"From CoLoR Require Import AArcticBasedInt Matrix OrdSemiRing VecUtil AMonAlg SN"
] | MatrixInt/AArcticInt.v | mkMatrixInt | null |
somewhere_finiteFs_ok :=
apply (fin_somewhere_finite _ _ Fs_ok);
(check_eq || fail 10 "invalid arctic interpretation"). | Ltac | MatrixInt | [
"From CoLoR Require Import AArcticBasedInt Matrix OrdSemiRing VecUtil AMonAlg SN"
] | MatrixInt/AArcticInt.v | somewhere_finite | null |
matrixInt:= @matrixInt A matrix. | Definition | MatrixInt | [
"From Stdlib Require Import Setoid",
"From CoLoR Require Import LogicUtil Matrix OrdSemiRing VecUtil AMonAlg"
] | MatrixInt/ABigMatrixInt.v | matrixInt | null |
mkMatrixInt:= @mkMatrixInt A matrix. | Definition | MatrixInt | [
"From Stdlib Require Import Setoid",
"From CoLoR Require Import LogicUtil Matrix OrdSemiRing VecUtil AMonAlg"
] | MatrixInt/ABigMatrixInt.v | mkMatrixInt | null |
matrixInt:= @matrixInt A matrix. | Definition | MatrixInt | [
"From Stdlib Require Import Setoid",
"From CoLoR Require Import LogicUtil Matrix OrdSemiRing VecUtil AMonAlg SN"
] | MatrixInt/AMatrixInt.v | matrixInt | null |
mkMatrixInt:= @mkMatrixInt A matrix. | Definition | MatrixInt | [
"From Stdlib Require Import Setoid",
"From CoLoR Require Import LogicUtil Matrix OrdSemiRing VecUtil AMonAlg SN"
] | MatrixInt/AMatrixInt.v | mkMatrixInt | null |
matrixInt:= @matrixInt A matrix. | Definition | MatrixInt | [
"From CoLoR Require Import ATropicalBasedInt Matrix OrdSemiRing VecUtil AMonAlg",
"From Stdlib Require Import Structures.Equalities"
] | MatrixInt/ATropicalInt.v | matrixInt | null |
mkMatrixInt:= @mkMatrixInt A matrix. | Definition | MatrixInt | [
"From CoLoR Require Import ATropicalBasedInt Matrix OrdSemiRing VecUtil AMonAlg",
"From Stdlib Require Import Structures.Equalities"
] | MatrixInt/ATropicalInt.v | mkMatrixInt | null |
somewhere_tfiniteFs_ok :=
apply (fin_somewhere_tfinite _ _ Fs_ok);
(check_eq || fail 10 "invalid tropical interpretation"). | Ltac | MatrixInt | [
"From CoLoR Require Import ATropicalBasedInt Matrix OrdSemiRing VecUtil AMonAlg",
"From Stdlib Require Import Structures.Equalities"
] | MatrixInt/ATropicalInt.v | somewhere_tfinite | null |
check_loopt' ds' p' :=
apply is_loop_correct with (t:=t') (ds:=ds') (p:=p');
(check_eq || fail 10 "not a loop"). | Ltac | NonTermin | [
"From Stdlib Require Import Euclid Wf_nat",
"From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil AMatching ListDec"
] | NonTermin/ALoop.v | check_loop | tactics |
loopt' ds' p' :=
match goal with
| |- EIS (red_mod ?E _) =>
remove_relative_rules E; check_loop t' ds' p'
| |- EIS (red _) => check_loop t' ds' p'
end. | Ltac | NonTermin | [
"From Stdlib Require Import Euclid Wf_nat",
"From CoLoR Require Import LogicUtil ATrs ListUtil RelUtil AMatching ListDec"
] | NonTermin/ALoop.v | loop | null |
loopt' mds' ds' p' :=
apply is_mod_loop_correct with (t:=t') (mds:=mds') (ds:=ds') (p:=p');
(check_eq || fail 10 "not a loop"). | Ltac | NonTermin | [
"From Stdlib Require Import Euclid Wf_nat",
"From CoLoR Require Import ATrs LogicUtil ALoop ListUtil RelUtil NatUtil"
] | NonTermin/AModLoop.v | loop | tactics |
var_condSig :=
(apply var_cond_mod || apply var_cond);
rewrite <- (ko (@brules_preserve_vars_ok Sig));
(check_eq || fail 10 "variable condition satisfied"). | Ltac | NonTermin | [
"From CoLoR Require Import LogicUtil ATrs AVariables BoolUtil EqUtil ListUtil RelUtil"
] | NonTermin/AVarCond.v | var_cond | null |
check_loopt' ds' p' :=
apply is_loop_correct with (t:=t') (ds:=ds') (p:=p');
(check_eq || fail 10 "not a loop"). | Ltac | NonTermin | [
"From Stdlib Require Import Euclid Wf_nat",
"From CoLoR Require Import LogicUtil Srs ListUtil EqUtil RelUtil ListDec NatUtil"
] | NonTermin/SLoop.v | check_loop | tactics |
loopt' ds' p' :=
match goal with
| |- EIS (red_mod ?E _) =>
remove_relative_rules E; check_loop t' ds' p'
| |- EIS (red _) => check_loop t' ds' p'
end. | Ltac | NonTermin | [
"From Stdlib Require Import Euclid Wf_nat",
"From CoLoR Require Import LogicUtil Srs ListUtil EqUtil RelUtil ListDec NatUtil"
] | NonTermin/SLoop.v | loop | null |
loopt' mds' ds' p' :=
apply is_mod_loop_correct with (t:=t') (mds:=mds') (ds:=ds') (p:=p');
(check_eq || fail 10 "not a loop"). | Ltac | NonTermin | [
"From Stdlib Require Import Euclid Wf_nat",
"From CoLoR Require Import Srs LogicUtil SLoop ListUtil NatUtil RelUtil"
] | NonTermin/SModLoop.v | loop | tactics |
PolyWeakMonotoneFs_ok :=
match goal with
| |- PolyWeakMonotone ?PI =>
apply (fin_PolyWeakMonotone PI Fs_ok);
(check_eq || fail 10 "could not prove monotony")
end. | Ltac | PolyInt | [
"From CoLoR Require Import ATerm ABterm ListUtil VecUtil"
] | PolyInt/APolyInt.v | PolyWeakMonotone | tactics |
Definitioncheck_mono (n : nat) (m : monomial) : option (Z * monom n) :=
match m with
| (coef, vars) =>
match eq_nat_dec n (List.length vars) with
| left _ => Some (coef, vec_of_list vars)
| right _ => None
end
end. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
check_poly(n : nat) (p : polynomial) : option (poly n) :=
map_opt (@check_mono n) p. | Definition | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | check_poly | null |
polyIntn := { p : poly n | pweak_monotone p }.
Notation symPI := (symInt Sig polyInt). | Definition | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | polyInt | null |
Definitionsymbol_poly_int (f : Sig) (p : polynomial) : option symPI :=
match check_poly (arity f) p with
| None => None
| Some fi =>
match pweak_monotone_check fi with
| None => None
| Some _ => Some (buildSymInt Sig polyInt f fi)
end
end. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
defaultPolyn : poly n :=
pconst n 1 ++ list_of_vec (Vbuild (fun i (ip : i < n) => (1%Z, mxi ip))). | Definition | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | defaultPoly | null |
defaultPoly_mxi_1n i (H : i < n) : List.In (1%Z, mxi H) (defaultPoly n).
Proof.
intros. right. simpl.
apply list_of_vec_in.
rewrite <- (Vbuild_nth (fun i (ip : i < n) => (1%Z, mxi ip)) H).
apply Vnth_in.
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | defaultPoly_mxi_1 | null |
defaultPoly_wmn : pweak_monotone (defaultPoly n).
Proof with simpl; auto with zarith.
intros. split...
apply lforall_intro. intros.
ded (in_list_of_vec H).
ded (Vbuild_in (fun i ip => (1%Z, mxi ip)) x H0).
decomp H1. subst...
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | defaultPoly_wm | null |
defaultPoly_smn : pstrong_monotone (defaultPoly n).
Proof.
split. apply defaultPoly_wm.
intros.
assert (HH : List.In (1%Z, mxi H) (defaultPoly n)).
apply defaultPoly_mxi_1.
set (w := coefPos_geC (defaultPoly n) (mxi H) 1 (defaultPoly_wm n) HH).
auto with zarith.
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | defaultPoly_sm | null |
DefinitiondefaultInt n : polyInt n := defaultPoly n.
Next Obligation.
Proof.
set (w := defaultPoly_wm). simpl in w. apply w.
Qed. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
Definitioninterpret n (fi : polyInt n) : naryFunction1 D n :=
@peval_D n fi _.
Next Obligation.
Proof.
destruct fi. hyp.
Qed. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
Definitionpoly_wm (fi : symPI) := True. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | weak and strong monotonicity checking |
Definitionpoly_sm (fi : symPI) := pstrong_monotone (projT2 fi). | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
sm_imp_wm(fi : symPI) : poly_sm fi -> poly_wm fi.
Proof.
fo.
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | sm_imp_wm | null |
Definitioncheck_wm (fi : symPI) : option (poly_wm fi) :=
Some _.
Next Obligation.
Proof.
fo.
Qed. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
wm_ok: forall fi, poly_wm fi -> Vmonotone1 (interpret (projT2 fi)) Dge.
Proof.
intros. apply Vmonotone_transp. apply coef_pos_monotone_peval_Dle.
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | wm_ok | null |
Definitioncheck_sm (fi : symPI) : option (poly_sm fi) :=
pstrong_monotone_check (projT2 fi). | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
sm_ok: forall fi, poly_sm fi -> Vmonotone1 (interpret (projT2 fi)) Dgt.
Proof.
intros. apply Vmonotone_transp.
apply pmonotone_imp_monotone_peval_Dlt. hyp.
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | sm_ok | null |
buildSymInt:= buildSymInt Sig polyInt. | Let | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | buildSymInt | null |
defaultIntForSymbol:= defaultIntForSymbol Sig polyInt defaultInt. | Let | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | defaultIntForSymbol | null |
default_sm: forall f, poly_sm (buildSymInt (defaultIntForSymbol f)).
Proof.
intros. apply defaultPoly_sm.
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | default_sm | null |
wm_spec:= Build_monSpec interpret poly_wm check_wm wm_ok. | Definition | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | wm_spec | null |
sm_spec:= Build_monSpec interpret poly_sm check_sm sm_ok. | Definition | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | sm_spec | null |
I:= makeI Sig D0 polyInt interpret i. | Definition | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | I | null |
succ:= IR I Dgt. | Let | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | succ | null |
succeq:= IR I Dge. | Let | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | succeq | null |
Definitioncheck_succ (r : rule Sig) : option (succ (lhs r) (rhs r)) :=
match coef_pos_check (rulePoly_gt i r) with
| None => None
| Some _ => Some _
end.
Next Obligation.
Proof with try discr; auto.
destruct_call coef_pos_check...
apply pi_compat_rule...
Qed. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
Definitioncheck_succeq (r : rule Sig) : option (succeq (lhs r) (rhs r)) :=
match coef_pos_check (rulePoly_ge i r) with
| None => None
| Some _ => Some _
end.
Next Obligation.
Proof with try discr; auto.
destruct_call coef_pos_check...
apply pi_compat_rule_weak...
Qed. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
succ_WF:= WF_Dgt. | Definition | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | succ_WF | null |
succ_succeq_compat: absorbs_left Dgt Dge.
Proof.
intros p q pq. destruct pq as [r [pr rq]].
unfold Dgt, Dlt, transp. apply Z.lt_le_trans with (val r); auto.
Qed. | Lemma | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | succ_succeq_compat | null |
DefinitionpolySolver := monotoneAlgebraSolver succ_WF
succ_succeq_compat defaultInt check_succ check_succeq wm_spec sm_spec
sm_imp_wm default_sm int symbol_poly_int. | Program | PolyInt | [
"From Stdlib Require Import Program",
"From CoLoR Require Import ListUtil LogicUtil ZUtil VecUtil Problem"
] | PolyInt/PolyChecker.v | Definition | null |
rawTrsInt:= trsInt Sig rawSymInt.
Variable arSymInt : nat -> Set.
Variable defaultInt : forall n, arSymInt n. | Definition | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | rawTrsInt | null |
funInt(f : Sig) := arSymInt (arity f). | Definition | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | funInt | null |
symInt:= { f : Sig & funInt f }. | Definition | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | symInt | null |
defaultIntForSymbolf := @defaultInt (@arity Sig f). | Definition | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | defaultIntForSymbol | null |
buildSymIntf (fi : funInt f) : symInt := existT fi.
Variable checkInt : Sig -> rawSymInt -> option symInt. | Definition | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | buildSymInt | null |
processInt(ri : rawTrsInt) : option (list symInt) :=
match ri with
| nil => Some nil
| cons i is =>
match checkInt (fst i) (snd i) with
| None => None
| Some fi =>
match processInt is with
| None => None
| Some fis => Some (fi :: fis)
end
end
end. | Fixpoint | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | processInt | null |
buildInt(i : list symInt) : forall f, funInt f :=
fun f => lookup_dep (el := f) (@eq_symb_dec Sig) defaultIntForSymbol i.
Variable P : symInt -> Prop.
Variable check_P : forall (i : symInt), option (P i).
Variable default_P : forall f, P (buildSymInt (defaultIntForSymbol f)). | Definition | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | buildInt | null |
FixpointcheckProp (i : list symInt) :
option (forall f, P (buildSymInt (buildInt i f))) :=
match lforall_opt P check_P i with
| None => None
| Some _ => Some _
end.
Next Obligation.
apply (lookup_dep_prop (P := fun _ fi => P (buildSymInt fi))); intros.
destruct_call lforall_opt; try discr.
ded (lforall_in l H). decomp H0. destruct x. hyp.
apply default_P.
Qed. | Program | ProofChecker | [
"From Stdlib Require Import Program",
"From CoLoR Require Import LogicUtil OptUtil ATrs NaryFunction"
] | ProofChecker/IntBasedChecker.v | Fixpoint | null |
trsIntsymInt := list (Sig * symInt).
(** [monomial] is a monomial in representation of a polynomial.
A monomial [C * x_0^i_0 * ... * x_n^i_n] is represented as:
[(C, i_0::...::i_n)]. *) | Definition | ProofChecker | [
"From Stdlib Require Import ZArith List",
"From CoLoR Require Import ASignature"
] | ProofChecker/Proof.v | trsInt | null |
monomial:= (Z * list nat)%type.
(** [polynomial] is a list of monomials so [m_0 + ... + m_n] becomes
[m_0::...::m_n]. *) | Definition | ProofChecker | [
"From Stdlib Require Import ZArith List",
"From CoLoR Require Import ASignature"
] | ProofChecker/Proof.v | monomial | null |
polynomial:= list monomial. | Definition | ProofChecker | [
"From Stdlib Require Import ZArith List",
"From CoLoR Require Import ASignature"
] | ProofChecker/Proof.v | polynomial | null |
TerminationProof:=
| TP_PolyInt (PI: trsInt polynomial) (Prf : TerminationProof)
| TP_ProblemEmpty. | Inductive | ProofChecker | [
"From Stdlib Require Import ZArith List",
"From CoLoR Require Import ASignature"
] | ProofChecker/Proof.v | TerminationProof | null |
TerminationAnalysisResult(P : Problem Sig) :=
| TerminationEstablished (Prf : Prob_WF P)
| Error.
Arguments TerminationEstablished [P] _.
Arguments Error {P}. | Inductive | ProofChecker | [
"From Stdlib Require Import Relations List",
"From CoLoR Require Import SN ATrs ARelation Problem Proof EmptyChecker"
] | ProofChecker/ProofChecker.v | TerminationAnalysisResult | null |
Fixpointcheck_proof (Pb : Problem Sig) (Prf : TerminationProof Sig)
: TerminationAnalysisResult Pb :=
match Prf with
| TP_PolyInt PI Prf' =>
match PolyChecker.polySolver PI Pb with
| None => Error
| Some Pb' =>
match check_proof Pb' Prf' with
| Error => Error
| TerminationEstablished _ => TerminationEstablished _
end
end
| TP_ProblemEmpty _ =>
match EmptyChecker.is_problem_empty Pb with
| true => TerminationEstablished _
| _ => Error
end
end.
Next Obligation.
Proof.
auto with rainbow.
Qed. | Program | ProofChecker | [
"From Stdlib Require Import Relations List",
"From CoLoR Require Import SN ATrs ARelation Problem Proof EmptyChecker"
] | ProofChecker/ProofChecker.v | Fixpoint | null |
status_name: Set :=
| lexicographic : status_name
| multiset : status_name. | Inductive | RPO | [
"From CoLoR Require Import VPrecedence MultisetListOrder ListLex VRPO_Type",
"From Stdlib Require Import Relations"
] | RPO/VRPO_Status.v | status_name | null |
enum_tuple2n : list (vector I n) :=
match n with
| 0 => nil
| S p =>
fold_left (fun e ds => fold_left (fun e d => Vcons d ds :: e) Is e)
(enum_tuple2 p) nil
end.*)
Definition enum R :=
flat_map (fun ds => map (lab_rule (val_of_vec I ds)) R)
(enum_tuple (S (maxvar_rules R))).
Lemma enum_correct : forall R a, In a (enum R) -> lab_rules (of_list R) a.
Proof.
intros. unfold enum in H. rewrite in_flat_map in H. do 2 destruct H.
rewrite in_map_iff in H0. do 2 destruct H0. exists x0.
exists (val_of_vec I x). intuition.
Qed.
Lemma enum_complete : forall R a, lab_rules (of_list R) a -> In a (enum R).
Proof.
intros. do 3 destruct H. set (n := maxvar_rules R).
unfold enum. rewrite in_flat_map. exists (vec_of_val x0 (S n)). split.
apply enum_tuple_complete. subst.
change (In (lab_rule x0 x) (map (lab_rule (fval x0 (S n))) R)).
rewrite map_lab_rule_fval. apply in_map. hyp. unfold n. lia.
Qed.
Infix "[=]" := equiv.
Lemma lab_rules_enum : forall R, lab_rules (of_list R) [=] of_list (enum R).
Proof. split. apply enum_complete. apply enum_correct. Qed.
(*REMARK: define a more efficient function? | Fixpoint | SemLab | [
"From CoLoR Require Import ATrs AInterpretation BoolUtil LogicUtil EqUtil",
"From Stdlib Require Import List"
] | SemLab/ASemLab.v | enum_tuple2 | null |
enum2R :=
let n := S (maxvar_rules R) in
fold_left (fun e ds =>
let v := val_of_vec I ds in
fold_left (fun e (a : rule) => let (l,r) := a in
mkRule (lab v l) (lab v r) :: e)
R e)
(enum_tuple n) nil.*)
Variable Fs : list Sig.
Variable Fs_ok : forall x, In x Fs.
Variable Ls : forall f, list (L f).
Variable Ls_ok : forall f (x : L f), In x (Ls f).
Definition Fs_lab := flat_map (fun f => map (@mk f) (Ls f)) Fs.
Lemma Fs_lab_ok : forall f : Sig', In f Fs_lab.
Proof.
intros [f l]. unfold Fs_lab. rewrite in_flat_map. exists f. split.
apply Fs_ok. rewrite in_map_iff. exists l. intuition.
Qed.
Variable L2s : forall f, list (L f * L f).
Variable L2s_ok : forall f (x y : L f), x >L y <-> In (x,y) (L2s f).
Definition enum_Decr := flat_map (fun f =>
map (fun x : L f * L f => let (a,b) := x in decr a b) (L2s f)) Fs.
Notation D' := enum_Decr.
Lemma enum_Decr_correct : forall x, In x D' -> Decr x.
Proof.
intros. unfold enum_Decr in H. rewrite in_flat_map in H.
destruct H as [f]. destruct H. rewrite in_map_iff in H0.
destruct H0 as [[a b]]. destruct H0. exists f. exists a. exists b.
rewrite L2s_ok. auto.
Qed.
Lemma enum_Decr_complete : forall x, Decr x -> In x D'.
Proof.
intros. destruct H as [f [a [b [h]]]]. unfold enum_Decr.
rewrite in_flat_map.
exists f. split. apply Fs_ok. rewrite in_map_iff. exists (a,b).
rewrite <- L2s_ok. auto.
Qed.
Lemma Rules_enum_Decr : of_list D' [=] Decr.
Proof.
unfold equiv. split; intro. apply enum_Decr_correct. hyp.
apply enum_Decr_complete. hyp.
Qed.
Import ATrs List. Notation rules := (rules Sig). Notation term := S.term.
Variable bge : term -> term -> bool.
Variable bge_ok : rel_of_bool bge << Ige.
Section bge.
Variable R : rules.
Variable bge_compat : forallb (brule bge) R = true.
Lemma ge_compat : forall l r, In (mkRule l r) R -> l >=I r.
Proof.
intros. apply bge_ok. unfold rel.
change (brule bge (mkRule l r) = true).
rewrite forallb_forall in bge_compat. apply bge_compat. hyp.
Qed.
End bge.
Arguments ge_compat [R] _ _ _ _ _.
Section red_mod.
Variables E R : rules.
Notation E' := (enum E). Notation R' := (enum R).
Variable bge_compatE : forallb (brule bge) E = true.
Variable bge_compatR : forallb (brule bge) R = true.
Notation ge_compatE := (ge_compat bge_compatE).
Notation ge_compatR := (ge_compat bge_compatR).
Lemma WF_red_lab_fin : WF (red R) <-> WF (red_mod D' R').
Proof.
rewrite <- red_Rules, <- red_mod_Rules, WF_red_lab.
2: apply ge_compatR. apply WF_same.
rewrite Rules_enum_Decr, lab_rules_enum. refl.
Qed.
Import List.
Lemma WF_red_mod_lab_fin :
WF (red_mod E R) <-> WF (red_mod (D' ++ E') R').
Proof.
rewrite <- !red_mod_Rules, WF_red_mod_lab.
2: apply ge_compatE. 2: apply ge_compatR. apply WF_same.
rewrite of_app, Rules_enum_Decr, !lab_rules_enum. refl.
Qed.
Lemma WF_hd_red_mod_lab_fin :
WF (hd_red_mod E R) <-> WF (hd_red_mod (D' ++ E') R').
Proof.
rewrite <- !hd_red_mod_Rules, WF_hd_red_mod_lab.
2: apply ge_compatE. apply WF_same.
rewrite of_app, Rules_enum_Decr, !lab_rules_enum. refl.
Qed.
End red_mod.
Lemma WF_red_unlab_fin : forall R,
WF (red (unlab_rules_fin R)) -> WF (red R).
Proof. intros. apply Fred_WF_fin with (S2:=Sig) (F:=F) (HF:=HF). hyp. Qed.
Lemma WF_red_mod_unlab_fin : forall E R,
WF (red_mod (unlab_rules_fin E) (unlab_rules_fin R)) -> WF (red_mod E R).
Proof.
intros. apply Fred_mod_WF_fin with (S2:=Sig) (F:=F) (HF:=HF). hyp.
Qed.
Lemma WF_hd_red_mod_unlab_fin : forall E R,
WF (hd_red_mod (unlab_rules_fin E) (unlab_rules_fin R))
-> WF (hd_red_mod E R).
Proof.
intros. apply Fhd_red_mod_WF_fin with (S2:=Sig) (F:=F) (HF:=HF). hyp.
Qed.
End enum. | Definition | SemLab | [
"From CoLoR Require Import ATrs AInterpretation BoolUtil LogicUtil EqUtil",
"From Stdlib Require Import List"
] | SemLab/ASemLab.v | enum2 | null |
validFs_ok := rewrite <- (bvalid_ok _ Fs_ok);
(check_eq || fail 10 "invalid simple projection"). | Ltac | SubtermCrit | [
"From CoLoR Require Import ATrs NatUtil BoolUtil VecUtil ListUtil LogicUtil"
] | SubtermCrit/ASimpleProj.v | valid | null |
ac_one_step_at_top: term -> term -> Prop :=
| a_axiom :
forall (f:symbol) (t1 t2 t3:term),
arity f = AC ->
ac_one_step_at_top
(Term f ((Term f (t1 :: t2 :: nil)) :: t3 :: nil))
(Term f (t1 :: ((Term f (t2 :: t3 :: nil)) :: nil)))
| c_axiom :
forall (f:symbol) (t1 t2:term),
arity f = C \/ arity f = AC ->
ac_one_step_at_top (Term f (t1 :: t2 :: nil))
(Term f (t2 :: t1 :: nil)).
#[global] Hint Constructors ac_one_step_at_top : core. | Inductive | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | ac_one_step_at_top | null |
ac:= th_eq ac_one_step_at_top. | Definition | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | ac | null |
flatten(f : symbol) (l : list term) : list term :=
match l with
| nil => nil
| (Var _ as t) :: tl => t :: (flatten f tl)
| (Term g ll as t) :: tl =>
if F.Symb.eq_bool f g
then ll ++ (flatten f tl)
else t :: (flatten f tl)
end. | Fixpoint | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | flatten | null |
canonical_form(t : term) : term :=
match t with
| Var _ => t
| Term f l =>
Term f
(match arity f with
| Free _ => map canonical_form l
| C => quicksort (map canonical_form l)
| AC => quicksort (flatten f (map canonical_form l))
end)
end.
Fixpoint well_formed_cf (t:term) : Prop :=
match t with
| Var _ => True
| Term f l =>
let wf_cf_list :=
(fix wf_cf_list (l:list term) : Prop :=
match l with
| nil => True
| h :: tl => well_formed_cf h /\ wf_cf_list tl
end) in
wf_cf_list l /\
(match arity f with
| Free n => length l = n
| C => length l = 2 /\ is_sorted l
| AC => length l >= 2 /\
is_sorted l /\
(forall s, In s l -> match s with
| Var _ => True
| Term g _ => f<>g
end)
end)
end. | Fixpoint | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | canonical_form | null |
build(f : symbol) l :=
match l with
| t :: nil => t
| _ => Term f (quicksort l)
end. | Definition | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | build | null |
well_formed_cf_substsigma :=
forall v, match find X.eq_bool v sigma with
| None => True
| Some t => well_formed_cf t
end.
Fixpoint apply_cf_subst (sigma : substitution) (t : term) {struct t} : term :=
match t with
| Var v =>
match find X.eq_bool v sigma with
| None => t
| Some v_sigma => v_sigma
end
| Term f l =>
let l_sigma :=
match arity f with
| AC => quicksort (flatten f (map (apply_cf_subst sigma) l))
| C => quicksort (map (apply_cf_subst sigma) l)
| Free _ => map (apply_cf_subst sigma) l
end in
Term f l_sigma
end.
Fixpoint ac_size (t:term) : nat :=
match t with
| Var v => 1
| Term f l =>
let ac_size_list :=
(fix ac_size_list (l : list term) {struct l} : nat :=
match l with
| nil => 0
| t :: lt => ac_size t + ac_size_list lt
end) in
(match arity f with
| AC => (length l) - 1
| C => 1
| Free _ => 1
end) + ac_size_list l
end.
Parameter l_assoc :
forall f t1 t2 t3, arity f = AC ->
ac (Term f (Term f (t1 :: t2 :: nil) :: t3 :: nil))
(Term f (t1 :: (Term f (t2 :: t3 :: nil)) :: nil)). | Definition | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | well_formed_cf_subst | null |
r_assoc:
forall f t1 t2 t3, arity f = AC ->
ac (Term f (t1 :: (Term f (t2 :: t3 :: nil)) :: nil))
(Term f (Term f (t1 :: t2 :: nil) :: t3 :: nil)). | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | r_assoc | null |
comm:
forall f t1 t2, arity f = C \/ arity f = AC ->
ac (Term f (t1 :: t2 :: nil)) (Term f (t2 :: t1 :: nil)). | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | comm | null |
ac_top_eq:
forall t1 t2, ac t1 t2 ->
match t1, t2 with
| Var x1, Var x2 => x1 = x2
| Term _ _, Var _ => False
| Var _, Term _ _ => False
| Term f1 _, Term f2 _ => f1 = f2
end. | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | ac_top_eq | null |
well_formed_cf_unfold: forall t,
well_formed_cf t -> match t with
| Var _ => True
| Term f l =>
(forall u, In u l -> well_formed_cf u) /\
(match arity f with
| AC => length l >= 2 /\ is_sorted l /\
(forall u, In u l -> match u with
| Var _ => True
| Term g _ => f <> g
end)
| C => length l = 2 /\ is_sorted l
| Free n => length l = n
end)
end. | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | well_formed_cf_unfold | null |
well_formed_cf_subterms:
forall f l, well_formed_cf (Term f l) -> (forall t, In t l -> well_formed_cf t). | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | well_formed_cf_subterms | null |
well_formed_cf_length:
forall f l, arity f = AC -> well_formed_cf (Term f l) -> 2 <= length l. | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | well_formed_cf_length | null |
well_formed_cf_sorted:
forall f l, arity f = AC -> well_formed_cf (Term f l) -> is_sorted l. | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | well_formed_cf_sorted | null |
well_formed_cf_alien:
forall f l, arity f = AC -> well_formed_cf (Term f l) ->
(forall t, In t l -> match t with
| Var _ => True
| Term g _ => f <> g
end).
Parameter flatten_app :
forall f l1 l2, flatten f (l1 ++ l2) = (flatten f l1) ++ (flatten f l2). | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | well_formed_cf_alien | null |
list_permut_flatten:
forall f l1 l2, permut l1 l2 -> permut (flatten f l1) (flatten f l2). | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | list_permut_flatten | null |
length_flatten_bis:
forall f, arity f = AC ->
forall l, (forall t, In t l -> well_formed_cf t) ->
(length l) <= length (flatten f l). | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | length_flatten_bis | null |
flatten_cf:
forall f t1 t2, arity f = AC -> well_formed_cf t1 -> well_formed_cf t2 ->
permut (flatten f (t1 :: nil)) (flatten f (t2 :: nil)) ->
t1 = t2.
Parameter flatten_cf_cf :
forall f t1 t2, arity f = AC -> well_formed t1 -> well_formed t2 ->
permut (flatten f (canonical_form t1 :: nil))
(flatten f (canonical_form t2 :: nil)) ->
canonical_form t1 = canonical_form t2. | Parameter | Coccinelle | [
"From Stdlib Require Import Relations List Arith Morphisms",
"From CoLoR Require Import more_list list_permut list_sort term_spec term_o equational_theory_spec equational_theory"
] | Coccinelle/ac_matching/ac.v | flatten_cf | null |
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