phanerozoic/Coq-CoLoR · Datasets at Hugging Face

statement stringlengths 1 9.63k	proof stringlengths 0 31.3k	type stringclasses 19 values	symbolic_name stringlengths 1 54	library stringclasses 43 values	filename stringclasses 263 values	imports listlengths 1 32	deps listlengths 0 64	docstring stringlengths 0 2.1k	source_url stringclasses 1 value	commit stringclasses 1 value
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...	Inductive	ac_one_step_at_top	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "symbol", "t1", "t2", "t3", "term" ]	** Definition of AC.	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac	:= th_eq ac_one_step_at_top.	Definition	ac	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_one_step_at_top", "th_eq" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	flatten	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "eq_bool", "symbol", "term" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	canonical_form	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "arity", "flatten", "map", "quicksort", "term" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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 ...	Fixpoint	well_formed_cf	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "is_sorted", "term" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
build (f : symbol) l	:= match l with \| t :: nil => t \| _ => Term f (quicksort l) end.	Definition	build	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "quicksort", "symbol" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_subst sigma	:= forall v, match find X.eq_bool v sigma with \| None => True \| Some t => well_formed_cf t end.	Definition	well_formed_cf_subst	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "eq_bool", "find", "well_formed_cf" ]	Definition well_formed_cf_subst	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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) \| ...	Fixpoint	apply_cf_subst	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "arity", "eq_bool", "find", "flatten", "map", "quicksort", "substitution", "term" ]	Definition apply_cf_subst	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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 ...	Fixpoint	ac_size	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "arity", "lt", "term" ]	Definition ac_size	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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)).		Parameter	l_assoc	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "ac", "arity", "t1", "t2", "t3" ]	Theorem l_assoc	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	r_assoc	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "ac", "arity", "t1", "t2", "t3" ]	Theorem r_assoc	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
comm : forall f t1 t2, arity f = C \/ arity f = AC -> ac (Term f (t1 :: t2 :: nil)) (Term f (t2 :: t1 :: nil)).		Parameter	comm	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "ac", "arity", "t1", "t2" ]	Theorem comm	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	ac_top_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "ac", "f1", "f2", "t1", "t2" ]	Theorem ac_top_eq	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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 /\ ...		Parameter	well_formed_cf_unfold	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "is_sorted", "well_formed_cf" ]	Theorem well_formed_cf_unfold	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_subterms : forall f l, well_formed_cf (Term f l) -> (forall t, In t l -> well_formed_cf t).		Parameter	well_formed_cf_subterms	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "well_formed_cf" ]	Theorem well_formed_cf_subterms	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_length : forall f l, arity f = AC -> well_formed_cf (Term f l) -> 2 <= length l.		Parameter	well_formed_cf_length	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "well_formed_cf" ]	Theorem well_formed_cf_length	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_sorted : forall f l, arity f = AC -> well_formed_cf (Term f l) -> is_sorted l.		Parameter	well_formed_cf_sorted	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "is_sorted", "well_formed_cf" ]	Theorem well_formed_cf_sorted	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	well_formed_cf_alien	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "well_formed_cf" ]	Theorem well_formed_cf_alien	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_app : forall f l1 l2, flatten f (l1 ++ l2) = (flatten f l1) ++ (flatten f l2).		Parameter	flatten_app	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "flatten" ]	Theorem flatten_app	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
list_permut_flatten : forall f l1 l2, permut l1 l2 -> permut (flatten f l1) (flatten f l2).		Parameter	list_permut_flatten	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "flatten", "permut" ]	Theorem list_permut_flatten	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	length_flatten_bis	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "arity", "flatten", "well_formed_cf" ]	Theorem length_flatten_bis	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "arity", "flatten", "permut", "t1", "t2", "well_formed_cf" ]	Theorem flatten_cf	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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	flatten_cf_cf	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "arity", "canonical_form", "flatten", "permut", "t1", "t2", "well_formed" ]	Theorem flatten_cf_cf	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
build_eq_Term : forall f l, 2 <= length l -> build f l = Term f (quicksort l).		Parameter	build_eq_Term	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "build", "quicksort" ]	Theorem build_eq_Term	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_build : forall f l, arity f = AC -> 1 <= length l -> (forall t, In t l -> well_formed_cf t) -> (forall t, In t l -> match t with \| Var _ => True \| Term g _ => f <> g end) -> well_formed_cf (build f l).		Parameter	well_formed_cf_build	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "build", "well_formed_cf" ]	Theorem well_formed_cf_build	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> well_formed_cf (build f l).		Parameter	well_formed_cf_build_cons	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "build", "well_formed_cf" ]	Theorem well_formed_cf_build_cons	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> well_formed_cf (build f (l1 ++ l2)).		Parameter	well_formed_cf_build_inside	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "build", "well_formed_cf" ]	Theorem well_formed_cf_build_inside	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_build : forall f l, arity f = AC -> (forall t, In t l -> match t with \| Var _ => True \| Term g _ => f <> g end) -> permut (flatten f ((build f l) :: nil)) l.		Parameter	flatten_build	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "build", "flatten", "permut" ]	Theorem flatten_build	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> permut (flatten f ((build f l) :: nil)) l.		Parameter	flatten_build_cons	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "build", "flatten", "permut", "well_formed_cf" ]	Theorem flatten_build_cons	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> permut (flatten f ((build f (l1 ++ l2)) :: nil)) (l1 ++ l2).		Parameter	flatten_build_inside	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "build", "flatten", "permut", "well_formed_cf" ]	Theorem flatten_build_inside	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_apply_cf_subst : forall sigma f l, arity f = AC -> permut (flatten f (map (apply_cf_subst sigma) l)) (flatten f (apply_cf_subst sigma (build f l) :: nil)).		Parameter	flatten_apply_cf_subst	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "apply_cf_subst", "arity", "build", "flatten", "map", "permut" ]	Theorem flatten_apply_cf_subst	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_apply_subst : forall sigma, well_formed_cf_subst sigma -> forall t, well_formed_cf t -> well_formed_cf (apply_cf_subst sigma t).		Parameter	well_formed_cf_apply_subst	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "apply_cf_subst", "well_formed_cf", "well_formed_cf_subst" ]	Theorem well_formed_cf_apply_subst	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
length_flatten_ter : forall f sigma, arity f = AC -> well_formed_cf_subst sigma -> forall l, (forall t, In t l -> well_formed_cf t) -> length l <= length (flatten f (map (apply_cf_subst sigma) l)).		Parameter	length_flatten_ter	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "apply_cf_subst", "arity", "flatten", "map", "well_formed_cf", "well_formed_cf_subst" ]	Theorem length_flatten_ter	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_cf_eq : forall t1 t2, ac t1 t2 -> canonical_form t1 = canonical_form t2.		Parameter	ac_cf_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac", "canonical_form", "t1", "t2" ]	Theorem ac_cf_eq	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_size_eq : forall t1 t2, ac t1 t2 -> size t1 = size t2.		Parameter	ac_size_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac", "size", "t1", "t2" ]	Theorem ac_size_eq	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_size_unfold : forall t, ac_size t = match t with \| Var _ => 1 \| Term f l => (match arity f with \| AC => (length l) - 1 \| C => 1 \| Free _ => 1 end) + list_size ac_size l end.		Parameter	ac_size_unfold	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "ac_size", "arity", "list_size" ]	Theorem ac_size_unfold	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
size_size_aux3 : forall f t, arity f = AC -> well_formed t -> 1 <= length (A:=term) (flatten f (canonical_form t :: nil)).		Parameter	size_size_aux3	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "arity", "canonical_form", "flatten", "term", "well_formed" ]	Theorem size_size_aux3	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
size_size : forall t, well_formed t -> size t = ac_size (canonical_form t).		Parameter	size_size	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_size", "canonical_form", "size", "well_formed" ]	Theorem size_size	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_size_ge_one : forall t, well_formed_cf t -> 1 <= ac_size t.		Parameter	ac_size_ge_one	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_size", "well_formed_cf" ]	Theorem ac_size_ge_one	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
no_need_of_instance : forall t1 t2, axiom (sym_refl ac_one_step_at_top) t1 t2 -> (sym_refl ac_one_step_at_top) t1 t2.	Proof. unfold sym_refl; intros t1 t2 H. inversion_clear H as [ u1 u2 sigma H']; destruct H' as [H' \| [H' \| H']]. inversion_clear H'. left; simpl; apply a_axiom; trivial. left; simpl; apply c_axiom; trivial. inversion_clear H'. right; left; simpl; apply a_axiom; trivial. right; left; simpl; apply c_axiom; trivial. righ...	Lemma	no_need_of_instance	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_one_step_at_top", "axiom", "subst", "sym_refl", "t1", "t2" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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)).	Proof. intros f t1 t2 t3 Af. unfold ac, th_eq. do 2 left. rewrite <- (empty_subst_is_id (Term f (Term f (t1 :: t2 :: nil) :: t3 :: nil))). rewrite <- (empty_subst_is_id (Term f (t1 :: Term f (t2 :: t3 :: nil) :: nil))). apply instance. left; apply a_axiom; trivial. Qed.	Lemma	l_assoc	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "ac", "arity", "empty_subst_is_id", "rewrite", "t1", "t2", "t3", "th_eq" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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)).	Proof. intros f t1 t2 t3 Af. unfold ac, th_eq. do 2 left. rewrite <- (empty_subst_is_id (Term f (Term f (t1 :: t2 :: nil) :: t3 :: nil))). rewrite <- (empty_subst_is_id (Term f (t1 :: Term f (t2 :: t3 :: nil) :: nil))). apply instance. right; left; apply a_axiom; trivial. Qed.	Lemma	r_assoc	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "ac", "arity", "empty_subst_is_id", "rewrite", "t1", "t2", "t3", "th_eq" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
comm : forall f t1 t2, arity f = C \/ arity f = AC -> ac (Term f (t1 :: t2 :: nil)) (Term f (t2 :: t1 :: nil)).	Proof. intros f t1 t2 Af. unfold ac, th_eq. do 2 left. rewrite <- (empty_subst_is_id (Term f (t1 :: t2 :: nil))). rewrite <- (empty_subst_is_id (Term f (t2 :: t1 :: nil))). apply instance. left; apply c_axiom; trivial. Qed.	Lemma	comm	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "ac", "arity", "empty_subst_is_id", "rewrite", "t1", "t2", "th_eq" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_one_step_at_top_top_eq : forall t1 t2, (sym_refl ac_one_step_at_top) 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.	Proof. unfold sym_refl; intros t1 t2 [Ac \| [Ac \| t1_eq_t2]]. inversion_clear Ac; simpl; trivial. inversion_clear Ac; simpl; trivial. subst t2; destruct t1; trivial. Qed.	Lemma	ac_one_step_at_top_top_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "ac_one_step_at_top", "f1", "f2", "subst", "sym_refl", "t1", "t2" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_one_step_top_eq : forall t1 t2 : term, one_step (sym_refl ac_one_step_at_top) 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.	Proof. intros t1 t2 Ac; inversion Ac; trivial. apply ac_one_step_at_top_top_eq; apply no_need_of_instance; trivial. Qed.	Lemma	ac_one_step_top_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "ac_one_step_at_top", "ac_one_step_at_top_top_eq", "f1", "f2", "no_need_of_instance", "one_step", "sym_refl", "t1", "t2", "term" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_top_eq : forall t1 t2 : term, 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.	Proof. intros t1 t2 Ac; induction Ac. (* one step ) inversion H as [ H1 H2 H' H4 H5 \| f l1 l2 H' H1 H2]; subst; trivial. apply ac_one_step_top_eq; assumption. ( th_trans *) generalize (ac_one_step_top_eq H). destruct x as [ v1 \| f1 l1 ]; destruct y as [ v2 \| f2 l2 ]; destruct z as [ v3 \| f3 l3 ]; trivial. intro; ...	Lemma	ac_top_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "H1", "Term", "Var", "ac", "ac_one_step_top_eq", "f1", "f2", "subst", "t1", "t2", "term" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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 /\ ...	Proof. intro t; destruct t as [v \| f l]; simpl; trivial; intros [Wl Ll]; split; trivial. clear Ll; induction l as [ \| t l]. contradiction. intros u [u_eq_t \| In_u]; subst; intuition. Qed.	Lemma	well_formed_cf_unfold	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "is_sorted", "split", "subst", "well_formed_cf" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_subterms : forall f l, well_formed_cf (Term f l) -> (forall t, In t l -> well_formed_cf t).	Proof. intros f l W; elim (well_formed_cf_unfold _ W); trivial. Qed.	Lemma	well_formed_cf_subterms	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "well_formed_cf", "well_formed_cf_unfold" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_length : forall f l, arity f = AC -> well_formed_cf (Term f l) -> 2 <= length l.	Proof. intros f l Af [_ Ll]; rewrite Af in Ll; intuition. Qed.	Lemma	well_formed_cf_length	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "rewrite", "well_formed_cf" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_sorted : forall f l, arity f = AC -> well_formed_cf (Term f l) -> is_sorted l.	Proof. intros f l Af [_ Ll]; rewrite Af in Ll; intuition. Qed.	Lemma	well_formed_cf_sorted	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "is_sorted", "rewrite", "well_formed_cf" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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).	Proof. intros f l Af [_ Ll]; rewrite Af in Ll; intuition. Qed.	Lemma	well_formed_cf_alien	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "rewrite", "well_formed_cf" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_fold : forall 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 /\ (fo...	Proof. intro t; destruct t as [v \| f l]; trivial. intros [Wl Hl]; split; trivial; clear Hl; induction l as [ \| t l]; intuition auto with *; apply IHl; intros; apply Wl; right; trivial. Qed.	Lemma	well_formed_cf_fold	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "is_sorted", "split", "well_formed_cf" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_app : forall f l1 l2, flatten f (l1 ++ l2) = (flatten f l1) ++ (flatten f l2).	Proof. intros f l1; induction l1 as [ \| [v1 \| g1 ll1]]; simpl; trivial. intros l2; rewrite IHl1; trivial. intros l2; rewrite IHl1; generalize (F.Symb.eq_bool_ok f g1); case (F.Symb.eq_bool f g1); [intro f_eq_g1 \| intro f_diff_g1]. rewrite app_ass; trivial. simpl; trivial. Qed.	Lemma	flatten_app	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "eq_bool", "eq_bool_ok", "flatten", "rewrite" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
list_permut_flatten : forall f l1 l2, permut l1 l2 -> permut (flatten f l1) (flatten f l2).	Proof. intros f l1; induction l1 as [ \| t1 l1]; intros l2 P. rewrite (permut_nil (permut_sym P)); apply permut_refl. assert (In_t1 : mem (@eq _) t1 l2). rewrite <- P; left; reflexivity. destruct (mem_split_set _ _ eq_bool_ok _ _ In_t1) as [t1' [l2' [l2'' [t1_eq_t1' [H _]]]]]; simpl in t1_eq_t1'; simpl in H; subst l2; ...	Lemma	list_permut_flatten	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "dummy", "eq", "eq_bool", "eq_bool_ok", "f1", "flatten", "flatten_app", "list_permut_app_app", "mem", "mem_split_set", "permut", "permut_app1", "permut_app2", "permut_cons_inside", "permut_nil", "permut_refl", "permut_sym", "rewrite", "subst", "t1" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_is_well_formed_cf : forall cf, well_formed_cf cf -> exists t, well_formed t /\ cf = canonical_form t.	Proof. intros cf Wcf; generalize (well_formed_cf_unfold _ Wcf); pattern cf; apply term_rec3; clear cf Wcf. intros v _; exists (Var v); unfold well_formed; simpl; split; trivial. intros f l Hrec [Wl Ll]; assert (Wl' : forall u, In u l -> exists s, well_formed s /\ u = canonical_form s). intros u In_u; apply Hrec; trivi...	Lemma	well_formed_cf_is_well_formed_cf	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "canonical_form", "destruct_arity", "eq_bool", "eq_bool_ok", "ge", "list_permut_app_app", "map", "permut_cons_inside", "permut_refl", "permut_sym", "quick_permut", "quick_permut_bis", "quick_sorted", "rewrite", "sort_is_unique", "sorted_tl_sorted", "split",...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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).	Proof. intros f Af l Wl; induction l as [ \| t l]. simpl; auto with arith. simpl; destruct t as [v \| g ll]. simpl; apply le_n_S; apply IHl; intros; apply Wl; right; trivial. generalize (F.Symb.eq_bool_ok f g); case (F.Symb.eq_bool f g); [intro f_eq_g \| intro f_diff_g]. rewrite length_app; replace (S (length l)) with (1 ...	Lemma	length_flatten_bis	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "arity", "eq_bool", "eq_bool_ok", "flatten", "length_app", "replace", "rewrite", "subst", "well_formed_cf" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
length_flatten : forall f, forall l, arity f = AC -> (forall u, In u l -> well_formed u) -> length l <= length (flatten f (map canonical_form l)).	Proof. intros f l; pattern l; apply (list_rec3 size); clear l; induction n; destruct l as [ \| t l]. simpl; trivial. simpl; intro S_l; absurd (1 <= 0); auto with arith; apply Nat.le_trans with (size t + list_size size l); trivial. apply Nat.le_trans with (2 := Nat.le_add_r _ _), size_ge_one. simpl; trivial. intros Sl A...	Lemma	length_flatten	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "arity", "canonical_form", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "le_trans", "length_app", "length_quicksort", "list_rec3", "list_size", "map", "map_app", "replace", "rewrite", "size", "size_ge_one", "size_unfold", "subst", "well_formed", "well_...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_is_well_formed_cf_conv : forall cf, (exists t, well_formed t /\ cf = canonical_form t) -> well_formed_cf cf.	Proof. intros cf [t [Wt Ct]]; subst; generalize Wt; clear Wt. pattern t; apply term_rec2; clear t; induction n as [ \| n ]; intros t St Wt. absurd (1 <= 0); auto with arith; apply Nat.le_trans with (size t); trivial; apply size_ge_one. apply well_formed_cf_fold; destruct t as [ v \| f l ]; simpl; trivial. generalize (we...	Lemma	well_formed_cf_is_well_formed_cf_conv	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "canonical_form", "destruct_arity", "eq_bool", "eq_bool_ok", "flatten_app", "ge", "in_quick_in", "le_trans", "length_flatten", "length_quicksort", "list_rec3", "list_size", "map", "quick_sorted", "rewrite", "size", "size_ge_one", "size_unfold", "split", "subst", "t1",...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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.	Proof. intros f t1 t2 Af Wt1 Wt2; destruct t1 as [v1 \| f1 l1]; destruct t2 as [v2 \| f2 l2]. (* t1 = Var v1; t2 = Var v2 ) simpl; intros; apply permut_length_1; trivial. ( t1 = Var v1; t2 = Term f2 l2 *) simpl; generalize (F.Symb.eq_bool_ok f f2); case (F.Symb.eq_bool f f2); [intros f_eq_f2; subst f2 \| intros f_diff_f...	Lemma	flatten_cf	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "eq_bool", "eq_bool_ok", "f1", "f2", "flatten", "permut", "permut_length", "permut_length_1", "rewrite", "sort_is_unique", "subst", "t1", "t2", "well_formed_cf", "well_formed_cf_length", "well_formed_cf_sorted" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
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.	Proof. intros f t1 t2 Af Wt1 Wt2 P; apply flatten_cf with f; trivial. apply well_formed_cf_is_well_formed_cf_conv; exists t1; split; trivial. apply well_formed_cf_is_well_formed_cf_conv; exists t2; split; trivial. Qed.	Lemma	flatten_cf_cf	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "arity", "canonical_form", "flatten", "flatten_cf", "permut", "split", "t1", "t2", "well_formed", "well_formed_cf_is_well_formed_cf_conv" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
build_eq_Term : forall f l, 2 <= length l -> build f l = Term f (quicksort l).	Proof. intros f l Ll; destruct l as [ \| u [ \| v l]]; simpl; trivial. simpl in Ll; absurd (2 <= 1); auto with arith. Qed.	Lemma	build_eq_Term	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "build", "quicksort" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_build : forall f l, arity f = AC -> 1 <= length l -> (forall t, In t l -> well_formed_cf t) -> (forall t, In t l -> match t with \| Var _ => True \| Term g _ => f <> g end) -> well_formed_cf (build f l).	Proof. intros f l Af Ll Wl Al; destruct l as [ \| t1 [ \| t2 l]]. simpl in Ll; absurd (1 <= 0); trivial; auto with arith. apply Wl; left; trivial. rewrite build_eq_Term; [idtac \| simpl; auto with arith]. apply well_formed_cf_fold; split. intros u In_u; rewrite <- in_quick_in in In_u. apply Wl; trivial. rewrite Af; split;...	Lemma	well_formed_cf_build	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "build", "build_eq_Term", "in_quick_in", "length_quicksort", "quick_sorted", "rewrite", "split", "t1", "t2", "well_formed_cf", "well_formed_cf_fold" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> well_formed_cf (build f l).	Proof. intros f t l Af W; apply well_formed_cf_build; trivial. apply le_S_n; replace (S (length l)) with (length (t :: l)); trivial; apply well_formed_cf_length with f; trivial. intros; apply (well_formed_cf_subterms W); right; trivial. intros; apply (well_formed_cf_alien Af W); right; trivial. Qed.	Lemma	well_formed_cf_build_cons	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "build", "replace", "well_formed_cf", "well_formed_cf_alien", "well_formed_cf_build", "well_formed_cf_length", "well_formed_cf_subterms" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> well_formed_cf (build f (l1 ++ l2)).	Proof. intros f t l1 l2 Af W; assert (H : forall u, In u (l1 ++ l2) -> In u (l1 ++ t :: l2)). intros u In_u; elim (in_app_or _ _ _ In_u); clear In_u; intro In_u; apply in_or_app; [ left \| right; right ]; trivial. apply well_formed_cf_build; trivial. apply le_S_n; replace (S (length (l1 ++ l2))) with (length (l1 ++ t ...	Lemma	well_formed_cf_build_inside	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "add_comm", "arity", "build", "length_app", "replace", "rewrite", "well_formed_cf", "well_formed_cf_alien", "well_formed_cf_build", "well_formed_cf_length", "well_formed_cf_subterms" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_build : forall f l, arity f = AC -> (forall t, In t l -> match t with \| Var _ => True \| Term g _ => f <> g end) -> permut (flatten f ((build f l) :: nil)) l.	Proof. intros f [ \| t1 [ \| t2 l]] Af Al. simpl; generalize (F.Symb.eq_bool_ok f f); case (F.Symb.eq_bool f f); [intros _; apply Pnil \| intro f_diff_f; apply False_rect; apply f_diff_f; reflexivity]. assert (Al_t1 := Al _ (or_introl _ (eq_refl _))). simpl; destruct t1 as [ \| g]; [auto \| idtac]. generalize (F.Symb.eq_boo...	Lemma	flatten_build	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "Term", "Var", "arity", "build", "eq_bool", "eq_bool_ok", "eq_refl", "flatten", "permut", "quick_permut_bis", "rewrite", "t1", "t2" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> permut (flatten f ((build f l) :: nil)) l.	Proof. intros f t l Af W; apply flatten_build; trivial. intros; apply (well_formed_cf_alien Af W); right; trivial. Qed.	Lemma	flatten_build_cons	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "build", "flatten", "flatten_build", "permut", "well_formed_cf", "well_formed_cf_alien" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> permut (flatten f ((build f (l1 ++ l2)) :: nil)) (l1 ++ l2).	Proof. intros f t l1 l2 Af W; apply flatten_build; trivial. intros u In_u; apply (well_formed_cf_alien Af W); elim (in_app_or _ _ _ In_u); clear In_u; intro In_u; apply in_or_app; [left \| right; right]; trivial. Qed.	Lemma	flatten_build_inside	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "arity", "build", "flatten", "flatten_build", "permut", "well_formed_cf", "well_formed_cf_alien" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
is_subst_canonical_form sigma sigma_cf	:= forall v, match find X.eq_bool v sigma with \| None => find X.eq_bool v sigma_cf = None \| Some v_sigma => find X.eq_bool v sigma_cf = Some (canonical_form v_sigma) end.	Definition	is_subst_canonical_form	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "canonical_form", "eq_bool", "find" ]	** Substitutions modulo AC	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
is_subst_cf_is_subst_cf : forall sigma, is_subst_canonical_form sigma (map_subst (fun _ t => canonical_form t) sigma).	Proof. intros sigma; unfold is_subst_canonical_form. intro v; rewrite subst_comp_is_subst_comp_aux1. destruct (find X.eq_bool v sigma); trivial. Qed.	Lemma	is_subst_cf_is_subst_cf	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "canonical_form", "eq_bool", "find", "is_subst_canonical_form", "map_subst", "rewrite", "subst_comp_is_subst_comp_aux1" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_subst_is_well_formed_cf_subst_aux : forall sigma, well_formed_cf_subst sigma -> (forall v, nb_occ X.eq_bool v sigma <= 1) -> exists sigma', well_formed_subst sigma' /\ is_subst_canonical_form sigma' sigma.	Proof. unfold well_formed_cf_subst, is_subst_canonical_form; intros sigma Wsigma Nb_occ_sigma; induction sigma as [ \| [v1 t1] sigma]. exists (nil : substitution); split; trivial; intro v; simpl; trivial. elim IHsigma. intros sigma' [Wsigma' Hsigma']; assert (Wt1 : well_formed_cf t1). generalize (Wsigma v1); simpl; ge...	Lemma	well_formed_cf_subst_is_well_formed_cf_subst_aux	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "eq_bool", "eq_bool_ok", "eq_refl", "find", "is_subst_canonical_form", "nb_occ", "some_nb_occ_Sn", "split", "subst", "substitution", "t1", "well_formed_cf", "well_formed_cf_is_well_formed_cf", "well_formed_cf_subst", "well_formed_subst" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_subst_is_well_formed_cf_subst : forall sigma, well_formed_cf_subst sigma -> exists sigma', well_formed_subst sigma' /\ is_subst_canonical_form sigma' sigma.	Proof. assert (E : equivalence _ (@eq variable)). repeat split. intros t1 t2 t3 H1 H2; subst; reflexivity. intros t1 t2 H; subst; reflexivity. intros sigma Wsigma; elim (reduce_assoc_list _ X.eq_bool_ok E sigma); intros sigma1 [Nb_occ H]; elim (well_formed_cf_subst_is_well_formed_cf_subst_aux (sigma := sigma1)); trivia...	Lemma	well_formed_cf_subst_is_well_formed_cf_subst	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "H1", "eq", "eq_bool_ok", "is_subst_canonical_form", "reduce_assoc_list", "rewrite", "split", "subst", "t1", "t2", "t3", "variable", "well_formed_cf_subst", "well_formed_cf_subst_is_well_formed_cf_subst_aux", "well_formed_subst" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
flatten_apply_cf_subst : forall sigma f l, arity f = AC -> permut (flatten f (map (apply_cf_subst sigma) l)) (flatten f (apply_cf_subst sigma (build f l) :: nil)).	Proof. intros sigma f l Af; induction l as [ \| t1 l]. simpl; generalize (F.Symb.eq_bool_ok f f); case (F.Symb.eq_bool f f); [intros _ \| intro f_diff_f; apply False_rect; apply f_diff_f; reflexivity]. rewrite Af; repeat rewrite quicksort_equation; simpl; auto. destruct l as [ \| t2 l]; auto. simpl; rewrite Af; generaliz...	Lemma	flatten_apply_cf_subst	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "apply_cf_subst", "arity", "build", "eq", "eq_bool", "eq_bool_ok", "flatten", "list_permut_flatten", "map", "permut", "permut_map", "permut_sym", "quick_permut", "quick_permut_bis", "quicksort_equation", "rewrite", "subst", "t1", "t2", "term" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
apply_cf_subst_is_sound : forall sigma sigma_cf, is_subst_canonical_form sigma sigma_cf -> forall t, apply_cf_subst sigma_cf (canonical_form t) = canonical_form (apply_subst sigma t).	Proof. intros sigma sigma_cf H t; pattern t; apply term_rec3; clear t. intro v; generalize (H v); simpl; destruct (find X.eq_bool v sigma) as [t \| ]; intro H_v; rewrite H_v; trivial. intros f l IH; assert (IHl : map (apply_cf_subst sigma_cf) (map canonical_form l) = map canonical_form (map (apply_subst sigma) l))...	Theorem	apply_cf_subst_is_sound	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "apply_cf_subst", "apply_subst", "canonical_form", "destruct_arity", "eq", "eq_bool", "eq_bool_ok", "find", "flatten", "flatten_app", "is_subst_canonical_form", "list_permut_flatten", "map", "map_app", "permut_app1", "permut_app2", "permut_map", "permut_sym", "quick_per...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
well_formed_cf_apply_subst : forall sigma, well_formed_cf_subst sigma -> forall t, well_formed_cf t -> well_formed_cf (apply_cf_subst sigma t).	Proof. intros sigma Wsigma t Wt; elim (well_formed_cf_is_well_formed_cf _ Wt); intros u [Wu Hu]; subst; elim (well_formed_cf_subst_is_well_formed_cf_subst Wsigma); intros tau [Wtau Htau]; rewrite apply_cf_subst_is_sound with tau sigma u; trivial; apply well_formed_cf_is_well_formed_cf_conv; exists (apply_subst tau u);...	Theorem	well_formed_cf_apply_subst	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "apply_cf_subst", "apply_cf_subst_is_sound", "apply_subst", "rewrite", "split", "subst", "tau", "well_formed_apply_subst", "well_formed_cf", "well_formed_cf_is_well_formed_cf", "well_formed_cf_is_well_formed_cf_conv", "well_formed_cf_subst", "well_formed_cf_subst_is_well_formed_cf_subst" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
length_flatten_ter : forall f sigma, arity f = AC -> well_formed_cf_subst sigma -> forall l, (forall t, In t l -> well_formed_cf t) -> length l <= length (flatten f (map (apply_cf_subst sigma) l)).	Proof. intros f sigma Af Wsigma l Wl; induction l as [ \| t l]; trivial. replace (t::l) with ((t::nil) ++ l); trivial; rewrite map_app; rewrite flatten_app; do 2 rewrite length_app; apply Nat.add_le_mono. assert (Wtsigma : well_formed_cf (apply_cf_subst sigma t)). apply well_formed_cf_apply_subst; trivial; apply Wl; le...	Lemma	length_flatten_ter	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "In", "apply_cf_subst", "arity", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "le_trans", "length_app", "map", "map_app", "replace", "rewrite", "subst", "well_formed_cf", "well_formed_cf_apply_subst", "well_formed_cf_length", "well_formed_cf_subst" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_one_step_at_top_cf_eq : forall t1 t2, ac_one_step_at_top t1 t2 -> canonical_form t1 = canonical_form t2.	Proof. assert (P12 : forall t1 t2, permut (t1 :: t2 :: nil) (t2 :: t1 :: nil)). intros t1 t2; replace (t1 :: t2 :: nil) with ((t1 :: nil) ++ (t2 :: nil)); trivial; replace (t2 :: t1 :: nil) with ((t2 :: nil) ++ (t1 :: nil)); trivial; apply list_permut_app_app. assert (Ccase : forall f, arity f = C \/ arity f = AC -> ...	Lemma	ac_one_step_at_top_cf_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "ac_one_step_at_top", "arity", "canonical_form", "eq_bool", "eq_bool_ok", "f1", "f2", "flatten", "flatten_app", "list_permut_app_app", "permut", "permut_app1", "permut_app2", "permut_cons", "permut_cons_inside", "permut_refl", "permut_sym", "quick_permut", "quick_permut...	** Canonical forms and equality modulo AC	https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
sym_refl_ac_one_step_at_top_cf_eq : forall t1 t2, (sym_refl ac_one_step_at_top) t1 t2 -> canonical_form t1 = canonical_form t2.	Proof. intros t1 t2; unfold sym_refl; intros [H \| [H \| H]]. apply ac_one_step_at_top_cf_eq; trivial. apply sym_eq; apply ac_one_step_at_top_cf_eq; trivial. subst; trivial. Qed.	Lemma	sym_refl_ac_one_step_at_top_cf_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_one_step_at_top", "ac_one_step_at_top_cf_eq", "canonical_form", "subst", "sym_refl", "t1", "t2" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_one_step_cf_eq : forall t1 t2, one_step (sym_refl ac_one_step_at_top) t1 t2 -> canonical_form t1 = canonical_form t2.	Proof. intros t1; pattern t1; apply term_rec3; clear t1. intros v1 t2 H; inversion_clear H as [H1 H2 H' \| f l1 l2 H']. apply sym_refl_ac_one_step_at_top_cf_eq; apply no_need_of_instance; trivial. intros f l1 IHl1 t2 H; inversion_clear H as [ H1 H2 Hroot \| H1 H2 l2 Hsub]. apply sym_refl_ac_one_step_at_top_cf_eq; apply...	Lemma	ac_one_step_cf_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "H1", "ac_one_step_at_top", "canonical_form", "map", "no_need_of_instance", "one_step", "rewrite", "sym_refl", "sym_refl_ac_one_step_at_top_cf_eq", "t1", "t2", "term_rec3" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_cf_eq : forall t1 t2, ac t1 t2 -> canonical_form t1 = canonical_form t2.	Proof. unfold ac, th_eq; intros t1 t2 H. refine (rwr_inv _ _ _ _ (@trans_eq term) _ _ _ H). clear t1 t2 H; intros t1 t2 H; apply ac_one_step_cf_eq; trivial. Qed.	Theorem	ac_cf_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac", "ac_one_step_cf_eq", "canonical_form", "rwr_inv", "t1", "t2", "term", "th_eq" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_one_step_at_top_size_eq : forall t1 t2, ac_one_step_at_top t1 t2 -> size t1 = size t2.	Proof. intros t1 t2 H; destruct H; simpl; repeat rewrite Nat.add_0_r; auto with arith; rewrite (Nat.add_comm (size t1) (S (size t2 + size t3))); simpl; apply (f_equal (fun n => S (S n))); rewrite <- (Nat.add_assoc (size t1) (size t2) (size t3)); apply Nat.add_comm. Qed.	Lemma	ac_one_step_at_top_size_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_one_step_at_top", "add_comm", "rewrite", "size", "t1", "t2", "t3" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
sym_refl_ac_one_step_at_top_size_eq : forall t1 t2, (sym_refl ac_one_step_at_top) t1 t2 -> size t1 = size t2.	Proof. intros t1 t2; unfold sym_refl; intros [H \| [H \| H]]. apply ac_one_step_at_top_size_eq; trivial. apply sym_eq; apply ac_one_step_at_top_size_eq; trivial. subst; trivial. Qed.	Lemma	sym_refl_ac_one_step_at_top_size_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_one_step_at_top", "ac_one_step_at_top_size_eq", "size", "subst", "sym_refl", "t1", "t2" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_one_step_size_eq : forall t1 t2, one_step (sym_refl ac_one_step_at_top) t1 t2 -> size t1 = size t2.	Proof. intros t1; pattern t1; apply term_rec3; clear t1. intros v1 t2 H; inversion_clear H as [H1 H2 H' \| f l1 l2 H']. apply sym_refl_ac_one_step_at_top_size_eq; apply no_need_of_instance; trivial. intros f l1 IHl1 t2 H; inversion_clear H as [ H1 H2 Hroot \| H1 H2 l2 Hsub]. apply sym_refl_ac_one_step_at_top_size_eq; a...	Lemma	ac_one_step_size_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "H1", "ac_one_step_at_top", "list_size", "no_need_of_instance", "one_step", "rewrite", "size", "size_unfold", "sym_refl", "sym_refl_ac_one_step_at_top_size_eq", "t1", "t2", "term_rec3" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_size_eq : forall t1 t2, ac t1 t2 -> size t1 = size t2.	Proof. intros t1 t2 H; refine (rwr_inv _ _ _ _ (@trans_eq nat) _ _ _ H). intros; apply ac_one_step_size_eq; trivial. Qed.	Lemma	ac_size_eq	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac", "ac_one_step_size_eq", "rwr_inv", "size", "t1", "t2" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_size_unfold : forall t, ac_size t = match t with \| Var _ => 1 \| Term f l => (match arity f with \| AC => (length l) - 1 \| C => 1 \| Free _ => 1 end) + list_size ac_size l end.	Proof. intro t; destruct t as [v \| f l]; trivial; simpl. apply (f_equal (fun n => (match arity f with \| AC => length l - 1 \| C => 1 \| Free _ => 1 end) + n)); ...	Lemma	ac_size_unfold	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Term", "Var", "ac_size", "arity", "list_size", "rewrite" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
size_size_aux2 : forall f t, arity f = AC -> well_formed t -> ac_size (canonical_form t) = list_size ac_size (flatten f (canonical_form t :: nil)) + (length (flatten f (canonical_form t :: nil))) - 1.	Proof. intros f t Af Wt; assert (Wu : well_formed_cf (canonical_form t)). apply well_formed_cf_is_well_formed_cf_conv; exists t; split; trivial. generalize (canonical_form t) Wu; clear t Wt Wu; intros u Wu. destruct u as [v \| g l]; trivial; simpl. simpl; generalize (F.Symb.eq_bool_ok f g); case (F.Symb.eq_bool f g); [i...	Lemma	size_size_aux2	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_size", "add_comm", "arity", "canonical_form", "eq_bool", "eq_bool_ok", "flatten", "list_size", "list_size_fold", "rewrite", "split", "subst", "well_formed", "well_formed_cf", "well_formed_cf_is_well_formed_cf_conv", "well_formed_cf_length" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
size_size_aux3 : forall f t, arity f = AC -> well_formed t -> 1 <= length (A:=term) (flatten f (canonical_form t :: nil)).	Proof. intros f t Ar Wt; apply (length_flatten (t :: nil) Ar); intros u In_u; elim In_u; clear In_u; intro In_u. subst u; trivial. contradiction. Qed.	Lemma	size_size_aux3	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "arity", "canonical_form", "flatten", "length_flatten", "subst", "term", "well_formed" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
size_size : forall t, well_formed t -> size t = ac_size (canonical_form t).	Proof. intros t; pattern t; apply term_rec3; clear t. intros v _; trivial. intros f l H Wt; destruct (well_formed_unfold Wt) as [Wl Ll]. assert (Hl : forall t, In t l -> size t = ac_size (canonical_form t)). intros; apply H; trivial; apply Wl; trivial. clear H Wt; assert (Hl' : list_size size l = list_size ac_size (map...	Lemma	size_size	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "Dummy", "In", "Term", "ac_size", "add_comm", "canonical_form", "destruct_arity", "eq", "eq_bool", "eq_bool_ok", "eq_refl", "f1", "f2", "flatten", "flatten_app", "length_app", "length_quicksort", "list_size", "list_size_app", "list_size_fold", "map", "permut_size", "plus"...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
ac_size_ge_one : forall t, well_formed_cf t -> 1 <= ac_size t.	Proof. intros t Wt; elim (well_formed_cf_is_well_formed_cf _ Wt); intros u [Wu Hu]; subst; rewrite <- size_size; trivial; apply size_ge_one. Qed.	Lemma	ac_size_ge_one	Coccinelle.ac_matching	Coccinelle/ac_matching/ac.v	[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]	[ "ac_size", "rewrite", "size_ge_one", "size_size", "subst", "well_formed_cf", "well_formed_cf_is_well_formed_cf" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
cf_eq_ac : forall t1 t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2.		Parameter	cf_eq_ac	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "ac", "canonical_form", "t1", "t2", "well_formed" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
split_cf : forall f, arity f = AC -> forall t u1 u2, well_formed t -> permut (flatten f (canonical_form t :: nil)) (u1 ++ u2) -> u1 = nil \/ u2 = nil \/ (exists t1, exists t2, well_formed t1 /\ well_formed t2 /\ ac t (Term f (t1 :: t2 :: nil)) /\ permut (flatten f (canonical_form t1 :: nil)) u1 /\...	Proof. intros f Af t; pattern t; apply term_rec3; clear t. (* t = Var _ *) intros v l1 l2 _ P; simpl in P; destruct l1 as [ \| t1 l1]; [left; trivial \| idtac]; destruct l2 as [ \| t2 l2]; [right; left; trivial \| idtac]; generalize (list_permut.permut_length P); rewrite length_app; simpl; rewrite Nat.add_comm; intro; dis...	Lemma	split_cf	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "H1", "Term", "ac", "ac_syntactic", "add_comm", "arity", "canonical_form", "comm", "context_in", "eq", "eq_bool", "eq_bool_ok", "eq_refl", "flatten", "flatten_app", "l_assoc", "length_app", "list_permut_app_app", "list_permut_flatten", "map", "permut", "permut_length", "p...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
syntactic_dec : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> ...	Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Wa3 Wa4 k1 k4 P1 P2 P3 P4. apply trans_clos_is_trans with (Term f (a3 :: a2 :: nil)). refine (context_in _ _ _ _ f nil (a2 :: nil)). apply IH; trivial; apply flatten_cf_cf with f; trivial; rewrite P1; rewrite P3; auto. refine (context_in _ _ _ _ f (a3 :: nil) nil). a...	Lemma	syntactic_dec	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "Term", "ac", "arity", "canonical_form", "context_in", "flatten", "flatten_cf_cf", "permut", "rewrite", "size", "t1", "t2", "trans_clos_is_trans", "well_formed" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
commutativity : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> wel...	Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Wa3 Wa4 k2 k3 P1 P2 P3 P4. apply trans_clos_is_trans with (Term f (a2 :: a1 :: nil)). apply comm; right; trivial. apply trans_clos_is_trans with (Term f (a3 :: a1 :: nil)). refine (context_in _ _ _ _ f nil (a1 :: nil)). apply IH; trivial; apply flatten_cf_cf with f; ...	Lemma	commutativity	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "Term", "ac", "arity", "canonical_form", "comm", "context_in", "flatten", "flatten_cf_cf", "permut", "rewrite", "size", "t1", "t2", "trans_clos_is_trans", "well_formed" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
associativity : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> si...	Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Sa3 Wa3 Sa4 Wa4 k1 k2 k4. destruct k1 as [ \| h1 k1]; [left; trivial \| right]; destruct k2 as [ \| h2 k2]; [left; trivial \| right]; destruct k4 as [ \| h4 k4]; [left; trivial \| right]; intros P1 P2 P3 P4; generalize (split_cf Af _ _ Wa1 P1); intros [Hk1 \| [Hk2 \| [t1 [t2 ...	Lemma	associativity	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "H1", "Term", "ac", "arity", "canonical_form", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h1", "h2", "l_assoc", "permut", "permut_refl", "quick_permut", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "th_sym", "t...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
swap_left : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> size a...	Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Sa3 Wa3 Sa4 Wa4 k2 k3 k4. destruct k2 as [ \| h2 k2]; [left; trivial \| right]; destruct k3 as [ \| h3 k3]; [left; trivial \| right]; destruct k4 as [ \| h4 k4]; [left; trivial \| right]; intros P1 P2 P3 P4; generalize (split_cf Af _ _ Wa2 P2); intros [Hk1 \| [Hk2 \| [t3 [t4 ...	Lemma	swap_left	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "Term", "ac", "arity", "canonical_form", "comm", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h2", "l_assoc", "permut", "quick_permut", "r_assoc", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "t3", "th_sym", "tra...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
swap_right : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> size ...	Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Sa3 Wa3 Sa4 Wa4 k1 k2 k3. destruct k1 as [ \| h1 k1]; [left; trivial \| right]; destruct k2 as [ \| h2 k2]; [left; trivial \| right]; destruct k3 as [ \| h3 k3]; [left; trivial \| right]; intros P1 P2 P3 P4; generalize (split_cf Af _ _ Wa1 P1); intros [Hk1 \| [Hk2 \| [t1 [t2...	Lemma	swap_right	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "H1", "Term", "ac", "arity", "canonical_form", "comm", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h1", "h2", "l_assoc", "permut", "quick_permut", "r_assoc", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "t3", ...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
middle_commutativity : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 ...	Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Sa3 Wa3 Sa4 Wa4 k1 k2 k3 k4. destruct k1 as [ \| h1 k1]; [left; trivial \| right]; destruct k2 as [ \| h2 k2]; [left; trivial \| right]; destruct k3 as [ \| h3 k3]; [left; trivial \| right]; destruct k4 as [ \| h4 k4]; [left; trivial \| right]; intros P1 P2 P3 P4; generaliz...	Lemma	middle_commutativity	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "H1", "Term", "ac", "arity", "canonical_form", "comm", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h1", "h2", "l_assoc", "permut", "quick_permut", "r_assoc", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "t3", ...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
length_morph : Proper (permut ==> eq) (length (A:=term)).	Proof. intros a b ab. eapply list_permut.permut_length. apply ab. Qed.	Instance	length_morph	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "eq", "permut", "permut_length", "term" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
cf_eq_ac : forall t1 t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2.	Proof. intro t1; pattern t1; apply term_rec2; clear t1; induction n as [ \| n]. intros t1 St1; absurd (1 <= 0); auto with arith; apply Nat.le_trans with (size t1); trivial; apply size_ge_one. intros [v1 \| f1 l1] St1 [v2 \| f l2] Wt1 Wt2 H; [simpl in H; rewrite H; unfold ac; apply th_refl \| discriminate \| discriminate \| ...	Theorem	cf_eq_ac	Coccinelle.ac_matching	Coccinelle/ac_matching/cf_eq_ac.v	[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]	[ "H1", "In", "Term", "ac", "ac_one_step_at_top", "ac_syntactic", "associativity", "canonical_form", "comm", "commutativity", "context_in", "destruct_arity", "eq", "eq_bool", "eq_bool_ok", "eq_refl", "f1", "flatten", "flatten_app", "general_context", "h1", "h2", "le_trans",...		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
partly_solved_term : Type	:= mk_pst { head_symb : symbol; new_var : variable; closed_term : term }.	Record	partly_solved_term	Coccinelle.ac_matching	Coccinelle/ac_matching/matching.v	[ "Stdlib", "Arith", "List", "CoLoR", "more_list", "list_sort", "term_spec", "ac", "cf_eq_ac", "Cf_eq_ac.Ac", "Sort", "EqTh", "EqTh.T", "F", "LPermut", "Cf_eq_ac1", "Ac", "T", "X" ]	[ "symbol", "term", "variable" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb
matching_problem : Type	:= mk_pb { existential_vars : list variable; unsolved_part : list (term * term); solved_part : substitution; partly_solved_part : list (variable * partly_solved_term) }.	Record	matching_problem	Coccinelle.ac_matching	Coccinelle/ac_matching/matching.v	[ "Stdlib", "Arith", "List", "CoLoR", "more_list", "list_sort", "term_spec", "ac", "cf_eq_ac", "Cf_eq_ac.Ac", "Sort", "EqTh", "EqTh.T", "F", "LPermut", "Cf_eq_ac1", "Ac", "T", "X" ]	[ "partly_solved_term", "substitution", "term", "variable" ]		https://github.com/fblanqui/color	9e82957ee5a398834098da7378735a7c94e2f2cb

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...

Inductive

ac_one_step_at_top

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "symbol", "t1", "t2", "t3", "term" ]

** Definition of AC.

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac

:= th_eq ac_one_step_at_top.

Definition

ac

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_one_step_at_top", "th_eq" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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

flatten

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "eq_bool", "symbol", "term" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

canonical_form (t : term) : term

Fixpoint

canonical_form

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "arity", "flatten", "map", "quicksort", "term" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf (t:term) : Prop

Fixpoint

well_formed_cf

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "is_sorted", "term" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

build (f : symbol) l

:= match l with | t :: nil => t | _ => Term f (quicksort l) end.

Definition

build

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "quicksort", "symbol" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_subst sigma

:= forall v, match find X.eq_bool v sigma with | None => True | Some t => well_formed_cf t end.

Definition

well_formed_cf_subst

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "eq_bool", "find", "well_formed_cf" ]

Definition well_formed_cf_subst

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

apply_cf_subst (sigma : substitution) (t : term) {struct t} : term

Fixpoint

apply_cf_subst

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "arity", "eq_bool", "find", "flatten", "map", "quicksort", "substitution", "term" ]

Definition apply_cf_subst

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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 ...

Fixpoint

ac_size

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "arity", "lt", "term" ]

Definition ac_size

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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)).

Parameter

l_assoc

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "ac", "arity", "t1", "t2", "t3" ]

Theorem l_assoc

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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

r_assoc

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "ac", "arity", "t1", "t2", "t3" ]

Theorem r_assoc

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

comm : forall f t1 t2, arity f = C \/ arity f = AC -> ac (Term f (t1 :: t2 :: nil)) (Term f (t2 :: t1 :: nil)).

Parameter

comm

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "ac", "arity", "t1", "t2" ]

Theorem comm

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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

ac_top_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "ac", "f1", "f2", "t1", "t2" ]

Theorem ac_top_eq

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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 /\ ...

Parameter

well_formed_cf_unfold

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "is_sorted", "well_formed_cf" ]

Theorem well_formed_cf_unfold

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_subterms : forall f l, well_formed_cf (Term f l) -> (forall t, In t l -> well_formed_cf t).

Parameter

well_formed_cf_subterms

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "well_formed_cf" ]

Theorem well_formed_cf_subterms

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_length : forall f l, arity f = AC -> well_formed_cf (Term f l) -> 2 <= length l.

Parameter

well_formed_cf_length

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "well_formed_cf" ]

Theorem well_formed_cf_length

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_sorted : forall f l, arity f = AC -> well_formed_cf (Term f l) -> is_sorted l.

Parameter

well_formed_cf_sorted

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "is_sorted", "well_formed_cf" ]

Theorem well_formed_cf_sorted

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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

well_formed_cf_alien

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "well_formed_cf" ]

Theorem well_formed_cf_alien

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_app : forall f l1 l2, flatten f (l1 ++ l2) = (flatten f l1) ++ (flatten f l2).

Parameter

flatten_app

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "flatten" ]

Theorem flatten_app

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

list_permut_flatten : forall f l1 l2, permut l1 l2 -> permut (flatten f l1) (flatten f l2).

Parameter

list_permut_flatten

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "flatten", "permut" ]

Theorem list_permut_flatten

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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

length_flatten_bis

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "arity", "flatten", "well_formed_cf" ]

Theorem length_flatten_bis

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "arity", "flatten", "permut", "t1", "t2", "well_formed_cf" ]

Theorem flatten_cf

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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

flatten_cf_cf

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "arity", "canonical_form", "flatten", "permut", "t1", "t2", "well_formed" ]

Theorem flatten_cf_cf

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

build_eq_Term : forall f l, 2 <= length l -> build f l = Term f (quicksort l).

Parameter

build_eq_Term

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "build", "quicksort" ]

Theorem build_eq_Term

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_build : forall f l, arity f = AC -> 1 <= length l -> (forall t, In t l -> well_formed_cf t) -> (forall t, In t l -> match t with | Var _ => True | Term g _ => f <> g end) -> well_formed_cf (build f l).

Parameter

well_formed_cf_build

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "build", "well_formed_cf" ]

Theorem well_formed_cf_build

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> well_formed_cf (build f l).

Parameter

well_formed_cf_build_cons

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "build", "well_formed_cf" ]

Theorem well_formed_cf_build_cons

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> well_formed_cf (build f (l1 ++ l2)).

Parameter

well_formed_cf_build_inside

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "build", "well_formed_cf" ]

Theorem well_formed_cf_build_inside

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_build : forall f l, arity f = AC -> (forall t, In t l -> match t with | Var _ => True | Term g _ => f <> g end) -> permut (flatten f ((build f l) :: nil)) l.

Parameter

flatten_build

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "build", "flatten", "permut" ]

Theorem flatten_build

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> permut (flatten f ((build f l) :: nil)) l.

Parameter

flatten_build_cons

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "build", "flatten", "permut", "well_formed_cf" ]

Theorem flatten_build_cons

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> permut (flatten f ((build f (l1 ++ l2)) :: nil)) (l1 ++ l2).

Parameter

flatten_build_inside

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "build", "flatten", "permut", "well_formed_cf" ]

Theorem flatten_build_inside

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_apply_cf_subst : forall sigma f l, arity f = AC -> permut (flatten f (map (apply_cf_subst sigma) l)) (flatten f (apply_cf_subst sigma (build f l) :: nil)).

Parameter

flatten_apply_cf_subst

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "apply_cf_subst", "arity", "build", "flatten", "map", "permut" ]

Theorem flatten_apply_cf_subst

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_apply_subst : forall sigma, well_formed_cf_subst sigma -> forall t, well_formed_cf t -> well_formed_cf (apply_cf_subst sigma t).

Parameter

well_formed_cf_apply_subst

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "apply_cf_subst", "well_formed_cf", "well_formed_cf_subst" ]

Theorem well_formed_cf_apply_subst

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

length_flatten_ter : forall f sigma, arity f = AC -> well_formed_cf_subst sigma -> forall l, (forall t, In t l -> well_formed_cf t) -> length l <= length (flatten f (map (apply_cf_subst sigma) l)).

Parameter

length_flatten_ter

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "apply_cf_subst", "arity", "flatten", "map", "well_formed_cf", "well_formed_cf_subst" ]

Theorem length_flatten_ter

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_cf_eq : forall t1 t2, ac t1 t2 -> canonical_form t1 = canonical_form t2.

Parameter

ac_cf_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac", "canonical_form", "t1", "t2" ]

Theorem ac_cf_eq

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_size_eq : forall t1 t2, ac t1 t2 -> size t1 = size t2.

Parameter

ac_size_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac", "size", "t1", "t2" ]

Theorem ac_size_eq

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

Parameter

ac_size_unfold

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "ac_size", "arity", "list_size" ]

Theorem ac_size_unfold

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

size_size_aux3 : forall f t, arity f = AC -> well_formed t -> 1 <= length (A:=term) (flatten f (canonical_form t :: nil)).

Parameter

size_size_aux3

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "arity", "canonical_form", "flatten", "term", "well_formed" ]

Theorem size_size_aux3

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

size_size : forall t, well_formed t -> size t = ac_size (canonical_form t).

Parameter

size_size

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_size", "canonical_form", "size", "well_formed" ]

Theorem size_size

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_size_ge_one : forall t, well_formed_cf t -> 1 <= ac_size t.

Parameter

ac_size_ge_one

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_size", "well_formed_cf" ]

Theorem ac_size_ge_one

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

no_need_of_instance : forall t1 t2, axiom (sym_refl ac_one_step_at_top) t1 t2 -> (sym_refl ac_one_step_at_top) t1 t2.

Proof. unfold sym_refl; intros t1 t2 H. inversion_clear H as [ u1 u2 sigma H']; destruct H' as [H' | [H' | H']]. inversion_clear H'. left; simpl; apply a_axiom; trivial. left; simpl; apply c_axiom; trivial. inversion_clear H'. right; left; simpl; apply a_axiom; trivial. right; left; simpl; apply c_axiom; trivial. righ...

Lemma

no_need_of_instance

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_one_step_at_top", "axiom", "subst", "sym_refl", "t1", "t2" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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)).

Proof. intros f t1 t2 t3 Af. unfold ac, th_eq. do 2 left. rewrite <- (empty_subst_is_id (Term f (Term f (t1 :: t2 :: nil) :: t3 :: nil))). rewrite <- (empty_subst_is_id (Term f (t1 :: Term f (t2 :: t3 :: nil) :: nil))). apply instance. left; apply a_axiom; trivial. Qed.

Lemma

l_assoc

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "ac", "arity", "empty_subst_is_id", "rewrite", "t1", "t2", "t3", "th_eq" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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)).

Proof. intros f t1 t2 t3 Af. unfold ac, th_eq. do 2 left. rewrite <- (empty_subst_is_id (Term f (Term f (t1 :: t2 :: nil) :: t3 :: nil))). rewrite <- (empty_subst_is_id (Term f (t1 :: Term f (t2 :: t3 :: nil) :: nil))). apply instance. right; left; apply a_axiom; trivial. Qed.

Lemma

r_assoc

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "ac", "arity", "empty_subst_is_id", "rewrite", "t1", "t2", "t3", "th_eq" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

comm : forall f t1 t2, arity f = C \/ arity f = AC -> ac (Term f (t1 :: t2 :: nil)) (Term f (t2 :: t1 :: nil)).

Proof. intros f t1 t2 Af. unfold ac, th_eq. do 2 left. rewrite <- (empty_subst_is_id (Term f (t1 :: t2 :: nil))). rewrite <- (empty_subst_is_id (Term f (t2 :: t1 :: nil))). apply instance. left; apply c_axiom; trivial. Qed.

Lemma

comm

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "ac", "arity", "empty_subst_is_id", "rewrite", "t1", "t2", "th_eq" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_one_step_at_top_top_eq : forall t1 t2, (sym_refl ac_one_step_at_top) 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.

Proof. unfold sym_refl; intros t1 t2 [Ac | [Ac | t1_eq_t2]]. inversion_clear Ac; simpl; trivial. inversion_clear Ac; simpl; trivial. subst t2; destruct t1; trivial. Qed.

Lemma

ac_one_step_at_top_top_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "ac_one_step_at_top", "f1", "f2", "subst", "sym_refl", "t1", "t2" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_one_step_top_eq : forall t1 t2 : term, one_step (sym_refl ac_one_step_at_top) 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.

Proof. intros t1 t2 Ac; inversion Ac; trivial. apply ac_one_step_at_top_top_eq; apply no_need_of_instance; trivial. Qed.

Lemma

ac_one_step_top_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "ac_one_step_at_top", "ac_one_step_at_top_top_eq", "f1", "f2", "no_need_of_instance", "one_step", "sym_refl", "t1", "t2", "term" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_top_eq : forall t1 t2 : term, 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.

Proof. intros t1 t2 Ac; induction Ac. (* one step *) inversion H as [ H1 H2 H' H4 H5 | f l1 l2 H' H1 H2]; subst; trivial. apply ac_one_step_top_eq; assumption. (* th_trans *) generalize (ac_one_step_top_eq H). destruct x as [ v1 | f1 l1 ]; destruct y as [ v2 | f2 l2 ]; destruct z as [ v3 | f3 l3 ]; trivial. intro; ...

Lemma

ac_top_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "H1", "Term", "Var", "ac", "ac_one_step_top_eq", "f1", "f2", "subst", "t1", "t2", "term" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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 /\ ...

Proof. intro t; destruct t as [v | f l]; simpl; trivial; intros [Wl Ll]; split; trivial. clear Ll; induction l as [ | t l]. contradiction. intros u [u_eq_t | In_u]; subst; intuition. Qed.

Lemma

well_formed_cf_unfold

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "is_sorted", "split", "subst", "well_formed_cf" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_subterms : forall f l, well_formed_cf (Term f l) -> (forall t, In t l -> well_formed_cf t).

Proof. intros f l W; elim (well_formed_cf_unfold _ W); trivial. Qed.

Lemma

well_formed_cf_subterms

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "well_formed_cf", "well_formed_cf_unfold" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_length : forall f l, arity f = AC -> well_formed_cf (Term f l) -> 2 <= length l.

Proof. intros f l Af [_ Ll]; rewrite Af in Ll; intuition. Qed.

Lemma

well_formed_cf_length

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "rewrite", "well_formed_cf" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_sorted : forall f l, arity f = AC -> well_formed_cf (Term f l) -> is_sorted l.

Proof. intros f l Af [_ Ll]; rewrite Af in Ll; intuition. Qed.

Lemma

well_formed_cf_sorted

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "is_sorted", "rewrite", "well_formed_cf" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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).

Proof. intros f l Af [_ Ll]; rewrite Af in Ll; intuition. Qed.

Lemma

well_formed_cf_alien

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "rewrite", "well_formed_cf" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_fold : forall 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 /\ (fo...

Proof. intro t; destruct t as [v | f l]; trivial. intros [Wl Hl]; split; trivial; clear Hl; induction l as [ | t l]; intuition auto with *; apply IHl; intros; apply Wl; right; trivial. Qed.

Lemma

well_formed_cf_fold

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "is_sorted", "split", "well_formed_cf" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_app : forall f l1 l2, flatten f (l1 ++ l2) = (flatten f l1) ++ (flatten f l2).

Proof. intros f l1; induction l1 as [ | [v1 | g1 ll1]]; simpl; trivial. intros l2; rewrite IHl1; trivial. intros l2; rewrite IHl1; generalize (F.Symb.eq_bool_ok f g1); case (F.Symb.eq_bool f g1); [intro f_eq_g1 | intro f_diff_g1]. rewrite app_ass; trivial. simpl; trivial. Qed.

Lemma

flatten_app

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "eq_bool", "eq_bool_ok", "flatten", "rewrite" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

list_permut_flatten : forall f l1 l2, permut l1 l2 -> permut (flatten f l1) (flatten f l2).

Proof. intros f l1; induction l1 as [ | t1 l1]; intros l2 P. rewrite (permut_nil (permut_sym P)); apply permut_refl. assert (In_t1 : mem (@eq _) t1 l2). rewrite <- P; left; reflexivity. destruct (mem_split_set _ _ eq_bool_ok _ _ In_t1) as [t1' [l2' [l2'' [t1_eq_t1' [H _]]]]]; simpl in t1_eq_t1'; simpl in H; subst l2; ...

Lemma

list_permut_flatten

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "dummy", "eq", "eq_bool", "eq_bool_ok", "f1", "flatten", "flatten_app", "list_permut_app_app", "mem", "mem_split_set", "permut", "permut_app1", "permut_app2", "permut_cons_inside", "permut_nil", "permut_refl", "permut_sym", "rewrite", "subst", "t1" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_is_well_formed_cf : forall cf, well_formed_cf cf -> exists t, well_formed t /\ cf = canonical_form t.

Proof. intros cf Wcf; generalize (well_formed_cf_unfold _ Wcf); pattern cf; apply term_rec3; clear cf Wcf. intros v _; exists (Var v); unfold well_formed; simpl; split; trivial. intros f l Hrec [Wl Ll]; assert (Wl' : forall u, In u l -> exists s, well_formed s /\ u = canonical_form s). intros u In_u; apply Hrec; trivi...

Lemma

well_formed_cf_is_well_formed_cf

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "canonical_form", "destruct_arity", "eq_bool", "eq_bool_ok", "ge", "list_permut_app_app", "map", "permut_cons_inside", "permut_refl", "permut_sym", "quick_permut", "quick_permut_bis", "quick_sorted", "rewrite", "sort_is_unique", "sorted_tl_sorted", "split",...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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).

Proof. intros f Af l Wl; induction l as [ | t l]. simpl; auto with arith. simpl; destruct t as [v | g ll]. simpl; apply le_n_S; apply IHl; intros; apply Wl; right; trivial. generalize (F.Symb.eq_bool_ok f g); case (F.Symb.eq_bool f g); [intro f_eq_g | intro f_diff_g]. rewrite length_app; replace (S (length l)) with (1 ...

Lemma

length_flatten_bis

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "arity", "eq_bool", "eq_bool_ok", "flatten", "length_app", "replace", "rewrite", "subst", "well_formed_cf" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

length_flatten : forall f, forall l, arity f = AC -> (forall u, In u l -> well_formed u) -> length l <= length (flatten f (map canonical_form l)).

Proof. intros f l; pattern l; apply (list_rec3 size); clear l; induction n; destruct l as [ | t l]. simpl; trivial. simpl; intro S_l; absurd (1 <= 0); auto with arith; apply Nat.le_trans with (size t + list_size size l); trivial. apply Nat.le_trans with (2 := Nat.le_add_r _ _), size_ge_one. simpl; trivial. intros Sl A...

Lemma

length_flatten

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "arity", "canonical_form", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "le_trans", "length_app", "length_quicksort", "list_rec3", "list_size", "map", "map_app", "replace", "rewrite", "size", "size_ge_one", "size_unfold", "subst", "well_formed", "well_...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_is_well_formed_cf_conv : forall cf, (exists t, well_formed t /\ cf = canonical_form t) -> well_formed_cf cf.

Proof. intros cf [t [Wt Ct]]; subst; generalize Wt; clear Wt. pattern t; apply term_rec2; clear t; induction n as [ | n ]; intros t St Wt. absurd (1 <= 0); auto with arith; apply Nat.le_trans with (size t); trivial; apply size_ge_one. apply well_formed_cf_fold; destruct t as [ v | f l ]; simpl; trivial. generalize (we...

Lemma

well_formed_cf_is_well_formed_cf_conv

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "canonical_form", "destruct_arity", "eq_bool", "eq_bool_ok", "flatten_app", "ge", "in_quick_in", "le_trans", "length_flatten", "length_quicksort", "list_rec3", "list_size", "map", "quick_sorted", "rewrite", "size", "size_ge_one", "size_unfold", "split", "subst", "t1",...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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.

Proof. intros f t1 t2 Af Wt1 Wt2; destruct t1 as [v1 | f1 l1]; destruct t2 as [v2 | f2 l2]. (* t1 = Var v1; t2 = Var v2 *) simpl; intros; apply permut_length_1; trivial. (* t1 = Var v1; t2 = Term f2 l2 *) simpl; generalize (F.Symb.eq_bool_ok f f2); case (F.Symb.eq_bool f f2); [intros f_eq_f2; subst f2 | intros f_diff_f...

Lemma

flatten_cf

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "eq_bool", "eq_bool_ok", "f1", "f2", "flatten", "permut", "permut_length", "permut_length_1", "rewrite", "sort_is_unique", "subst", "t1", "t2", "well_formed_cf", "well_formed_cf_length", "well_formed_cf_sorted" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

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.

Proof. intros f t1 t2 Af Wt1 Wt2 P; apply flatten_cf with f; trivial. apply well_formed_cf_is_well_formed_cf_conv; exists t1; split; trivial. apply well_formed_cf_is_well_formed_cf_conv; exists t2; split; trivial. Qed.

Lemma

flatten_cf_cf

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "arity", "canonical_form", "flatten", "flatten_cf", "permut", "split", "t1", "t2", "well_formed", "well_formed_cf_is_well_formed_cf_conv" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

build_eq_Term : forall f l, 2 <= length l -> build f l = Term f (quicksort l).

Proof. intros f l Ll; destruct l as [ | u [ | v l]]; simpl; trivial. simpl in Ll; absurd (2 <= 1); auto with arith. Qed.

Lemma

build_eq_Term

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "build", "quicksort" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_build : forall f l, arity f = AC -> 1 <= length l -> (forall t, In t l -> well_formed_cf t) -> (forall t, In t l -> match t with | Var _ => True | Term g _ => f <> g end) -> well_formed_cf (build f l).

Proof. intros f l Af Ll Wl Al; destruct l as [ | t1 [ | t2 l]]. simpl in Ll; absurd (1 <= 0); trivial; auto with arith. apply Wl; left; trivial. rewrite build_eq_Term; [idtac | simpl; auto with arith]. apply well_formed_cf_fold; split. intros u In_u; rewrite <- in_quick_in in In_u. apply Wl; trivial. rewrite Af; split;...

Lemma

well_formed_cf_build

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "build", "build_eq_Term", "in_quick_in", "length_quicksort", "quick_sorted", "rewrite", "split", "t1", "t2", "well_formed_cf", "well_formed_cf_fold" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> well_formed_cf (build f l).

Proof. intros f t l Af W; apply well_formed_cf_build; trivial. apply le_S_n; replace (S (length l)) with (length (t :: l)); trivial; apply well_formed_cf_length with f; trivial. intros; apply (well_formed_cf_subterms W); right; trivial. intros; apply (well_formed_cf_alien Af W); right; trivial. Qed.

Lemma

well_formed_cf_build_cons

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "build", "replace", "well_formed_cf", "well_formed_cf_alien", "well_formed_cf_build", "well_formed_cf_length", "well_formed_cf_subterms" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> well_formed_cf (build f (l1 ++ l2)).

Proof. intros f t l1 l2 Af W; assert (H : forall u, In u (l1 ++ l2) -> In u (l1 ++ t :: l2)). intros u In_u; elim (in_app_or _ _ _ In_u); clear In_u; intro In_u; apply in_or_app; [ left | right; right ]; trivial. apply well_formed_cf_build; trivial. apply le_S_n; replace (S (length (l1 ++ l2))) with (length (l1 ++ t ...

Lemma

well_formed_cf_build_inside

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "add_comm", "arity", "build", "length_app", "replace", "rewrite", "well_formed_cf", "well_formed_cf_alien", "well_formed_cf_build", "well_formed_cf_length", "well_formed_cf_subterms" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_build : forall f l, arity f = AC -> (forall t, In t l -> match t with | Var _ => True | Term g _ => f <> g end) -> permut (flatten f ((build f l) :: nil)) l.

Proof. intros f [ | t1 [ | t2 l]] Af Al. simpl; generalize (F.Symb.eq_bool_ok f f); case (F.Symb.eq_bool f f); [intros _; apply Pnil | intro f_diff_f; apply False_rect; apply f_diff_f; reflexivity]. assert (Al_t1 := Al _ (or_introl _ (eq_refl _))). simpl; destruct t1 as [ | g]; [auto | idtac]. generalize (F.Symb.eq_boo...

Lemma

flatten_build

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "Term", "Var", "arity", "build", "eq_bool", "eq_bool_ok", "eq_refl", "flatten", "permut", "quick_permut_bis", "rewrite", "t1", "t2" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_build_cons : forall f t l, arity f = AC -> well_formed_cf (Term f (t :: l)) -> permut (flatten f ((build f l) :: nil)) l.

Proof. intros f t l Af W; apply flatten_build; trivial. intros; apply (well_formed_cf_alien Af W); right; trivial. Qed.

Lemma

flatten_build_cons

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "build", "flatten", "flatten_build", "permut", "well_formed_cf", "well_formed_cf_alien" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_build_inside : forall f t l1 l2, arity f = AC -> well_formed_cf (Term f (l1 ++ t :: l2)) -> permut (flatten f ((build f (l1 ++ l2)) :: nil)) (l1 ++ l2).

Proof. intros f t l1 l2 Af W; apply flatten_build; trivial. intros u In_u; apply (well_formed_cf_alien Af W); elim (in_app_or _ _ _ In_u); clear In_u; intro In_u; apply in_or_app; [left | right; right]; trivial. Qed.

Lemma

flatten_build_inside

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "arity", "build", "flatten", "flatten_build", "permut", "well_formed_cf", "well_formed_cf_alien" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

is_subst_canonical_form sigma sigma_cf

:= forall v, match find X.eq_bool v sigma with | None => find X.eq_bool v sigma_cf = None | Some v_sigma => find X.eq_bool v sigma_cf = Some (canonical_form v_sigma) end.

Definition

is_subst_canonical_form

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "canonical_form", "eq_bool", "find" ]

** Substitutions modulo AC

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

is_subst_cf_is_subst_cf : forall sigma, is_subst_canonical_form sigma (map_subst (fun _ t => canonical_form t) sigma).

Proof. intros sigma; unfold is_subst_canonical_form. intro v; rewrite subst_comp_is_subst_comp_aux1. destruct (find X.eq_bool v sigma); trivial. Qed.

Lemma

is_subst_cf_is_subst_cf

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "canonical_form", "eq_bool", "find", "is_subst_canonical_form", "map_subst", "rewrite", "subst_comp_is_subst_comp_aux1" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_subst_is_well_formed_cf_subst_aux : forall sigma, well_formed_cf_subst sigma -> (forall v, nb_occ X.eq_bool v sigma <= 1) -> exists sigma', well_formed_subst sigma' /\ is_subst_canonical_form sigma' sigma.

Proof. unfold well_formed_cf_subst, is_subst_canonical_form; intros sigma Wsigma Nb_occ_sigma; induction sigma as [ | [v1 t1] sigma]. exists (nil : substitution); split; trivial; intro v; simpl; trivial. elim IHsigma. intros sigma' [Wsigma' Hsigma']; assert (Wt1 : well_formed_cf t1). generalize (Wsigma v1); simpl; ge...

Lemma

well_formed_cf_subst_is_well_formed_cf_subst_aux

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "eq_bool", "eq_bool_ok", "eq_refl", "find", "is_subst_canonical_form", "nb_occ", "some_nb_occ_Sn", "split", "subst", "substitution", "t1", "well_formed_cf", "well_formed_cf_is_well_formed_cf", "well_formed_cf_subst", "well_formed_subst" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_subst_is_well_formed_cf_subst : forall sigma, well_formed_cf_subst sigma -> exists sigma', well_formed_subst sigma' /\ is_subst_canonical_form sigma' sigma.

Proof. assert (E : equivalence _ (@eq variable)). repeat split. intros t1 t2 t3 H1 H2; subst; reflexivity. intros t1 t2 H; subst; reflexivity. intros sigma Wsigma; elim (reduce_assoc_list _ X.eq_bool_ok E sigma); intros sigma1 [Nb_occ H]; elim (well_formed_cf_subst_is_well_formed_cf_subst_aux (sigma := sigma1)); trivia...

Lemma

well_formed_cf_subst_is_well_formed_cf_subst

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "H1", "eq", "eq_bool_ok", "is_subst_canonical_form", "reduce_assoc_list", "rewrite", "split", "subst", "t1", "t2", "t3", "variable", "well_formed_cf_subst", "well_formed_cf_subst_is_well_formed_cf_subst_aux", "well_formed_subst" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

flatten_apply_cf_subst : forall sigma f l, arity f = AC -> permut (flatten f (map (apply_cf_subst sigma) l)) (flatten f (apply_cf_subst sigma (build f l) :: nil)).

Proof. intros sigma f l Af; induction l as [ | t1 l]. simpl; generalize (F.Symb.eq_bool_ok f f); case (F.Symb.eq_bool f f); [intros _ | intro f_diff_f; apply False_rect; apply f_diff_f; reflexivity]. rewrite Af; repeat rewrite quicksort_equation; simpl; auto. destruct l as [ | t2 l]; auto. simpl; rewrite Af; generaliz...

Lemma

flatten_apply_cf_subst

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "apply_cf_subst", "arity", "build", "eq", "eq_bool", "eq_bool_ok", "flatten", "list_permut_flatten", "map", "permut", "permut_map", "permut_sym", "quick_permut", "quick_permut_bis", "quicksort_equation", "rewrite", "subst", "t1", "t2", "term" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

apply_cf_subst_is_sound : forall sigma sigma_cf, is_subst_canonical_form sigma sigma_cf -> forall t, apply_cf_subst sigma_cf (canonical_form t) = canonical_form (apply_subst sigma t).

Proof. intros sigma sigma_cf H t; pattern t; apply term_rec3; clear t. intro v; generalize (H v); simpl; destruct (find X.eq_bool v sigma) as [t | ]; intro H_v; rewrite H_v; trivial. intros f l IH; assert (IHl : map (apply_cf_subst sigma_cf) (map canonical_form l) = map canonical_form (map (apply_subst sigma) l))...

Theorem

apply_cf_subst_is_sound

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "apply_cf_subst", "apply_subst", "canonical_form", "destruct_arity", "eq", "eq_bool", "eq_bool_ok", "find", "flatten", "flatten_app", "is_subst_canonical_form", "list_permut_flatten", "map", "map_app", "permut_app1", "permut_app2", "permut_map", "permut_sym", "quick_per...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

well_formed_cf_apply_subst : forall sigma, well_formed_cf_subst sigma -> forall t, well_formed_cf t -> well_formed_cf (apply_cf_subst sigma t).

Proof. intros sigma Wsigma t Wt; elim (well_formed_cf_is_well_formed_cf _ Wt); intros u [Wu Hu]; subst; elim (well_formed_cf_subst_is_well_formed_cf_subst Wsigma); intros tau [Wtau Htau]; rewrite apply_cf_subst_is_sound with tau sigma u; trivial; apply well_formed_cf_is_well_formed_cf_conv; exists (apply_subst tau u);...

Theorem

well_formed_cf_apply_subst

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "apply_cf_subst", "apply_cf_subst_is_sound", "apply_subst", "rewrite", "split", "subst", "tau", "well_formed_apply_subst", "well_formed_cf", "well_formed_cf_is_well_formed_cf", "well_formed_cf_is_well_formed_cf_conv", "well_formed_cf_subst", "well_formed_cf_subst_is_well_formed_cf_subst" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

length_flatten_ter : forall f sigma, arity f = AC -> well_formed_cf_subst sigma -> forall l, (forall t, In t l -> well_formed_cf t) -> length l <= length (flatten f (map (apply_cf_subst sigma) l)).

Proof. intros f sigma Af Wsigma l Wl; induction l as [ | t l]; trivial. replace (t::l) with ((t::nil) ++ l); trivial; rewrite map_app; rewrite flatten_app; do 2 rewrite length_app; apply Nat.add_le_mono. assert (Wtsigma : well_formed_cf (apply_cf_subst sigma t)). apply well_formed_cf_apply_subst; trivial; apply Wl; le...

Lemma

length_flatten_ter

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "In", "apply_cf_subst", "arity", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "le_trans", "length_app", "map", "map_app", "replace", "rewrite", "subst", "well_formed_cf", "well_formed_cf_apply_subst", "well_formed_cf_length", "well_formed_cf_subst" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_one_step_at_top_cf_eq : forall t1 t2, ac_one_step_at_top t1 t2 -> canonical_form t1 = canonical_form t2.

Proof. assert (P12 : forall t1 t2, permut (t1 :: t2 :: nil) (t2 :: t1 :: nil)). intros t1 t2; replace (t1 :: t2 :: nil) with ((t1 :: nil) ++ (t2 :: nil)); trivial; replace (t2 :: t1 :: nil) with ((t2 :: nil) ++ (t1 :: nil)); trivial; apply list_permut_app_app. assert (Ccase : forall f, arity f = C \/ arity f = AC -> ...

Lemma

ac_one_step_at_top_cf_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "ac_one_step_at_top", "arity", "canonical_form", "eq_bool", "eq_bool_ok", "f1", "f2", "flatten", "flatten_app", "list_permut_app_app", "permut", "permut_app1", "permut_app2", "permut_cons", "permut_cons_inside", "permut_refl", "permut_sym", "quick_permut", "quick_permut...

** Canonical forms and equality modulo AC

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

sym_refl_ac_one_step_at_top_cf_eq : forall t1 t2, (sym_refl ac_one_step_at_top) t1 t2 -> canonical_form t1 = canonical_form t2.

Proof. intros t1 t2; unfold sym_refl; intros [H | [H | H]]. apply ac_one_step_at_top_cf_eq; trivial. apply sym_eq; apply ac_one_step_at_top_cf_eq; trivial. subst; trivial. Qed.

Lemma

sym_refl_ac_one_step_at_top_cf_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_one_step_at_top", "ac_one_step_at_top_cf_eq", "canonical_form", "subst", "sym_refl", "t1", "t2" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_one_step_cf_eq : forall t1 t2, one_step (sym_refl ac_one_step_at_top) t1 t2 -> canonical_form t1 = canonical_form t2.

Proof. intros t1; pattern t1; apply term_rec3; clear t1. intros v1 t2 H; inversion_clear H as [H1 H2 H' | f l1 l2 H']. apply sym_refl_ac_one_step_at_top_cf_eq; apply no_need_of_instance; trivial. intros f l1 IHl1 t2 H; inversion_clear H as [ H1 H2 Hroot | H1 H2 l2 Hsub]. apply sym_refl_ac_one_step_at_top_cf_eq; apply...

Lemma

ac_one_step_cf_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "H1", "ac_one_step_at_top", "canonical_form", "map", "no_need_of_instance", "one_step", "rewrite", "sym_refl", "sym_refl_ac_one_step_at_top_cf_eq", "t1", "t2", "term_rec3" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_cf_eq : forall t1 t2, ac t1 t2 -> canonical_form t1 = canonical_form t2.

Proof. unfold ac, th_eq; intros t1 t2 H. refine (rwr_inv _ _ _ _ (@trans_eq term) _ _ _ H). clear t1 t2 H; intros t1 t2 H; apply ac_one_step_cf_eq; trivial. Qed.

Theorem

ac_cf_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac", "ac_one_step_cf_eq", "canonical_form", "rwr_inv", "t1", "t2", "term", "th_eq" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_one_step_at_top_size_eq : forall t1 t2, ac_one_step_at_top t1 t2 -> size t1 = size t2.

Proof. intros t1 t2 H; destruct H; simpl; repeat rewrite Nat.add_0_r; auto with arith; rewrite (Nat.add_comm (size t1) (S (size t2 + size t3))); simpl; apply (f_equal (fun n => S (S n))); rewrite <- (Nat.add_assoc (size t1) (size t2) (size t3)); apply Nat.add_comm. Qed.

Lemma

ac_one_step_at_top_size_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_one_step_at_top", "add_comm", "rewrite", "size", "t1", "t2", "t3" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

sym_refl_ac_one_step_at_top_size_eq : forall t1 t2, (sym_refl ac_one_step_at_top) t1 t2 -> size t1 = size t2.

Proof. intros t1 t2; unfold sym_refl; intros [H | [H | H]]. apply ac_one_step_at_top_size_eq; trivial. apply sym_eq; apply ac_one_step_at_top_size_eq; trivial. subst; trivial. Qed.

Lemma

sym_refl_ac_one_step_at_top_size_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_one_step_at_top", "ac_one_step_at_top_size_eq", "size", "subst", "sym_refl", "t1", "t2" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_one_step_size_eq : forall t1 t2, one_step (sym_refl ac_one_step_at_top) t1 t2 -> size t1 = size t2.

Proof. intros t1; pattern t1; apply term_rec3; clear t1. intros v1 t2 H; inversion_clear H as [H1 H2 H' | f l1 l2 H']. apply sym_refl_ac_one_step_at_top_size_eq; apply no_need_of_instance; trivial. intros f l1 IHl1 t2 H; inversion_clear H as [ H1 H2 Hroot | H1 H2 l2 Hsub]. apply sym_refl_ac_one_step_at_top_size_eq; a...

Lemma

ac_one_step_size_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "H1", "ac_one_step_at_top", "list_size", "no_need_of_instance", "one_step", "rewrite", "size", "size_unfold", "sym_refl", "sym_refl_ac_one_step_at_top_size_eq", "t1", "t2", "term_rec3" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_size_eq : forall t1 t2, ac t1 t2 -> size t1 = size t2.

Proof. intros t1 t2 H; refine (rwr_inv _ _ _ _ (@trans_eq nat) _ _ _ H). intros; apply ac_one_step_size_eq; trivial. Qed.

Lemma

ac_size_eq

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac", "ac_one_step_size_eq", "rwr_inv", "size", "t1", "t2" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

Proof. intro t; destruct t as [v | f l]; trivial; simpl. apply (f_equal (fun n => (match arity f with | AC => length l - 1 | C => 1 | Free _ => 1 end) + n)); ...

Lemma

ac_size_unfold

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Term", "Var", "ac_size", "arity", "list_size", "rewrite" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

size_size_aux2 : forall f t, arity f = AC -> well_formed t -> ac_size (canonical_form t) = list_size ac_size (flatten f (canonical_form t :: nil)) + (length (flatten f (canonical_form t :: nil))) - 1.

Proof. intros f t Af Wt; assert (Wu : well_formed_cf (canonical_form t)). apply well_formed_cf_is_well_formed_cf_conv; exists t; split; trivial. generalize (canonical_form t) Wu; clear t Wt Wu; intros u Wu. destruct u as [v | g l]; trivial; simpl. simpl; generalize (F.Symb.eq_bool_ok f g); case (F.Symb.eq_bool f g); [i...

Lemma

size_size_aux2

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_size", "add_comm", "arity", "canonical_form", "eq_bool", "eq_bool_ok", "flatten", "list_size", "list_size_fold", "rewrite", "split", "subst", "well_formed", "well_formed_cf", "well_formed_cf_is_well_formed_cf_conv", "well_formed_cf_length" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

size_size_aux3 : forall f t, arity f = AC -> well_formed t -> 1 <= length (A:=term) (flatten f (canonical_form t :: nil)).

Proof. intros f t Ar Wt; apply (length_flatten (t :: nil) Ar); intros u In_u; elim In_u; clear In_u; intro In_u. subst u; trivial. contradiction. Qed.

Lemma

size_size_aux3

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "arity", "canonical_form", "flatten", "length_flatten", "subst", "term", "well_formed" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

size_size : forall t, well_formed t -> size t = ac_size (canonical_form t).

Proof. intros t; pattern t; apply term_rec3; clear t. intros v _; trivial. intros f l H Wt; destruct (well_formed_unfold Wt) as [Wl Ll]. assert (Hl : forall t, In t l -> size t = ac_size (canonical_form t)). intros; apply H; trivial; apply Wl; trivial. clear H Wt; assert (Hl' : list_size size l = list_size ac_size (map...

Lemma

size_size

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "Dummy", "In", "Term", "ac_size", "add_comm", "canonical_form", "destruct_arity", "eq", "eq_bool", "eq_bool_ok", "eq_refl", "f1", "f2", "flatten", "flatten_app", "length_app", "length_quicksort", "list_size", "list_size_app", "list_size_fold", "map", "permut_size", "plus"...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

ac_size_ge_one : forall t, well_formed_cf t -> 1 <= ac_size t.

Proof. intros t Wt; elim (well_formed_cf_is_well_formed_cf _ Wt); intros u [Wu Hu]; subst; rewrite <- size_size; trivial; apply size_ge_one. Qed.

Lemma

ac_size_ge_one

Coccinelle.ac_matching

Coccinelle/ac_matching/ac.v

[ "Stdlib", "Relations", "List", "Arith", "Morphisms", "CoLoR", "more_list", "list_permut", "list_sort", "term_spec", "term_o", "equational_theory_spec", "equational_theory", "T", "F", "LPermut", "EqTh", "T1", "T1.F" ]

[ "ac_size", "rewrite", "size_ge_one", "size_size", "subst", "well_formed_cf", "well_formed_cf_is_well_formed_cf" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

cf_eq_ac : forall t1 t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2.

Parameter

cf_eq_ac

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "ac", "canonical_form", "t1", "t2", "well_formed" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

split_cf : forall f, arity f = AC -> forall t u1 u2, well_formed t -> permut (flatten f (canonical_form t :: nil)) (u1 ++ u2) -> u1 = nil \/ u2 = nil \/ (exists t1, exists t2, well_formed t1 /\ well_formed t2 /\ ac t (Term f (t1 :: t2 :: nil)) /\ permut (flatten f (canonical_form t1 :: nil)) u1 /\...

Proof. intros f Af t; pattern t; apply term_rec3; clear t. (* t = Var _ *) intros v l1 l2 _ P; simpl in P; destruct l1 as [ | t1 l1]; [left; trivial | idtac]; destruct l2 as [ | t2 l2]; [right; left; trivial | idtac]; generalize (list_permut.permut_length P); rewrite length_app; simpl; rewrite Nat.add_comm; intro; dis...

Lemma

split_cf

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "H1", "Term", "ac", "ac_syntactic", "add_comm", "arity", "canonical_form", "comm", "context_in", "eq", "eq_bool", "eq_bool_ok", "eq_refl", "flatten", "flatten_app", "l_assoc", "length_app", "list_permut_app_app", "list_permut_flatten", "map", "permut", "permut_length", "p...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

syntactic_dec : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> ...

Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Wa3 Wa4 k1 k4 P1 P2 P3 P4. apply trans_clos_is_trans with (Term f (a3 :: a2 :: nil)). refine (context_in _ _ _ _ f nil (a2 :: nil)). apply IH; trivial; apply flatten_cf_cf with f; trivial; rewrite P1; rewrite P3; auto. refine (context_in _ _ _ _ f (a3 :: nil) nil). a...

Lemma

syntactic_dec

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "Term", "ac", "arity", "canonical_form", "context_in", "flatten", "flatten_cf_cf", "permut", "rewrite", "size", "t1", "t2", "trans_clos_is_trans", "well_formed" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

commutativity : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> wel...

Proof. intros n IH f Af a1 a2 a3 a4 Sa1 Wa1 Sa2 Wa2 Wa3 Wa4 k2 k3 P1 P2 P3 P4. apply trans_clos_is_trans with (Term f (a2 :: a1 :: nil)). apply comm; right; trivial. apply trans_clos_is_trans with (Term f (a3 :: a1 :: nil)). refine (context_in _ _ _ _ f nil (a1 :: nil)). apply IH; trivial; apply flatten_cf_cf with f; ...

Lemma

commutativity

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "Term", "ac", "arity", "canonical_form", "comm", "context_in", "flatten", "flatten_cf_cf", "permut", "rewrite", "size", "t1", "t2", "trans_clos_is_trans", "well_formed" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

associativity : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> si...

Lemma

associativity

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "H1", "Term", "ac", "arity", "canonical_form", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h1", "h2", "l_assoc", "permut", "permut_refl", "quick_permut", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "th_sym", "t...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

swap_left : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> size a...

Lemma

swap_left

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "Term", "ac", "arity", "canonical_form", "comm", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h2", "l_assoc", "permut", "quick_permut", "r_assoc", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "t3", "th_sym", "tra...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

swap_right : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 -> size ...

Lemma

swap_right

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "H1", "Term", "ac", "arity", "canonical_form", "comm", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h1", "h2", "l_assoc", "permut", "quick_permut", "r_assoc", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "t3", ...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

middle_commutativity : forall n, (forall t1, size t1 <= n -> forall t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2) -> forall f, arity f = AC -> forall a1 a2 a3 a4, size a1 <= n -> well_formed a1 -> size a2 <= n -> well_formed a2 ...

Lemma

middle_commutativity

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "H1", "Term", "ac", "arity", "canonical_form", "comm", "context_in", "eq_bool", "eq_bool_ok", "flatten", "flatten_app", "flatten_cf_cf", "h1", "h2", "l_assoc", "permut", "quick_permut", "r_assoc", "rewrite", "size", "split", "split_cf", "subst", "t1", "t2", "t3", ...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

length_morph : Proper (permut ==> eq) (length (A:=term)).

Proof. intros a b ab. eapply list_permut.permut_length. apply ab. Qed.

Instance

length_morph

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "eq", "permut", "permut_length", "term" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

cf_eq_ac : forall t1 t2, well_formed t1 -> well_formed t2 -> canonical_form t1 = canonical_form t2 -> ac t1 t2.

Proof. intro t1; pattern t1; apply term_rec2; clear t1; induction n as [ | n]. intros t1 St1; absurd (1 <= 0); auto with arith; apply Nat.le_trans with (size t1); trivial; apply size_ge_one. intros [v1 | f1 l1] St1 [v2 | f l2] Wt1 Wt2 H; [simpl in H; rewrite H; unfold ac; apply th_refl | discriminate | discriminate | ...

Theorem

cf_eq_ac

Coccinelle.ac_matching

Coccinelle/ac_matching/cf_eq_ac.v

[ "Stdlib", "Setoid", "Arith", "List", "Morphisms", "CoLoR", "closure", "more_list", "weaved_relation", "list_sort", "term_spec", "ac", "Ac.EqTh.T", "Ac1", "EqTh", "Sort", "T", "F", "X", "LPermut" ]

[ "H1", "In", "Term", "ac", "ac_one_step_at_top", "ac_syntactic", "associativity", "canonical_form", "comm", "commutativity", "context_in", "destruct_arity", "eq", "eq_bool", "eq_bool_ok", "eq_refl", "f1", "flatten", "flatten_app", "general_context", "h1", "h2", "le_trans",...

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

partly_solved_term : Type

:= mk_pst { head_symb : symbol; new_var : variable; closed_term : term }.

Record

partly_solved_term

Coccinelle.ac_matching

Coccinelle/ac_matching/matching.v

[ "Stdlib", "Arith", "List", "CoLoR", "more_list", "list_sort", "term_spec", "ac", "cf_eq_ac", "Cf_eq_ac.Ac", "Sort", "EqTh", "EqTh.T", "F", "LPermut", "Cf_eq_ac1", "Ac", "T", "X" ]

[ "symbol", "term", "variable" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

matching_problem : Type

:= mk_pb { existential_vars : list variable; unsolved_part : list (term * term); solved_part : substitution; partly_solved_part : list (variable * partly_solved_term) }.

Record

matching_problem

Coccinelle.ac_matching

Coccinelle/ac_matching/matching.v

[ "Stdlib", "Arith", "List", "CoLoR", "more_list", "list_sort", "term_spec", "ac", "cf_eq_ac", "Cf_eq_ac.Ac", "Sort", "EqTh", "EqTh.T", "F", "LPermut", "Cf_eq_ac1", "Ac", "T", "X" ]

[ "partly_solved_term", "substitution", "term", "variable" ]

https://github.com/fblanqui/color

9e82957ee5a398834098da7378735a7c94e2f2cb

Type	Count
Lemma	5,854
Definition	1,597
Instance	593
Notation	553
Parameter	408
Fixpoint	364
Ltac	214
Inductive	171
Hypothesis	85
Let	59
Function	41
Theorem	39
Record	35
Coercion	15
Axiom	4
Class	4
Parameters	4
Scheme	3
Hypotheses	3

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Column	Type	Description
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has none
type	string	Declaration keyword
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted