fact stringlengths 12 2.45k | type stringclasses 9
values | library stringclasses 1
value | imports listlengths 1 4 | filename stringlengths 21 66 | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
veldman_theorem_vtree_upto : afs_veldman_vtree_upto. Proof. exact afs_vtree_upto_embed. Qed. | Theorem | theories | [
"Require Import base statements."
] | theories/conversions.v | veldman_theorem_vtree_upto | |
higman_theorem_dtree_afs : afs_higman_dtree. Proof. apply veldman_vtree_upto_afs_to_higman_dtree_afs, veldman_theorem_vtree_upto. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree."
] | theories/higman_theorems.v | higman_theorem_dtree_afs | |
higman_theorem_dtree_af : af_higman_dtree. Proof. apply higman_dtree_afs_to_af, higman_theorem_dtree_afs. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree."
] | theories/higman_theorems.v | higman_theorem_dtree_af | |
higman_theorem_atree_af : af_higman_atree. Proof. apply higman_theorem_dtree_atree_af, higman_theorem_dtree_af. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree."
] | theories/higman_theorems.v | higman_theorem_atree_af | |
higman_lemma_list_af : af_higman_list. Proof. apply higman_dtree_to_list, higman_theorem_dtree_af. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree."
] | theories/higman_theorems.v | higman_lemma_list_af | |
l : vtree X := ⟨x|∅⟩. | Let | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree."
] | theories/higman_theorems.v | l | |
t n : vtree X := ⟨x|vec_set (λ _ : idx (S n), l)⟩. Local Fact embed_l r : r ≤ₚ l → arity r = 0. Proof. intros [ (p & ?) | H ]%dtree_product_embed_inv. + idx invert p. + destruct r as [ n y w ]. now destruct H as (-> & _). Qed. (* The only way for t n to embed into t m is n = m *) Local Fact embed_t n m : t n ≤ₚ t m → n... | Let | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree."
] | theories/higman_theorems.v | t | |
not_af_product_embed : af (dtree_product_embed R) → False. Proof. intros (? & ? & ? & ? & ?%embed_t)%(af_good_pair t); tlia. Qed. | Lemma | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree."
] | theories/higman_theorems.v | not_af_product_embed | |
kruskal_theorem_vtree_afs : afs_kruskal_vtree. Proof. apply kruskal_theorem_vtree_afs, veldman_theorem_vtree_upto. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | kruskal_theorem_vtree_afs | |
kruskal_theorem_vtree_af : af_kruskal_vtree. Proof. apply kruskal_vtree_afs_to_af, kruskal_theorem_vtree_afs. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | kruskal_theorem_vtree_af | |
kruskal_theorem_ltree_af : af_kruskal_ltree. Proof. apply kruskal_vtree_to_ltree, kruskal_theorem_vtree_af. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | kruskal_theorem_ltree_af | |
kruskal_theorem_ltree_afs : afs_kruskal_ltree. Proof. apply kruskal_ltree_af_to_afs, kruskal_theorem_ltree_af. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | kruskal_theorem_ltree_afs | |
kruskal_theorem_atree_af : af_kruskal_atree. Proof. apply kruskal_theorem_vtree_atree_af, kruskal_theorem_vtree_af. Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | kruskal_theorem_atree_af | |
afs_vtree_homeo_embed : afs X R → afs (wft (λ _, X)) (vtree_homeo_embed R). Proof. exact (@kruskal_theorem_vtree_afs _ _ _). Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | afs_vtree_homeo_embed | |
af_vtree_homeo_embed : af R → af (vtree_homeo_embed R). Proof. exact (@kruskal_theorem_vtree_af _ _). Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | af_vtree_homeo_embed | |
afs_ltree_homeo_embed : afs X R → afs (ltree_fall (λ x _, X x)) (ltree_homeo_embed R). Proof. exact (@kruskal_theorem_ltree_afs _ _ _). Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | afs_ltree_homeo_embed | |
af_ltree_homeo_embed : af R → af (ltree_homeo_embed R). Proof. exact (@kruskal_theorem_ltree_af _ _). Qed. | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree ltree."
] | theories/kruskal_theorems.v | af_ltree_homeo_embed | |
le_pirr x y (h₁ h₂ : x ≤ y) { struct h₁ } : h₁ = h₂. Proof. destruct h₁ as [ | y h₁ ]. + apply le_inv_eq_dep with (e := eq_refl). + specialize (le_pirr _ _ h₁). (* Freeze the recursive call on h₁ *) destruct (le_inv_le_dep h₂) as [ | (? & []) ]. * exfalso; lia. * now f_equal. Qed. | Fixpoint | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics."
] | theories/le_lt_pirr.v | le_pirr | |
lt_pirr x y (h₁ h₂ : x < y) : h₁ = h₂ := le_pirr h₁ h₂. | Definition | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics."
] | theories/le_lt_pirr.v | lt_pirr | |
af_higman_list := ∀ X (R : rel₂ X), af R → af (list_embed R). (** The statement of Higman's theorem for dependent roses trees: - sons are collected in vectors at each arity - the type of nodes can vary depending on the arity - the relation on nodes can vary depending on the arity - the type of nodes of arity greater th... | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | af_higman_list | |
af_higman_dtree := ∀ (k : nat) (X : nat → Type) (R : ∀n, rel₂ (X n)), (∀n, k ≤ n → X n → False) → (∀n, n < k → af (R n)) → af (dtree_product_embed R). (** The statement of Higman's theorem for vector based roses trees: - each node (x : X) can only be used with arity (a x : nat) - the relation R : nat → rel₂ X between n... | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | af_higman_dtree | |
af_higman_atree := ∀ (k : nat) X (a : X → nat) (R : nat → rel₂ X), (∀x, a x < k) → (∀n, n < k → af (R n)⇓(λ x, n = a x)) → af (atree_product_embed a R). (** The statement of Kruskal's theorem for vector based uniform roses trees: - the type of nodes is independent of the arity - the relation between nodes is independen... | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | af_higman_atree | |
af_kruskal_vtree := ∀ X (R : rel₂ X), af R → af (vtree_homeo_embed R). (** The statement of Kruskal's theorem for vector based roses trees: - each node (x : X) can only be used with arity (a x : nat) - the relation between nodes does not depend on the arity In that case, the homeomorphic embedding is AF. *) | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | af_kruskal_vtree | |
af_kruskal_atree := ∀ X (a : X → nat) (R : rel₂ X), af R → af (atree_homeo_embed a R). (** The statement of Kruskal's theorem for list based roses trees *) | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | af_kruskal_atree | |
af_kruskal_ltree := ∀ X (R : rel₂ X), af R → af (ltree_homeo_embed R). (** The statement of Veldman's theorem for uniform well formed rose trees, as established in the Kruskal-Veldman project: - sons are collected in vectors - the type of nodes is independent of the arity - but the sub-type of allowed nodes depends on ... | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | af_kruskal_ltree | |
afs_veldman_vtree_upto := ∀ (k : nat) A (X : nat → rel₁ A) (R : nat → rel₂ A), (∀n, k ≤ n → X n = X k) → (∀n, k ≤ n → R n = R k) → (∀n, n ≤ k → afs (X n) (R n)) → afs (wft X) (vtree_upto_embed k R). (** Below are afs versions of the above statements, that is when variations on types is replaced by variations on sub-typ... | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | afs_veldman_vtree_upto | |
afs_higman_dtree := ∀ k U (X : nat → rel₁ U) (R : nat → rel₂ U), (∀ n x, k ≤ n → X n x → False) → (∀n, n < k → afs (X n) (R n)) → afs (wft X) (dtree_product_embed R). | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | afs_higman_dtree | |
afs_kruskal_vtree := ∀ U (X : rel₁ U) (R : rel₂ U), afs X R → afs (wft (λ _, X)) (vtree_homeo_embed R). | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | afs_kruskal_vtree | |
afs_kruskal_ltree := ∀ U (X : rel₁ U) (R : rel₂ U), afs X R → afs (ltree_fall (λ x _, X x)) (ltree_homeo_embed R). (** The statement of Vazsonyi's conjecture for vector based undecorated rose trees, of breadth bounded by k *) #[local] Notation "x ∊ v" := (@vec_in _ x _ v) (at level 70, no associativity, format "x ∊ v")... | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | afs_kruskal_ltree | |
vazsonyi_conjecture_bounded := ∀ k (R : rel₂ (btree k)), (∀ s t n (h : n < k) v, t ∊ v → R s t → R s ⟨v|h⟩ᵥ) → (∀ n v m w (hₙ : n < k) (hₘ : m < k), vec_forall2 R v w → R ⟨v|hₙ⟩ᵥ ⟨w|hₘ⟩ᵥ) → ∀f, ∃ₜ n, ∃ i j, i < j < n ∧ R (f i) (f j). (** The statement of Vazsonyi's conjecture for list based (decorated) rose trees, but ... | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | vazsonyi_conjecture_bounded | |
vazsonyi_conjecture := ∀ X (R : rel₂ (ltree X)), (∀ s t x l, t ∈ l → R s t → R s ⟨x|l⟩ₗ) → (∀ x l y m, list_embed R l m → R ⟨x|l⟩ₗ ⟨y|m⟩ₗ) → ∀f, ∃ₜ n, ∃ i j, i < j < n ∧ R (f i) (f j). | Definition | theories | [
"From Coq\n Require Import List Utf8.",
"Require Import base dtree_embed vtree_embed ltree_embed atree_embed."
] | theories/statements.v | vazsonyi_conjecture | |
vazsonyi_theorem_bounded : vazsonyi_conjecture_bounded. Proof. apply higman_dtree_to_vazsonyi_bounded, higman_theorem_dtree_af. Qed. (** See statements.v for the statement of the "conjecture" *) | Theorem | theories | [
"From Coq\n Require Import Arith Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec vtree ltree."
] | theories/vazsonyi_theorems.v | vazsonyi_theorem_bounded | |
vazsonyi_theorem : vazsonyi_conjecture. Proof. apply kruskal_ltree_to_vazsonyi, kruskal_theorem_ltree_af. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec vtree ltree."
] | theories/vazsonyi_theorems.v | vazsonyi_theorem | |
Y := sigT X. | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree.",
"Require Import base dtree_embed statements."
] | theories/conversions/higman_dtree_afs_to_af.v | Y | |
T : nat → Y → Prop := λ n s, n = projT1 s. Local Definition dtree_vtree (t : dtree X) : vtree Y. Proof. induction t as [ n x v f ]. exact ⟨existT _ n x|vec_set f⟩. Defined. Local Fact dtree_vtree_fix n (x : X n) (v : vec _ n) : dtree_vtree ⟨x|v⟩ = ⟨existT _ n x|vec_map dtree_vtree v⟩. Proof. rewrite <- vec_set_map; aut... | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree.",
"Require Import base dtree_embed statements."
] | theories/conversions/higman_dtree_afs_to_af.v | T | |
Y := (sigT X). | Notation | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree.",
"Require Import base dtree_embed statements."
] | theories/conversions/higman_dtree_afs_to_af.v | Y | |
T n (y : Y) := n = projT1 y. Local Fact T_empty n x : k ≤ n → T n x → False. Proof. unfold T; intros H; destruct x as (j,x); simpl; intros; subst; revert x; apply HX; auto. Qed. | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree.",
"Require Import base dtree_embed statements."
] | theories/conversions/higman_dtree_afs_to_af.v | T | |
R' n (u v : Y) := match u, v with | existT _ _ x, existT _ _ y => exists e f, @R n x↺e y↺f end. Local Fact R'_afs n : n < k → afs (T n) (@R' n). Proof. intros Hn; apply afs_iff_af_sub_rel; generalize (HR Hn). af rel morph (fun (x : X n) (y : sig (T n)) => match proj1_sig y with | existT _ i a => exists e, x↺e = a end);... | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree.",
"Require Import base dtree_embed statements."
] | theories/conversions/higman_dtree_afs_to_af.v | R' | |
higman_dtree_afs_to_af : afs_higman_dtree → af_higman_dtree. Proof. intros ? ? ? ?; apply higman_afs_to_higman_af_at; auto. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree.",
"Require Import base dtree_embed statements."
] | theories/conversions/higman_dtree_afs_to_af.v | higman_dtree_afs_to_af | |
Y := unary_family. | Notation | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils idx vec dtree.",
"Require Import base notations dtree_embed statements."
] | theories/conversions/higman_dtree_to_list.v | Y | |
T := (dtree Y). Local Fixpoint dtree_list (t : T) : list X := match t with | @dtree_cons _ 0 _ _ => [] | @dtree_cons _ 1 x v => x :: dtree_list v⦃idx₀⦄ | @dtree_cons _ _ x _ => @Empty_set_rect _ x end. Local Fixpoint list_dtree l : T := match l with | [] => ⟨tt|∅⟩ | x::l => ⟨x|list_dtree l##∅⟩ end. Local Fact dtree_lis... | Notation | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils idx vec dtree.",
"Require Import base notations dtree_embed statements."
] | theories/conversions/higman_dtree_to_list.v | T | |
Y := (unary_family X). | Notation | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils idx vec dtree.",
"Require Import base notations dtree_embed statements."
] | theories/conversions/higman_dtree_to_list.v | Y | |
T := (dtree Y). | Notation | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils idx vec dtree.",
"Require Import base notations dtree_embed statements."
] | theories/conversions/higman_dtree_to_list.v | T | |
RY n : rel₂ (Y n) := match n with | 1 => R | _ => ⊤₂ end. Local Lemma higman_lemma_af : af R → af (list_embed R). Proof. intros H. cut (af (dtree_product_embed RY)). + clear H. af rel morph (fun x y => dtree_list x = y). * intros l; exists (list_dtree l); rewrite dtree_list_dtree; auto. * intros r t ? ? <- <-. inductio... | Let | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils idx vec dtree.",
"Require Import base notations dtree_embed statements."
] | theories/conversions/higman_dtree_to_list.v | RY | |
higman_dtree_to_list : af_higman_dtree → af_higman_list. Proof. intros ? ? ? ?; apply higman_lemma_af; auto. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils idx vec dtree.",
"Require Import base notations dtree_embed statements."
] | theories/conversions/higman_dtree_to_list.v | higman_dtree_to_list | |
X := dtree_bounded. Local Definition tt' n : n < k → X n. Proof. refine (match le_lt_dec k n as d return _ → if d then Empty_set else unit with | left _ => λ _, match _ : False with end | right _ => λ _, tt end); tlia. Defined. Local Fact X_uniq n : ∀ a b : X n, a = b. Proof. unfold X; destruct (le_lt_dec k n); intros ... | Notation | theories | [
"From Coq\n Require Import Arith Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree ltree btree."
] | theories/conversions/higman_dtree_to_vazsonyi_bounded.v | X | |
higman_dtree_to_vazsonyi_bounded : af_higman_dtree → vazsonyi_conjecture_bounded. Proof. intros Hk k R HR1 HR2 f. apply vazsonyi_conjecture_bounded_strong; eauto. Qed. Print vazsonyi_conjecture_bounded. (** Notation for binary relations/predicates *) Locate "rel₂ _". (** Inductive definition vec X n for vectors of leng... | Theorem | theories | [
"From Coq\n Require Import Arith Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree ltree btree."
] | theories/conversions/higman_dtree_to_vazsonyi_bounded.v | higman_dtree_to_vazsonyi_bounded | |
A := (λ n x, n = a x). | Notation | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | A | |
atree_dtree (t : atree a) : dtree (λ n, sig (A n)) := match t with | ⟨x|v⟩ₐ => ⟨exist _ x eq_refl|vec_map atree_dtree v⟩ end. | Fixpoint | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | atree_dtree | |
dtree_atree (s : dtree (λ n, sig (A n))) : atree a := match s with | ⟨exist _ x e|v⟩ => ⟨x|vec_map dtree_atree v↺e⟩ₐ end. | Fixpoint | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | dtree_atree | |
dtree_atree_dtree t : dtree_atree (atree_dtree t) = t. Proof. induction t; simpl; f_equal. rewrite vec_map_map; now vec ext; vec rew. Qed. | Fact | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | dtree_atree_dtree | |
atree_dtree_atree s : atree_dtree (dtree_atree s) = s. Proof. induction s as [ ? [] ]; simpl; eq refl; f_equal. rewrite vec_map_map; now vec ext; vec rew. Qed. | Fact | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | atree_dtree_atree | |
va_tree_eq : vtree X → atree a → Prop := | in_va_tree_eq x (v : vec _ (a x)) w : vec_fall2 va_tree_eq v w → va_tree_eq ⟨x|v⟩ ⟨x|w⟩ₐ. | Inductive | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | va_tree_eq | |
va_tree_eq_surj t : { s | va_tree_eq s t }. Proof. induction t as [ x v IHv ]. vec reif IHv as [ w Hw ]. exists ⟨x|w⟩; now constructor. Qed. | Fact | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | va_tree_eq_surj | |
va_tree_eq_wft s t : va_tree_eq s t → wft A s. Proof. induction 1; split; auto. Qed. | Remark | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | va_tree_eq_wft | |
va_tree_eq_total s : wft A s → { t | va_tree_eq s t }. Proof. induction s as [ n x v IHv ]; intros (Hx & Hv)%wft_fix. specialize (fun p => IHv _ (Hv p)). vec reif IHv as [ w Hw ]. subst n. exists (atree_cons x w). now constructor. Qed. (* This is the critical inversion lemma *) | Remark | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | va_tree_eq_total | |
va_tree_eq_invl s t : va_tree_eq s t → match s with | @dtree_cons _ n x v => ∃ (e : n = a x) w, t = atree_cons x w↺e ∧ vec_fall2 va_tree_eq v w end. Proof. intros []; eexists eq_refl, _; simpl; eauto. Qed. | Lemma | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | va_tree_eq_invl | |
A := (λ n x, n = a x). Hypothesis (Hk : ∀x, a x < k) (HR : ∀n, n < k → af (R n)⇓(A n)). | Notation | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | A | |
Y n := sig (A n). | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | Y | |
T n : rel₂ (Y n) := (R n)⇓(A n). Local Fact af_dtree_T : af (dtree_product_embed T). Proof. apply higman_dtree with (k := k). + intros n Hn (x & Hx). specialize (Hk x). rewrite <- Hx in Hk; tlia. + apply HR. Qed. Local Theorem higman_dtree_atree_af_local : af (atree_product_embed a R). Proof. generalize af_dtree_T. af ... | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | T | |
higman_theorem_dtree_atree_af : af_higman_dtree → af_higman_atree. Proof. exact higman_dtree_atree_af_local. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | higman_theorem_dtree_atree_af | |
A := (λ n x, n = a x). Hypothesis (HR : af R). Local Fact af_vtree : af (vtree_homeo_embed R). Proof. apply kruskal_vtree, HR. Qed. Local Theorem kruskal_vtree_atree_af_local : af (atree_homeo_embed a R). Proof. generalize af_vtree. af rel morph (@va_tree_eq _ a). + intros t. destruct (va_tree_eq_surj t); eauto. + intr... | Notation | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | A | |
kruskal_theorem_vtree_atree_af : af_kruskal_vtree → af_kruskal_atree. Proof. exact kruskal_vtree_atree_af_local. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/higman_kruskal_dtree_to_atree.v | kruskal_theorem_vtree_atree_af | |
kruskal_ltree_afs_to_af : afs_kruskal_ltree → af_kruskal_ltree. Proof. intros K X R H%af_iff_afs_True%K; apply af_iff_afs_True; revert H. apply afs_mono; auto. intros t _. induction t; apply ltree_fall_fix; auto. Qed. | Theorem | theories | [
"From Coq\n Require Import List Utf8.",
"From KruskalTrees\n Require Import list_utils ltree.",
"Require Import base statements."
] | theories/conversions/kruskal_ltree_afs_to_af.v | kruskal_ltree_afs_to_af | |
forget := (@ltree_sig_forget _ _). Hypothesis kruskal : af_kruskal_ltree. | Notation | theories | [
"From Coq\n Require Import List Utf8.",
"From KruskalTrees\n Require Import list_utils ltree.",
"Require Import base statements."
] | theories/conversions/kruskal_ltree_af_to_afs.v | forget | |
kruskal_ltree_af_to_afs : afs_kruskal_ltree. Proof. intros U X R H%afs_iff_af_sub_rel%kruskal. apply afs_iff_af_sub_rel; revert H. af rel morph (fun x y => proj1_sig y = forget x). + intros (t & Ht); simpl; revert t Ht. induction 1 as [ x l H1 H2 IH2 ] using ltree_fall_rect. Forall reif IH2 as (m & Hm). exists ⟨exist _... | Theorem | theories | [
"From Coq\n Require Import List Utf8.",
"From KruskalTrees\n Require Import list_utils ltree.",
"Require Import base statements."
] | theories/conversions/kruskal_ltree_af_to_afs.v | kruskal_ltree_af_to_afs | |
kruskal_ltree_to_vazsonyi : af_kruskal_ltree → vazsonyi_conjecture. Proof. intros H X R HR1 HR2 f. apply af_good_pair. generalize (H _ _ (@af_True X)); apply af_mono. induction 1; simpl; eauto. Qed. Print list. Print ltree. Locate "_ ∈ _". Print "∈". Print list_embed. Check kruskal_ltree_to_vazsonyi. Print Assumptions ... | Theorem | theories | [
"From Coq\n Require Import List Utf8.",
"From KruskalTrees\n Require Import list_utils ltree.",
"Require Import base ltree_embed statements."
] | theories/conversions/kruskal_ltree_to_vazsonyi.v | kruskal_ltree_to_vazsonyi | |
kruskal_vtree_afs_to_af : afs_kruskal_vtree → af_kruskal_vtree. Proof. intros K X R H%af_iff_afs_True%K. apply af_iff_afs_True; revert H. apply afs_mono; auto. intros t _; induction t; apply wft_fix; auto. Qed. | Theorem | theories | [
"From Coq \n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree.",
"Require Import base statements."
] | theories/conversions/kruskal_vtree_afs_to_af.v | kruskal_vtree_afs_to_af | |
V := (idx k * U)%type. | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/kruskal_vtree_afs_to_higman_dtree_afs.v | V | |
Y : V → Prop := λ '(p,x), X (idx2nat p) x. | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/kruskal_vtree_afs_to_higman_dtree_afs.v | Y | |
T : V → V → Prop := λ '(p,x) '(q,y), p = q ∧ R (idx2nat p) x y. | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/kruskal_vtree_afs_to_higman_dtree_afs.v | T | |
W : nat → V → Prop := λ _, Y. Local Fact af_R (p : idx k) : af (R (idx2nat p))⇓(X (idx2nat p)). Proof. apply afs_iff_af_sub_rel; apply HXR, idx2nat_lt. Qed. (* Using finite dependent sum, T is afs over Y *) Local Fact afs_YT : afs Y T. Proof. generalize (af_dep_sum _ _ af_R). intros H; apply afs_iff_af_sub_rel; revert ... | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/kruskal_vtree_afs_to_higman_dtree_afs.v | W | |
K : ∀n, V → vec (vtree V) n → Prop := λ n '(p,x) _, n = idx2nat p ∧ X n x. Local Fact K_inc_W : ∀ n x v, @K n x v → W n x. Proof. intros ? [] _ (-> & ?); simpl; auto. Qed. Local Fact dtree_fall_K_inc_wft_W : dtree_fall K ⊆₁ wft W. Proof. apply dtree_fall_mono, K_inc_W. Qed. (** Homeomorphic embedding of T is afs over w... | Let | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/kruskal_vtree_afs_to_higman_dtree_afs.v | K | |
kruskal_vtree_afs_to_higman_dtree_afs : afs_kruskal_vtree → afs_higman_dtree. Proof. intros ? ? ? ? ? ?; apply kruskal_afs_to_higman_afs_at; auto. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree vtree."
] | theories/conversions/kruskal_vtree_afs_to_higman_dtree_afs.v | kruskal_vtree_afs_to_higman_dtree_afs | |
forget := (@vtree_sig_forget _ _). Hypothesis kruskal : af_kruskal_vtree. | Notation | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree.",
"Require Import base statements."
] | theories/conversions/kruskal_vtree_af_to_afs.v | forget | |
kruskal_vtree_af_to_afs : afs_kruskal_vtree. Proof. intros U X R H%afs_iff_af_sub_rel%kruskal. apply afs_iff_af_sub_rel; revert H. af rel morph (fun x y => forget x = proj1_sig y). + intros (t & Ht); revert t Ht; simpl. induction 1 as [ n x v Hx Hv IHv ] using wft_rect. vec reif IHv as (w & Hw). exists ⟨exist _ x Hx|w⟩... | Theorem | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import idx vec vtree.",
"Require Import base statements."
] | theories/conversions/kruskal_vtree_af_to_afs.v | kruskal_vtree_af_to_afs | |
kruskal_vtree_to_ltree : af_kruskal_vtree → af_kruskal_ltree. Proof. intros K X R HR; generalize (K _ _ HR); clear HR. af rel morph (fun x y => vtree_ltree x = y). + intros t; destruct (vtree_ltree_surj t); eauto. + intros t1 t2 ? ? <- <-. induction 1 as [ t n x v p H1 IH1 | n x v m y w H1 H2 IH2 ]. * rewrite vtree_ltr... | Theorem | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec vtree ltree.",
"Require Import base statements."
] | theories/conversions/kruskal_vtree_to_ltree.v | kruskal_vtree_to_ltree | |
veldman_vtree_upto_afs_to_higman_dtree_afs : afs_veldman_vtree_upto → afs_higman_dtree. Proof. intros Kr k U X R H1 H2. set (X' i := if le_lt_dec k i then ⊥₁ else X i). set (R' i := if le_lt_dec k i then ⊥₂ else R i). cut (afs (wft X') (vtree_upto_embed k R')). af rel morph eq. + intros t Ht; exists t; split; auto. rev... | Theorem | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import dtree vtree."
] | theories/conversions/veldman_vtree_upto_afs_to_higman_dtree_afs.v | veldman_vtree_upto_afs_to_higman_dtree_afs | |
kruskal_theorem_vtree_afs : afs_veldman_vtree_upto → afs_kruskal_vtree. Proof. intros V U X R H. cut (afs (wft (fun _ => X)) (vtree_upto_embed 0 (fun _ => R))). + apply afs_mono; auto. intros ? ? ? ?; apply vtree_upto_homeo_uniform; auto. + apply afs_vtree_upto_embed; auto. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec vtree ltree.",
"Require Import base vtree_embed ltree_embed statements."
] | theories/conversions/veldman_vtree_upto_afs_to_kruskal_vtree_afs.v | kruskal_theorem_vtree_afs | |
atree : Type := | atree_cons x : vec atree (a x) → atree. Set Elimination Schemes. Arguments atree_cons : clear implicits. | Inductive | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree | |
atree_rect t : P t := match t with | ⟨x|v⟩ₐ => P_cons v (λ i, atree_rect v⦃i⦄) end. | Fixpoint | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_rect | |
atree_rect_fix x v : atree_rect ⟨x|v⟩ₐ = P_cons v (λ i, atree_rect v⦃i⦄). Proof. reflexivity. Qed. | Fact | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_rect_fix | |
atree_rec (P : _ → Set) := atree_rect P. | Definition | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_rec | |
atree_ind (P : _ → Prop) := atree_rect P. Unset Elimination Schemes. Variable (R : nat → rel₂ X) (T : rel₂ X). | Definition | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_ind | |
atree_product_embed : atree → atree → Prop := | dtree_embed_subt x v s i : s ≤ₚ v⦃i⦄ → s ≤ₚ ⟨x|v⟩ₐ | dtree_embed_root x v y w : R (a x) x y → vec_forall2 atree_product_embed v w → ⟨x|v⟩ₐ ≤ₚ ⟨y|w⟩ₐ where "s ≤ₚ t" := (atree_product_embed s t). | Inductive | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_product_embed | |
atree_homeo_embed : atree → atree → Prop := | vtree_homeo_embed_subt t x v i : t ≤ₕ v⦃i⦄ → t ≤ₕ ⟨x|v⟩ₐ | vtree_homeo_embed_root x v y w : T x y → vec_embed atree_homeo_embed v w → ⟨x|v⟩ₐ ≤ₕ ⟨y|w⟩ₐ where "s ≤ₕ t" := (atree_homeo_embed s t). Set Elimination Schemes. | Inductive | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_homeo_embed | |
atree_product_embed_ind : ∀ s t, s ≤ₚ t → P s t. Proof. refine (fix loop s t D { struct D } := _). destruct D as [ t x v p H1 | x v y w H1 H2 ]. + apply HT1 with (1 := H1); trivial. apply loop, H1. + apply HT2; trivial. revert v w H2; generalize (a x) (a y). clear x y H1. induction 1; eauto with vec_db. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_product_embed_ind | |
atree_homeo_embed_ind : ∀ s t, s ≤ₕ t → P s t. Proof. refine (fix loop s t D { struct D } := _). destruct D as [ t x v p H1 | x v y w H1 H2 ]. + apply HT1 with (1 := H1); trivial. apply loop, H1. + apply HT2; trivial. revert v w H2; generalize (a x) (a y). clear x y H1. induction 1; eauto with vec_db. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import list_utils idx vec dtree vtree.",
"Require Import base."
] | theories/embeddings/atree_embed.v | atree_homeo_embed_ind | |
dtree_product_embed : dtree X → dtree X → Prop := | dtree_embed_subt k x (v : vec _ k) s i : s ≤ₚ v⦃i⦄ → s ≤ₚ ⟨x|v⟩ | dtree_embed_root k x (v : vec _ k) y w : R x y → vec_fall2 dtree_product_embed v w → ⟨x|v⟩ ≤ₚ ⟨y|w⟩ where "s ≤ₚ t" := (dtree_product_embed s t). | Inductive | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree.",
"Require Import base."
] | theories/embeddings/dtree_embed.v | dtree_product_embed | |
dtree_product_embed_inv r t : r ≤ₚ t → match t with | @dtree_cons _ m y v => (∃i, r ≤ₚ v⦃i⦄) ∨ match r with | @dtree_cons _ n x u => ∃ e : n = m, R x↺e y ∧ vec_fall2 dtree_product_embed u↺e v end end. Proof. intros []; simpl; [ left | right ]; eauto; exists eq_refl; auto. Qed. | Fact | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree.",
"Require Import base."
] | theories/embeddings/dtree_embed.v | dtree_product_embed_inv | |
dtree_product_embed_mono (R T : ∀k, X k → X k → Prop) : (∀k, R k ⊆₂ T k) → dtree_product_embed R ⊆₂ dtree_product_embed T. Proof. induction 2; eauto. Qed. | Fact | theories | [
"From Coq\n Require Import Utf8.",
"From KruskalTrees\n Require Import tactics idx vec dtree.",
"Require Import base."
] | theories/embeddings/dtree_embed.v | dtree_product_embed_mono | |
ltree_product_embed : ltree X → ltree X → Prop := | ltree_product_embed_subt {s t x l} : t ∈ l → s ≤ₚ t → s ≤ₚ ⟨x|l⟩ₗ | ltree_product_embed_root {x l y m} : R x y → Forall2 ltree_product_embed l m → ⟨x|l⟩ₗ ≤ₚ ⟨y|m⟩ₗ where "s ≤ₚ t" := (ltree_product_embed s t). (* This is the homeomorphic embedding for Kruskal's tree th... | Inductive | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils ltree.",
"Require Import base."
] | theories/embeddings/ltree_embed.v | ltree_product_embed | |
ltree_homeo_embed : ltree X → ltree X → Prop := | homeo_embed_subt s t x l : t ∈ l → s ≤ₕ t → s ≤ₕ ⟨x|l⟩ₗ | homeo_embed_root x l y m : R x y → list_embed ltree_homeo_embed l m → ⟨x|l⟩ₗ ≤ₕ ⟨y|m⟩ₗ where "s ≤ₕ t" := (ltree_homeo_embed s t). Set Elimination Schemes. Hint Constructors ltree_product_embed ltree_homeo_embed :... | Inductive | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils ltree.",
"Require Import base."
] | theories/embeddings/ltree_embed.v | ltree_homeo_embed | |
ltree_product_embed_ind s t (Hst : s ≤ₚ t) { struct Hst } : P s t := match Hst with | ltree_product_embed_subt H1 H2 => HT1 H1 H2 (ltree_product_embed_ind H2) | ltree_product_embed_root H1 H2 => HT2 H1 H2 (Forall2_mono ltree_product_embed_ind H2) end. | Fixpoint | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils ltree.",
"Require Import base."
] | theories/embeddings/ltree_embed.v | ltree_product_embed_ind | |
ltree_homeo_embed_ind s t (Hst : s ≤ₕ t) : P s t. Proof. destruct Hst as [ s t x ll H1 H2 | x ll y mm H1 H2 ]. + apply HT1 with (1 := H1); trivial. apply ltree_homeo_embed_ind, H2. + apply HT2; trivial. now apply list_embed_mono with (1 := ltree_homeo_embed_ind). Qed. | Fixpoint | theories | [
"From Coq\n Require Import Arith List Lia Utf8.",
"From KruskalTrees\n Require Import tactics list_utils ltree.",
"Require Import base."
] | theories/embeddings/ltree_embed.v | ltree_homeo_embed_ind | |
vtree_homeo_embed : vtree A → vtree A → Prop := | vtree_homeo_embed_subt t n x (v : vec _ n) i : t ≤ₕ v⦃i⦄ → t ≤ₕ ⟨x|v⟩ | vtree_homeo_embed_root n x (v : vec _ n) m y (w : vec _ m) : Rₕ x y → vec_embed vtree_homeo_embed v w → ⟨x|v⟩ ≤ₕ ⟨y|w⟩ where "s ≤ₕ t" := (vtree_homeo_embed s t). Set Elimination Schemes. Hint Constr... | Inductive | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree.",
"Require Import base.",
"Require Export dtree_embed."
] | theories/embeddings/vtree_embed.v | vtree_homeo_embed | |
vtree_homeo_embed_ind : ∀ s t, s ≤ₕ t → P s t. Proof. refine (fix loop s t D { struct D } := _). destruct D as [ t n x v p H1 | n x v m y w H1 H2 ]. + apply HT1 with (1 := H1); trivial. apply loop, H1. + apply HT2; trivial. clear x y H1; revert H2. induction 1; eauto with vec_db. Qed. | Theorem | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree.",
"Require Import base.",
"Require Export dtree_embed."
] | theories/embeddings/vtree_embed.v | vtree_homeo_embed_ind | |
vtree_product_upto k (R : nat → rel₂ A) : (∀ n, k ≤ n → R n ⊆₂ R k) → dtree_product_embed R ⊆₂ vtree_upto_embed k R. Proof. intros H. induction 1 as [ | n ]; eauto. destruct (le_lt_dec k n); eauto. Qed. | Fact | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree.",
"Require Import base.",
"Require Export dtree_embed."
] | theories/embeddings/vtree_embed.v | vtree_product_upto | |
vtree_product_upto_iff k (X : nat → rel₁ A) (R : nat → rel₂ A) : (∀ n x, k ≤ n → ¬ X n x) → ∀ s t, wft X s → dtree_product_embed R s t ↔ vtree_upto_embed k R s t. Proof. intros HX s t H; split; intros H1; revert H1 H. + induction 1 as [ t n x v p H1 IH1 | n x v y w H1 H2 IH2 ]; intros H; eauto. rewrite wft_fix in H; de... | Fact | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree.",
"Require Import base.",
"Require Export dtree_embed."
] | theories/embeddings/vtree_embed.v | vtree_product_upto_iff | |
vtree_upto_homeo k (R : nat → rel₂ A) : (∀n, n ≤ k → R n ⊆₂ Rₕ) → vtree_upto_embed k R ⊆₂ vtree_homeo_embed. Proof. intros H. induction 1 as [ | n ? ? ? ? Hn | ]; eauto. generalize (lt_le_weak _ _ Hn); intro; eauto. Qed. | Fact | theories | [
"From Coq\n Require Import Arith Lia Utf8.",
"From KruskalTrees\n Require Import tactics idx vec vtree.",
"Require Import base.",
"Require Export dtree_embed."
] | theories/embeddings/vtree_embed.v | vtree_upto_homeo |
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