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Coq-Kruskal

statement stringlengths 1 322	proof stringlengths 0 2.19k	type stringclasses 10 values	symbolic_name stringlengths 1 42	library stringclasses 3 values	filename stringclasses 24 values	imports listlengths 2 20	deps listlengths 0 9	docstring stringclasses 33 values	source_url stringclasses 1 value	commit stringclasses 1 value
veldman_theorem_vtree_upto : afs_veldman_vtree_upto.	Proof. exact afs_vtree_upto_embed. Qed.	Theorem	veldman_theorem_vtree_upto	Root	theories/conversions.v	[ "base", "statements", "veldman_vtree_upto_afs_to_kruskal_vtree_afs", "veldman_vtree_upto_afs_to_higman_dtree_afs", "kruskal_vtree_afs_to_af", "kruskal_vtree_to_ltree", "kruskal_ltree_to_vazsonyi", "kruskal_ltree_af_to_afs", "kruskal_vtree_afs_to_higman_dtree_afs", "higman_dtree_to_list", "higman...	[ "afs_veldman_vtree_upto" ]	Since afs_veldman_vtree_upto is already proved as the main theorem of the Kruskal-Veldman project, all the following results derive more or less easily from it. In conversions/*.v, we establish the following implications using easy relational morphisms afs_veldman_vtree_up...	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_theorem_dtree_afs : afs_higman_dtree.	Proof. apply veldman_vtree_upto_afs_to_higman_dtree_afs, veldman_theorem_vtree_upto. Qed.	Theorem	higman_theorem_dtree_afs	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[ "afs_higman_dtree", "veldman_theorem_vtree_upto", "veldman_vtree_upto_afs_to_higman_dtree_afs" ]	We use conversion and Veldman's theorem afs_vtree_upto_embed	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_theorem_dtree_af : af_higman_dtree.	Proof. apply higman_dtree_afs_to_af, higman_theorem_dtree_afs. Qed.	Theorem	higman_theorem_dtree_af	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[ "af_higman_dtree", "higman_dtree_afs_to_af", "higman_theorem_dtree_afs" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_theorem_atree_af : af_higman_atree.	Proof. apply higman_theorem_dtree_atree_af, higman_theorem_dtree_af. Qed.	Theorem	higman_theorem_atree_af	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[ "af_higman_atree", "higman_theorem_dtree_af", "higman_theorem_dtree_atree_af" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_lemma_list_af : af_higman_list.	Proof. apply higman_dtree_to_list, higman_theorem_dtree_af. Qed.	Theorem	higman_lemma_list_af	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[ "af_higman_list", "higman_dtree_to_list", "higman_theorem_dtree_af" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
arity {X} (r : dtree X)	:= match r with \| @dtree_cons _ n _ _ => n end.	Definition	arity	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
l : vtree X	:= ⟨x\|∅⟩.	Let	l	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[]	l is a leaf of height 0 and t is a tree of height 1 with 1+n sons	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
t n : vtree X	:= ⟨x\|vec_set (λ _ : idx (S n), l)⟩.	Let	t	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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.	Fact	embed_l	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[ "arity", "dtree_product_embed_inv" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
embed_t n m : t n ≤ₚ t m → n = m.	Proof. intros [ (p & H) \| (e & _) ]%dtree_product_embed_inv. + rewrite vec_prj_set in H. now apply embed_l in H. + tlia. Qed.	Fact	embed_t	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[ "dtree_product_embed_inv", "embed_l" ]	The only way for t n to embed into t m is n = m	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
not_af_product_embed : af (dtree_product_embed R) → False.	Proof. intros (? & ? & ? & ? & ?%embed_t)%(af_good_pair t); tlia. Qed.	Lemma	not_af_product_embed	Root	theories/higman_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "vtree", "base", "vtree_embed", "statements", "conversions", "idx_notations", "vec_notations", "vtree_notations" ]	[ "dtree_product_embed", "embed_t" ]	If X is inhabited then (dtree_product_embed R) is never almost-full when branching is unbounded	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
kruskal_theorem_vtree_afs : afs_kruskal_vtree.	Proof. apply kruskal_theorem_vtree_afs, veldman_theorem_vtree_upto. Qed.	Theorem	kruskal_theorem_vtree_afs	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "afs_kruskal_vtree", "veldman_theorem_vtree_upto" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
kruskal_theorem_vtree_af : af_kruskal_vtree.	Proof. apply kruskal_vtree_afs_to_af, kruskal_theorem_vtree_afs. Qed.	Theorem	kruskal_theorem_vtree_af	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "af_kruskal_vtree", "kruskal_theorem_vtree_afs", "kruskal_vtree_afs_to_af" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
kruskal_theorem_ltree_af : af_kruskal_ltree.	Proof. apply kruskal_vtree_to_ltree, kruskal_theorem_vtree_af. Qed.	Theorem	kruskal_theorem_ltree_af	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "af_kruskal_ltree", "kruskal_theorem_vtree_af", "kruskal_vtree_to_ltree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
kruskal_theorem_ltree_afs : afs_kruskal_ltree.	Proof. apply kruskal_ltree_af_to_afs, kruskal_theorem_ltree_af. Qed.	Theorem	kruskal_theorem_ltree_afs	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "afs_kruskal_ltree", "kruskal_ltree_af_to_afs", "kruskal_theorem_ltree_af" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
kruskal_theorem_atree_af : af_kruskal_atree.	Proof. apply kruskal_theorem_vtree_atree_af, kruskal_theorem_vtree_af. Qed.	Theorem	kruskal_theorem_atree_af	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "af_kruskal_atree", "kruskal_theorem_vtree_af", "kruskal_theorem_vtree_atree_af" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
afs_vtree_homeo_embed : afs X R → afs (wft (λ _, X)) (vtree_homeo_embed R).	Proof. exact (@kruskal_theorem_vtree_afs _ _ _). Qed.	Theorem	afs_vtree_homeo_embed	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "kruskal_theorem_vtree_afs", "vtree_homeo_embed" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
af_vtree_homeo_embed : af R → af (vtree_homeo_embed R).	Proof. exact (@kruskal_theorem_vtree_af _ _). Qed.	Theorem	af_vtree_homeo_embed	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "kruskal_theorem_vtree_af", "vtree_homeo_embed" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	afs_ltree_homeo_embed	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "kruskal_theorem_ltree_afs", "ltree_homeo_embed" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
af_ltree_homeo_embed : af R → af (ltree_homeo_embed R).	Proof. exact (@kruskal_theorem_ltree_af _ _). Qed.	Theorem	af_ltree_homeo_embed	Root	theories/kruskal_theorems.v	[ "Coq", "Utf8", "KruskalTrees", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "vec_notations", "vtree_notations", "af_notations" ]	[ "kruskal_theorem_ltree_af", "ltree_homeo_embed" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
le_inv_eq_dep x y (h : x ≤ y) : ∀e : y = x, le_n x = eq_rect y (le x) h _ e.	Proof. destruct h as [ \| y h' ]; intros e. + now rewrite (eq_refl_nat e). + exfalso; lia. Qed.	Fact	le_inv_eq_dep	Root	theories/le_lt_pirr.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics" ]	[]	le and lt are proof irrelevant	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
le_inv_le_dep x y' (h : x ≤ y') : match y' return x ≤ y' → Prop with \| 0 => λ _, True \| S y => λ h, S y = x ∨ ∃h', le_S x y h' = h end h.	Proof. destruct h; [ destruct x \| ]; eauto. Qed.	Fact	le_inv_le_dep	Root	theories/le_lt_pirr.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	le_pirr	Root	theories/le_lt_pirr.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics" ]	[ "le_inv_eq_dep", "le_inv_le_dep" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
lt_pirr x y (h₁ h₂ : x < y) : h₁ = h₂	:= le_pirr h₁ h₂.	Definition	lt_pirr	Root	theories/le_lt_pirr.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics" ]	[ "le_pirr" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
"⊤₁"	:= (λ _, True).	Notation	⊤₁	Root	theories/notations.v	[ "Coq", "Utf8" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
"⊤₂"	:= (λ _ _, True).	Notation	⊤₂	Root	theories/notations.v	[ "Coq", "Utf8" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
af_higman_list	:= ∀ X (R : rel₂ X), af R → af (list_embed R).	Definition	af_higman_list	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[]	The statement of Higman's lemma for lists	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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).	Definition	af_higman_dtree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "dtree_product_embed" ]	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 than k (fixed) should be empty hence...	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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).	Definition	af_higman_atree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "atree_product_embed" ]	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 nodes depends on the arity - the arity is bounded by k : a _ < k - for any arities n < k, R n restricted to (λ x, n = a x) is AF ...	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
af_kruskal_vtree	:= ∀ X (R : rel₂ X), af R → af (vtree_homeo_embed R).	Definition	af_kruskal_vtree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "vtree_homeo_embed" ]	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 independent of the arity In that case, the homeomorphic embedding is AF.	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
af_kruskal_atree	:= ∀ X (a : X → nat) (R : rel₂ X), af R → af (atree_homeo_embed a R).	Definition	af_kruskal_atree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "atree_homeo_embed" ]	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.	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
af_kruskal_ltree	:= ∀ X (R : rel₂ X), af R → af (ltree_homeo_embed R).	Definition	af_kruskal_ltree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "ltree_homeo_embed" ]	The statement of Kruskal's theorem for list based roses trees	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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).	Definition	afs_veldman_vtree_upto	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[]	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 the arity - the relation on nodes can vary depe...	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	afs_higman_dtree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "dtree_product_embed" ]	Below are afs versions of the above statements, that is when variations on types is replaced by variations on sub-types	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
afs_kruskal_vtree	:= ∀ U (X : rel₁ U) (R : rel₂ U), afs X R → afs (wft (λ _, X)) (vtree_homeo_embed R).	Definition	afs_kruskal_vtree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "vtree_homeo_embed" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
afs_kruskal_ltree	:= ∀ U (X : rel₁ U) (R : rel₂ U), afs X R → afs (ltree_fall (λ x _, X x)) (ltree_homeo_embed R).	Definition	afs_kruskal_ltree	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[ "ltree_homeo_embed" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
"x ∊ v"	:= (@vec_in _ x _ v) (at level 70, no associativity, format "x ∊ v").	Notation	x ∊ v	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[]	The statement of Vazsonyi's conjecture for vector based undecorated rose trees, of breadth bounded by k	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
"⟨ v \| h ⟩ᵥ"	:= (btree_cons v h) (at level 0, v at level 200, format "⟨ v \| h ⟩ᵥ").	Notation	⟨ v \| h ⟩ᵥ	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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).	Definition	vazsonyi_conjecture_bounded	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	vazsonyi_conjecture	Root	theories/statements.v	[ "Coq", "List", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "dtree", "vtree", "ltree", "btree", "base", "dtree_embed", "vtree_embed", "ltree_embed", "atree_embed", "af_notations", "idx_notations", "vec_notations", "ltree_notations" ]	[]	The statement of Vazsonyi's conjecture for list based (decorated) rose trees, but the decoration is ignored as if X = unit.	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
vazsonyi_theorem_bounded : vazsonyi_conjecture_bounded.	Proof. apply higman_dtree_to_vazsonyi_bounded, higman_theorem_dtree_af. Qed.	Theorem	vazsonyi_theorem_bounded	Root	theories/vazsonyi_theorems.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "kruskal_theorems", "higman_theorems", "vec_notations", "vtree_notations", "af_notations" ]	[ "higman_dtree_to_vazsonyi_bounded", "higman_theorem_dtree_af", "vazsonyi_conjecture_bounded" ]	See statements.v for the statement of the "conjecture"	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
vazsonyi_theorem : vazsonyi_conjecture.	Proof. apply kruskal_ltree_to_vazsonyi, kruskal_theorem_ltree_af. Qed.	Theorem	vazsonyi_theorem	Root	theories/vazsonyi_theorems.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "list_utils", "idx", "vec", "vtree", "ltree", "base", "vtree_embed", "ltree_embed", "statements", "conversions", "kruskal_theorems", "higman_theorems", "vec_notations", "vtree_notations", "af_notations" ]	[ "kruskal_ltree_to_vazsonyi", "kruskal_theorem_ltree_af", "vazsonyi_conjecture" ]	See statements.v for the statement of the "conjecture"	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
Y	:= sigT X.	Let	Y	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
T : nat → Y → Prop	:= λ n s, n = projT1 s.	Let	T	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_vtree (t : dtree X) : vtree Y.	Proof. induction t as [ n x v f ]. exact ⟨existT _ n x\|vec_set f⟩. Defined.	Definition	dtree_vtree	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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; auto. Qed.	Fact	dtree_vtree_fix	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_vtree_inj s t : dtree_vtree s = dtree_vtree t → s = t.	Proof. revert t; induction s as [ n x v IH ]; intros [ m y w ]. rewrite !dtree_vtree_fix, dtree_cons_inj. intros (? & H1 & H2); eq refl; simpl in *. apply eq_sigT_inj in H1 as (e & H1); eq refl; subst; clear e. f_equal; vec ext; apply IH. apply f_equal with (f := fun v => v⦃p⦄) in H2. now re...	Fact	dtree_vtree_inj	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree", "dtree_vtree_fix" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_vtree_wf t : wft T (dtree_vtree t).	Proof. unfold T. induction t. rewrite dtree_vtree_fix, wft_fix; simpl; split; auto. now intro; vec rew. Qed.	Fact	dtree_vtree_wf	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree", "dtree_vtree_fix" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_vtree_surj t' : wft T t' → { t \| dtree_vtree t = t' }.	Proof. unfold T. induction 1 as [ n (j,x) v H1 H2 IH2 ] using wft_rect. vec reif IH2 as (w & Hw). simpl in H1; subst j. exists ⟨x\|w⟩. rewrite dtree_vtree_fix; f_equal. now vec ext; vec rew. Qed.	Fact	dtree_vtree_surj	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree", "dtree_vtree_fix" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_vtree_vec_surj n (v : vec _ n) : vec_fall (wft T) v → { w \| vec_map dtree_vtree w = v }.	Proof. apply vec_cond_reif, dtree_vtree_surj. Qed.	Fact	dtree_vtree_vec_surj	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree", "dtree_vtree_surj" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
vtree_dtree t' Ht'	:= proj1_sig (dtree_vtree_surj t' Ht').	Definition	vtree_dtree	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree_surj" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_vtree_dtree t' Ht' : dtree_vtree (@vtree_dtree t' Ht') = t'.	Proof. apply (proj2_sig (dtree_vtree_surj t' Ht')). Qed.	Fact	dtree_vtree_dtree	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree", "dtree_vtree_surj", "vtree_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
vtree_dtree_vtree t H : vtree_dtree (dtree_vtree t) H = t.	Proof. apply dtree_vtree_inj; rewrite dtree_vtree_dtree; auto. Qed.	Fact	vtree_dtree_vtree	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree", "dtree_vtree_dtree", "dtree_vtree_inj", "vtree_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
vtree_dtree_fix n x (w : vec (dtree X) n) H : vtree_dtree ⟨existT _ n x\|vec_map dtree_vtree w⟩ H = ⟨x\|w⟩.	Proof. apply dtree_vtree_inj; rewrite dtree_vtree_dtree, dtree_vtree_fix; auto. Qed.	Fact	vtree_dtree_fix	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "dtree_vtree", "dtree_vtree_dtree", "dtree_vtree_fix", "dtree_vtree_inj", "vtree_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
Y	:= (sigT X).	Notation	Y	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
T n (y : Y)	:= n = projT1 y.	Let	T	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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.	Fact	T_empty	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
R' n (u v : Y)	:= match u, v with \| existT _ _ x, existT _ _ y => exists e f, @R n x↺e y↺f end.	Let	R'	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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); unfold T. + intros ((j,x),e); simpl in *; subst; exists x, eq_refl; auto. + intros x1 x2 ((i1,y1),e...	Fact	R'_afs	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "HR", "R'" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_afs_to_higman_af_at : af (dtree_product_embed R).	Proof. cut (afs (wft T) (dtree_product_embed R')). 2: { apply higman with k; eauto. } equiv with afs_iff_af_sub_rel. af rel morph (fun x y => vtree_dtree (proj1_sig x) (proj2_sig x) = y ). + intros t. induction t as [ n x v IH ]. assert (Hw : forall p, ∃ₜ t (Ht : wft _ t), vtree_dtree t ...	Lemma	higman_afs_to_higman_af_at	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "R'", "dtree_product_embed", "dtree_vtree_dtree", "dtree_vtree_fix", "dtree_vtree_inj", "dtree_vtree_vec_surj", "vtree_dtree", "vtree_dtree_fix", "vtree_dtree_vtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_dtree_afs_to_af : afs_higman_dtree → af_higman_dtree.	Proof. intros ? ? ? ?; apply higman_afs_to_higman_af_at; auto. Qed.	Theorem	higman_dtree_afs_to_af	conversions	theories/conversions/higman_dtree_afs_to_af.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "vtree_notations", "af_notations" ]	[ "af_higman_dtree", "afs_higman_dtree", "higman_afs_to_higman_af_at" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
unary_family n : Type	:= match n with \| 0 => unit \| 1 => X \| _ => Empty_set end.	Definition	unary_family	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
Y	:= unary_family.	Notation	Y	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "unary_family" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
T	:= (dtree Y).	Notation	T	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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.	Fixpoint	dtree_list	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
list_dtree l : T	:= match l with \| [] => ⟨tt\|∅⟩ \| x::l => ⟨x\|list_dtree l##∅⟩ end.	Fixpoint	list_dtree	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_list_dtree l : dtree_list (list_dtree l) = l.	Proof. induction l; simpl; f_equal; auto. Qed.	Fact	dtree_list_dtree	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "dtree_list", "list_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
list_dtree_list_not_needed t : list_dtree (dtree_list t) = t.	Proof. induction t as [ [ \| [ \| n ] ] x v IHv ]; simpl in *; try easy; f_equal. + now destruct x. + now vec invert v. + vec invert v as ? v; vec invert v. now rewrite IHv. Qed.	Fact	list_dtree_list_not_needed	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "dtree_list", "list_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_dtree : af_higman_dtree.		Hypothesis	higman_dtree	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "af_higman_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
Y	:= (unary_family X).	Notation	Y	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "unary_family" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
RY n : rel₂ (Y n)	:= match n with \| 1 => R \| _ => ⊤₂ end.	Let	RY	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "⊤₂" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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 ? ? <- <-. induction 1 as [ [\|[]] x t v p H IH \| [\|[]] x v y w H IH ]; simpl; auto; (...	Lemma	higman_lemma_af	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "RY", "dtree_list", "dtree_list_dtree", "dtree_product_embed", "higman_dtree", "list_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
higman_dtree_to_list : af_higman_dtree → af_higman_list.	Proof. intros ? ? ? ?; apply higman_lemma_af; auto. Qed.	Theorem	higman_dtree_to_list	conversions	theories/conversions/higman_dtree_to_list.v	[ "Coq", "Arith", "List", "Lia", "Utf8", "KruskalTrees", "tactics", "list_utils", "idx", "vec", "dtree", "base", "notations", "dtree_embed", "statements", "ListNotations", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "af_higman_dtree", "af_higman_list", "higman_lemma_af" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
btree_f_equal k n v w h h' : v = w → @btree_cons k n v h = @btree_cons k n w h'.	Proof. intros ->; f_equal; apply lt_pirr. Qed.	Fact	btree_f_equal	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "lt_pirr" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_bounded n	:= if le_lt_dec k n then Empty_set else unit.	Definition	dtree_bounded	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
X	:= dtree_bounded.	Notation	X	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "dtree_bounded" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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.	Definition	tt'	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
X_uniq n : ∀ a b : X n, a = b.	Proof. unfold X; destruct (le_lt_dec k n); intros [] []; auto. Qed.	Fact	X_uniq	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
btree_dtree (t : btree k) : dtree X	:= match t with \| btree_cons v h => ⟨tt' h\|vec_map btree_dtree v⟩ end.	Fixpoint	btree_dtree	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
btree_dtree_fix n v h : btree_dtree (@btree_cons k n v h) = ⟨tt' h\|vec_map btree_dtree v⟩.	Proof. reflexivity. Qed.	Fact	btree_dtree_fix	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "btree_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
btree_dtree_inj s t : btree_dtree s = btree_dtree t → s = t.	Proof. revert t; induction s as [ n v hv IH ]; intros [ m w hw ]; simpl. rewrite dtree_cons_inj. intros (e & H1 & H2); eq refl; simpl in *. apply btree_f_equal. vec ext. apply f_equal with (f := fun v => v⦃p⦄) in H2. rewrite !vec_prj_map in H2; auto. Qed.	Fact	btree_dtree_inj	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "btree_dtree", "btree_f_equal" ]	Hint Resolve lt_pirr : core.	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_btree_pwc (t : dtree X) : { r \| btree_dtree r = t }.	Proof. induction t as [ n x v IH ]. unfold X in x. case_eq (le_lt_dec k n); intros Hn E. + exfalso; rewrite E in x; destruct x. + vec reif IH as (w & Hw). exists (btree_cons w Hn); simpl; f_equal. * apply X_uniq. * now vec ext; vec rew. Qed.	Fact	dtree_btree_pwc	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "X_uniq", "btree_dtree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_btree t	:= proj1_sig (dtree_btree_pwc t).	Definition	dtree_btree	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "dtree_btree_pwc" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
btree_dtree_btree t : btree_dtree (dtree_btree t) = t.	Proof. apply (proj2_sig (dtree_btree_pwc t)). Qed.	Fact	btree_dtree_btree	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "btree_dtree", "dtree_btree", "dtree_btree_pwc" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_btree_dtree t : dtree_btree (btree_dtree t) = t.	Proof. apply btree_dtree_inj; now rewrite btree_dtree_btree. Qed.	Fact	dtree_btree_dtree	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "btree_dtree", "btree_dtree_btree", "btree_dtree_inj", "dtree_btree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_btree_fix n x (v : vec _ n) : { h : n < k \| dtree_btree ⟨x\|v⟩ = btree_cons (vec_map dtree_btree v) h }.	Proof. unfold X in x. case_eq (le_lt_dec k n); intros Hn E. + exfalso; rewrite E in x; destruct x. + exists Hn. apply btree_dtree_inj; rewrite btree_dtree_btree, btree_dtree_fix. f_equal. * apply X_uniq. * vec ext; vec rew. now rewrite btree_dtree_btree. Qed.	Fact	dtree_btree_fix	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "X_uniq", "btree_dtree_btree", "btree_dtree_fix", "btree_dtree_inj", "dtree_btree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
dtree_btree_morph t₁ t₂ : dtree_product_embed (λ _, ⊤₂) t₁ t₂ → R (dtree_btree t₁) (dtree_btree t₂).	Proof. induction 1 as [ n x v t p H IH \| n x v y w _ H IH ]. + destruct (dtree_btree_fix x v) as (Hn & ->). apply HR1 with (1 := IH). apply vec_in_iff_prj; exists p; vec rew; auto. + destruct (dtree_btree_fix x v) as (H1 & ->). destruct (dtree_btree_fix y w) as (H2 & ->). apply HR2, ...	Fact	dtree_btree_morph	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "dtree_btree", "dtree_btree_fix", "dtree_product_embed", "⊤₂" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
vazsonyi_conjecture_bounded_strong (f : nat → btree k) : ∃ₜ n, ∃ i j, i < j < n ∧ R (f i) (f j).	Proof. apply af_good_pair. cut (af (dtree_product_embed (fun n (_ _ : dtree_bounded k n) => True))). + af rel morph (fun x y => y = dtree_btree x). * intros y; exists (btree_dtree y); rewrite dtree_btree_dtree; auto. * intros t1 t2 ? ? -> ->; apply dtree_btree_morph. + apply higman_theorem w...	Theorem	vazsonyi_conjecture_bounded_strong	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "btree_dtree", "dtree_bounded", "dtree_btree", "dtree_btree_dtree", "dtree_btree_morph", "dtree_product_embed" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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.	Theorem	higman_dtree_to_vazsonyi_bounded	conversions	theories/conversions/higman_dtree_to_vazsonyi_bounded.v	[ "Coq", "Arith", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "ltree", "btree", "base", "notations", "le_lt_pirr", "dtree_embed", "ltree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations" ]	[ "Hk", "af_higman_dtree", "vazsonyi_conjecture_bounded", "vazsonyi_conjecture_bounded_strong" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
A	:= (λ n x, n = a x).	Notation	A	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
"⟨ x \| v ⟩ₐ"	:= (atree_cons x v).	Notation	⟨ x \| v ⟩ₐ	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	atree_dtree	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "atree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	dtree_atree	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "atree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	dtree_atree_dtree	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "atree_dtree", "dtree_atree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	atree_dtree_atree	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "atree_dtree", "dtree_atree" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	va_tree_eq	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "atree" ]	Writing the correspondence between { s : vtree X \| wft A s } <~~> atree is too complicated because of the wft A proofs part. The wft A part is required because otherwise, we do not have a corresponding atree when arities do not respect A. We prefer to describe this bijective c...	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	va_tree_eq_surj	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "va_tree_eq" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
va_tree_eq_wft s t : va_tree_eq s t → wft A s.	Proof. induction 1; split; auto. Qed.	Remark	va_tree_eq_wft	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "va_tree_eq" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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.	Remark	va_tree_eq_total	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "va_tree_eq" ]		https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71
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	va_tree_eq_invl	conversions	theories/conversions/higman_kruskal_dtree_to_atree.v	[ "Coq", "Arith", "Lia", "Utf8", "KruskalTrees", "tactics", "idx", "vec", "dtree", "vtree", "base", "dtree_embed", "vtree_embed", "atree_embed", "statements", "idx_notations", "vec_notations", "dtree_notations", "af_notations" ]	[ "va_tree_eq" ]	This is the critical inversion lemma	https://github.com/DmxLarchey/Kruskal-Theorems	e6b7c0c93bc4f68b7ef6af9fad64220111febf71

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Coq-Kruskal

Structured dataset from Kruskal-Theorems — Kruskal tree theorem formalization.

Source

Repository: https://github.com/DmxLarchey/Kruskal-Theorems
Commit: e6b7c0c93bc4f68b7ef6af9fad64220111febf71
Files: 27
License: mpl-2.0

Schema

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

Statistics

Entries: 167
With proof: 160 (95.8%)
With docstring: 33 (19.8%)
Libraries: 3

By type

Type	Count
Fact	47
Theorem	36
Definition	24
Let	14
Notation	13
Fixpoint	11
Inductive	8
Hypothesis	7
Lemma	5
Remark	2

Example

higman_theorem_dtree_afs : afs_higman_dtree.
Proof. apply veldman_vtree_upto_afs_to_higman_dtree_afs, veldman_theorem_vtree_upto. Qed.

type: Theorem | symbolic_name: higman_theorem_dtree_afs | theories/higman_theorems.v

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{coq_kruskal_dataset,
  title  = {Coq-Kruskal},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/DmxLarchey/Kruskal-Theorems, commit e6b7c0c93bc4},
  url    = {https://huggingface.co/datasets/phanerozoic/Coq-Kruskal}
}

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