hcju/isabelleps huggingface dataset:
split_sel: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'e"   and sum :: "'f + 'g" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((sum = Inl (projl sum) \<longrightarrow> P (f1 (projl sum))) \<and> (sum = Inr (projr sum) \<longrightarrow> P (f2 (projr sum))))"
poly_condD_1: fixes p :: "int poly" assumes "poly_cond p" shows "irreducible p"
stutter_extend_ex_def: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "stutter_extend_ex ex \<equiv> \<lambda>x. case x of None \<Rightarrow> id | Some a \<Rightarrow> ex a"
UPFDefs_25: fixes S :: "'a set"   and p :: "'a \<Rightarrow> 'b decision option" shows "S \<triangleleft> p \<equiv> \<lambda>x. if x \<in> S then p x else \<bottom>"
stabilizer_def: fixes m :: "'a set" shows "conjP.stabilizer m = {g \<in> carrier (G\<lparr>carrier := P\<rparr>). conjugation_action (p ^ a) g m = m}"
assms_12: shows "getC state \<noteq> [opposite (getCl state)]"
ag_eq_sol2: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<in> carrier M"   and "x \<in> carrier M"   and "b \<in> carrier M"   and "x \<plusminus> a = b" shows "x = b \<plusminus> -\<^sub>a a"
listrel_sym: fixes r :: "('a \<times> 'a) set" assumes "sym r" shows "sym (listrel r)"
intro: fixes M1 :: "'d measure"   and M2 :: "'e measure" assumes "sigma_finite_measure M1"   and "sigma_finite_measure M2" shows "pair_sigma_finite M1 M2"
subformulas\<^sub>\<mu>\<^sub>\<nu>_disjoint: fixes \<phi> :: "'a ltln" shows "subformulas\<^sub>\<mu> \<phi> \<inter> subformulas\<^sub>\<nu> \<phi> = {}"
inj1: shows "\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>"
minim_least: fixes B :: "'b set"   and b :: "'b" assumes "B \<subseteq> Field s"   and "b \<in> B" shows "(s.minim B, b) \<in> s"
visits_base: fixes target :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option" shows "visits target e s r [] = (s = target)"
graphI: fixes x :: "'c"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "x \<in> F" shows "(x, f x) \<in> graph F f"
lformula_NFAND: fixes \<Phi> :: "('a, 'k) aformula set" assumes "finite \<Phi>" shows "lformula (NFAND \<Phi>) = Ball \<Phi> lformula"
card_number_partitions_with_atmost_k_parts: fixes n :: "nat"   and x :: "nat" shows "card {N. number_partition n N \<and> size N \<le> x} = Partition (n + x) x"
sorted_on_single: fixes r :: "('a \<times> 'a) set"   and x :: "'a" assumes "Linear_order r" shows "sorted_on r [x] = (x \<in> Field r)"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Can (t \<^bold>\<cdot> u) = (Can t \<and> Can u \<and> Dom t = Cod u)"
star_left_induct_mult: fixes star :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.left_kleene_algebra star one times sup less_eq less bot"   and "less_eq (times x y) y" shows "less_eq (times (star x) y) y"
abs_uexpr_def: fixes u :: "('e, 'f) uexpr" shows "\<bar>u\<bar> = uop abs u"
aristo_conversion1: fixes A :: "'b set"   and B :: "'b set" assumes "A E B" shows "B E A"
funcset_mem: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and x :: "'d" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
strict_prefixI: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys"   and "xs \<noteq> ys" shows "strict_prefix xs ys"
unit_not_pair: fixes A :: "'b trm"   and B :: "'b trm" shows "Unit \<noteq> SimplyTyped.Pair A B"
strong_induct: fixes x1 :: "(Vars.var \<times> exp) list"   and x2 :: "exp"   and x3 :: "Vars.var list"   and x4 :: "(Vars.var \<times> exp) list"   and x5 :: "exp"   and P :: "'a \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Vars.var list \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> bool"   and c :: "'a" assumes "x1 : x2 \<Down>\<^bsub>x3\<^esub> x4 : x5"   and "\<And>\<Gamma> x e L c. P c \<Gamma> (Lam [x]. e) L \<Gamma> (Lam [x]. e)"   and "\<And>y \<Gamma> e x L \<Delta> \<Theta> z e' c. \<lbrakk>{atom y} \<sharp>* c; atom y \<sharp> (\<Gamma>, e, x, L, \<Delta>, \<Theta>, z); \<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : Lam [y]. e'; \<And>c. P c \<Gamma> e L \<Delta> (Lam [y]. e'); \<Delta> : e'[y::=x] \<Down>\<^bsub>L\<^esub> \<Theta> : z; \<And>c. P c \<Delta> e'[y::=x] L \<Theta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (App e x) L \<Theta> z"   and "\<And>\<Gamma> x e L \<Delta> z c. \<lbrakk>map_of \<Gamma> x = Some e; delete x \<Gamma> : e \<Down>\<^bsub>x # L\<^esub> \<Delta> : z; \<And>c. P c (delete x \<Gamma>) e (x # L) \<Delta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) L ((x, z) # \<Delta>) z"   and "\<And>\<Delta> \<Gamma> L body \<Theta> z c. \<lbrakk>atom ` domA \<Delta> \<sharp>* (\<Gamma>, L); \<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z; \<And>c. P c (\<Delta> @ \<Gamma>) body L \<Theta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (Terms.Let \<Delta> body) L \<Theta> z"   and "\<And>\<Gamma> b L c. P c \<Gamma> (Bool b) L \<Gamma> (Bool b)"   and "\<And>\<Gamma> scrut L \<Delta> b e\<^sub>1 e\<^sub>2 \<Theta> z c. \<lbrakk>\<Gamma> : scrut \<Down>\<^bsub>L\<^esub> \<Delta> : Bool b; \<And>c. P c \<Gamma> scrut L \<Delta> (Bool b); \<Delta> : (if b then e\<^sub>1 else e\<^sub>2) \<Down>\<^bsub>L\<^esub> \<Theta> : z; \<And>c. P c \<Delta> (if b then e\<^sub>1 else e\<^sub>2) L \<Theta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (scrut ? e\<^sub>1 : e\<^sub>2) L \<Theta> z" shows "P c x1 x2 x3 x4 x5"
homotopic_with_id2: fixes X :: "'b topology"   and g :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x" shows "homotopic_with (\<lambda>x. True) X X (g \<circ> f) id"
trms\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('e, 'f) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
sup_assoc: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
list_eq2_equiv: fixes l :: "'b list"   and r :: "'b list" shows "(l \<approx> r) = l \<approx>2 r"
chamber_system_def: fixes X :: "'c set set" assumes "ChamberComplex X" shows "ChamberComplex.chamber_system X \<equiv> {C. SimplicialComplex.maxsimp X C}"
language_alphabet: fixes A :: "('c, 'd) nba" shows "language A \<subseteq> streams (alphabet A)"
chain_def: fixes Y :: "nat \<Rightarrow> 'd upred" shows "chain Y = (Y 0 = false \<and> (\<forall>i. Y (Suc i) \<sqsubseteq> Y i))"
preserves_arr: fixes f :: "'b" assumes "D.arr f" shows "D.arr (D.R f)"
simplePath_length_less_V: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "u \<in> V"   and "isSimplePath u p v" shows "length p < card V"
sfirst_Stream: fixes P :: "'c stream \<Rightarrow> bool"   and s :: "'c"   and x :: "'c stream" shows "sfirst P (s ## x) = (if P (s ## x) then 0 else eSuc (sfirst P x))"
inject_3: fixes x41 :: "'b formula"   and x42 :: "'b formula"   and y41 :: "'b formula"   and y42 :: "'b formula" shows "(x41 \<^bold>\<and> x42 = y41 \<^bold>\<and> y42) = (x41 = y41 \<and> x42 = y42)"
pair_measureI: fixes x :: "'b set"   and A :: "'b measure"   and y :: "'c set"   and B :: "'c measure" assumes "x \<in> sets A"   and "y \<in> sets B" shows "x \<times> y \<in> sets (A \<Otimes>\<^sub>M B)"
ring_iso_trans: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and Q :: "('g, 'h) ring_scheme" assumes "R \<simeq> S"   and "S \<simeq> Q" shows "R \<simeq> Q"
fold_map: fixes g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "fold g (map f xs) = fold (g \<circ> f) xs"
prems_3: shows "set A \<subseteq> set (ops @ [a])"
empty_iff: fixes c :: "'d" shows "(c \<in> {}) = False"
smultfunset_def: fixes smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set" shows "aezfun_scalar_mult.smultfunset smult G FH \<equiv> {f. \<forall>a. \<forall>g\<in>G. \<forall>x\<in>FH. f (a \<delta>\<delta> g * x) = smult (a \<delta>\<delta> g) (f x)}"
elims_3: fixes x :: "ct"   and xa :: "ct" assumes "\<not> ct_eq x xa" obtains t :: "t" and t' :: "t" where "x = TSome t" and "xa = TSome t'" and "t \<noteq> t'"
seq_nos_incs: fixes i :: "nat" shows "ptoy i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). nos_inc \<xi> \<xi>')"
nat_pow_closed: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier (UP R)" shows "[n] \<cdot>\<^bsub>UP R\<^esub> x \<in> carrier (UP R)"
has_dom_def: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set" shows "has_dom f S \<equiv> \<forall>s. s \<notin> S \<longrightarrow> f s = s"
comp_cod_arr: fixes f :: "nat"   and b :: "nat" assumes "J.arr f"   and "J.cod f = b" shows "J.comp b f = f"
nonpos_Ints_subset_Ints: shows "\<int>\<^sub>\<le>\<^sub>0 \<subseteq> \<int>"
pbeta_Upd': fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and f' :: "Label \<Rightarrow> sterm option"   and L :: "char list set"   and t :: "sterm"   and t' :: "sterm"   and l :: "Label" assumes "Obj f T => Obj f' T"   and "finite L"   and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> t' = \<sigma>[s,p] t'')"   and "l \<in> dom f"   and "lc (Obj f T)"   and "body t" shows "Upd (Obj f T) l t => Obj (f'(l \<mapsto> t')) T"
nn_integral_stream_space: fixes f :: "'a stream \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (stream_space M)" shows "integral\<^sup>N (stream_space M) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ X. f (x ## X) \<partial>stream_space M \<partial>M"
min_gallery_least_length: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D" shows "SimplicialComplex.min_maxsimpchain X (C # (ARG_MIN length Cs. SimplicialComplex.maxsimpchain X (C # Cs @ [D])) @ [D])"
wf_design_iff: fixes bl :: "'b set" assumes "bl \<in># \<B>'" shows "design \<V>' \<B>' = (bl \<subseteq> \<V>' \<and> finite \<V>' \<and> bl \<noteq> {})"
dist_along_geodesic_wrt_endpoint: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and u :: "'b"   and v :: "'b" assumes "geodesic_segment_between G x y"   and "u \<in> G"   and "v \<in> G" shows "dist u v = \<bar>dist u x - dist v x\<bar>"
Rep_unit: fixes x :: "Product_Type.unit" shows "Product_Type.unit.Rep_unit x \<in> {True}"
5: shows "Blinfun ((*) (inverse z)) o\<^sub>L Blinfun ((*) (exp (Ln z))) = id_blinfun"
Arr_in_Hom_1: fixes t :: "'a local.term" assumes "Arr t" shows "t \<in> HHom (Src t) (Trg t)"
unit: fixes p :: "fm"   and q :: "fm"   and B :: "MIR.num list"   and d :: "int"   and bs :: "real list"   and i :: "int" assumes "qfree p"   and "unit p = (q, B, d)" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = (\<exists>x. Ifm (real_of_int x # bs) q) \<and> Inum (real_of_int i # bs) ` set B = Inum (real_of_int i # bs) ` set (\<beta> q) \<and> d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q (real_of_int i # bs) \<and> (\<forall>b\<in>set B. numbound0 b)"
drop_eq_map_nth: fixes n :: "nat"   and xs :: "'a list" shows "drop n xs = map ((!) xs) [n..<length xs]"
ap_tree_homomorphism: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "pure f \<diamondop> pure x = pure (f x)"
closure: shows "(S\<^sup>* O (R \<union> R') O S\<^sup>*) `` X \<subseteq> X"
C_def: shows "C = rat_of_int (2 ^ log_ceiling 2 m_up)"
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<tau>'\<sigma>'.map a = B (\<sigma>' a) (\<tau>' a)"
GroupByPresentationInducedFun_S_P_id: shows "GroupByPresentationInducedFun S P id"
hA: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) A = Av"
arr_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f \<and> partial_magma.arr (free_monoidal_category.comp C) g \<and> partial_magma.dom (free_monoidal_category.comp C) f = partial_magma.dom (free_monoidal_category.comp C) g \<and> partial_magma.cod (free_monoidal_category.comp C) f = partial_magma.cod (free_monoidal_category.comp C) g"   and "t \<in> f"   and "u \<in> g"   and "monoidal_language.Diagonalize C t = monoidal_language.Diagonalize C u" shows "f = g"
divide_rat: fixes a :: "int"   and b :: "int"   and c :: "int"   and d :: "int" shows "Fract a b / Fract c d = Fract (a * d) (b * c)"
cast_chunk_assemble_id: fixes n :: "nat"   and m :: "nat"   and x :: "'c word" assumes "n = LENGTH('b)"   and "m = LENGTH('c)"   and "n * 2 = m" shows "UCAST('b \<rightarrow> 'c) (UCAST('c \<rightarrow> 'b) x) OR (UCAST('b \<rightarrow> 'c) (UCAST('c \<rightarrow> 'b) (x >> n)) << n) = x"
extend_to_affine_basis_nonempty: fixes S :: "'a set"   and V :: "'a set" assumes "\<not> affine_dependent S"   and "S \<subseteq> V"   and "S \<noteq> {}" shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V"
rel_interior_substd_simplex_nonempty: fixes D :: "'a set" assumes "D \<noteq> {}"   and "D \<subseteq> Basis" obtains a :: "'a" where "a \<in> rel_interior (convex hull insert (0::'a) D)"
round_altdef: fixes x :: "'b" shows "round x = (if (1::'b) / (2::'b) \<le> frac x then \<lceil>x\<rceil> else \<lfloor>x\<rfloor>)"
abs_eq: fixes x :: "rat" shows "mau_of_rat x = Abs_mini_alg_unique (ma_of_rat x)"
subdomain_def: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "subdomain H R \<equiv> subcring H R \<and> subdomain_axioms H R"
euclidean_eqI: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and x :: "'b"   and y :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and sum :: "'b + 'c" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Inr x2 \<longrightarrow> P (f2 x2)))"
hash: shows "SIDE_GEN_ALGO (is_bounded_hashcode node_rel node_eq_impl node_hash_impl)"
image_eq: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on g A" shows "\<Sum> (g ` A) = sum g A"
br_iq_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br_iq \<delta> \<equiv> {lhs r |r. r \<in> \<delta> \<and> rhsq r = []}"
functor_axioms: shows "functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.L"
ctr_1: fixes n :: "enat"   and xs :: "'a llist" assumes "n = 0 \<or> lnull xs" shows "ltake n xs = []\<^sub>l"
match_sel_def: fixes simple_rule :: "'a simple_rule" shows "match_sel simple_rule = (case simple_rule of SimpleRule x1 x2 \<Rightarrow> x1)"
ActualConjE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>(\<phi> \<^bold>& \<psi>) in v]" shows "[\<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi> in v]"
assms: shows "<\<nu>x>x{y}.P' \<in> summands P"
Afix_least_below: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "ABinds \<Gamma>\<cdot>ae' \<sqsubseteq> ae'"   and "ae \<sqsubseteq> ae'" shows "Afix \<Gamma>\<cdot>ae \<sqsubseteq> ae'"
dist_sq_mtx_def: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" shows "dist A B = \<parallel>A - B\<parallel>"
cases: fixes a1 :: "'b set"   and a2 :: "'b \<Rightarrow> nat"   and a3 :: "'b \<Rightarrow> intv"   and a4 :: "'b \<Rightarrow> 'b \<Rightarrow> intv'"   and a5 :: "('b \<times> 'b) set" assumes "valid_region a1 a2 a3 a4 a5" obtains X\<^sub>0 :: "'b set" and X :: "'b set" and I :: "'b \<Rightarrow> intv" and r :: "('b \<times> 'b) set" and k :: "'b \<Rightarrow> nat" and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'" where "a1 = X" and "a2 = k" and "a3 = I" and "a4 = J" and "a5 = r" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}" and "refl_on X\<^sub>0 r" and "trans r" and "total_on X\<^sub>0 r" and "\<forall>x\<in>X. valid_intv (k x) (I x)" and "\<forall>x\<in>X. \<forall>y\<in>X. isGreater (I x) \<or> isGreater (I y) \<longrightarrow> valid_intv' (int (k y)) (int (k x)) (J x y)"
n2_type2: fixes t1 :: "'c bro"   and h :: "nat"   and t2 :: "'c bro"   and a :: "'c" assumes "t1 \<in> B h"   and "t2 \<in> Um h" shows "n2 t1 a t2 \<in> T (Suc h)"
scc_of_tail_neq_head: shows "H.scc_of (tail G a) \<noteq> H.scc_of (head G a)"
rep_eq: fixes x :: "'b ttree" shows "possible x = (\<lambda>xa. \<exists>xs. xa # xs \<in> Rep_ttree x)"
pc: shows "poly_cond (cf_pos_poly (poly_mult_rat r1 p2))"
conv_tail1_1: fixes s :: "'a"   and d :: "bool"   and \<alpha> :: "'b"   and t :: "'a"   and xs :: "(bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, (d, \<alpha>, t) # xs) \<in> conv ars" shows "(t, xs) \<in> conv ars"
sorted_list_of_set_1: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
assms_2: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>2"
gen_bg_impl_rel_eext_def_internal: fixes Rm :: "('f \<times> 'h) set"   and Rf :: "('e \<times> 'g set) set" shows "gen_bg_impl_rel_eext Rm Rf \<equiv> {(\<lparr>bgi_F = Fi, \<dots> = mi\<rparr>, \<lparr>bg_F = F, \<dots> = m\<rparr>) |Fi mi F m. (Fi, F) \<in> Rf \<and> (mi, m) \<in> Rm}"
varsPrefixValuation: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" assumes "isPrefix valuation1 valuation2" shows "vars valuation1 \<subseteq> vars valuation2"
general_U_def: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and p :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>" shows "general_U outcomes \<R> p = (if some_best outcomes \<R> \<approx>[\<R>] some_worst outcomes \<R> then 1 else my_U outcomes \<R> p)"
FreeGroup_Group: fixes S :: "'c set" shows "Group (FreeGroup S)"
minus_times_le_abs: fixes err :: "real"   and B :: "real" assumes "- 1 \<le> err"   and "err \<le> 1" shows "- (err * B) \<le> \<bar>B\<bar>"
evaluation_map_axioms: shows "evaluation_map (\<cdot>\<^sub>C) (\<cdot>) T \<alpha> \<iota> V"
degree_aform_uminus_aform: fixes X :: "'a \<times> 'a pdevs" shows "degree_aform (uminus_aform X) = degree_aform X"
I_def: shows "I \<equiv> \<lambda>X. X \<subseteq> carrier \<and> (\<forall>x\<in>X. x \<notin> cl (X - {x}))"
sup_aci_1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas_axioms: shows "l_put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas ShadowRootClass.type_wf"
bounds: fixes i :: "nat" assumes "p (i + 1) \<noteq> 0" shows "1 \<le> i \<and> i \<le> m - k"
distinct_1: fixes x2 :: "'b" shows "Null \<noteq> Ref x2"
to_fun_const: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_fun (monom P a 0) b = a"
minim_least: fixes B :: "'a set"   and b :: "'a" assumes "B \<subseteq> Field r"   and "b \<in> B" shows "(minim B, b) \<in> r"
times_div_less_eq_dividend: fixes n :: "nat"   and m :: "nat" shows "n * (m div n) \<le> m"
simps_5: fixes b :: "bexp"   and c :: "com" shows "rvars (WHILE b DO c) = vars b \<union> rvars c"
Units_pow_closed: fixes x :: "'a"   and d :: "nat" assumes "x \<in> Units \<G>" shows "x [^] d \<in> Units \<G>"
set_alternating_list2: fixes n :: "nat"   and s :: "'b"   and t :: "'b" assumes "order.greater_eq n 2" shows "set (alternating_list n s t) = {s, t}"
lem: fixes x :: "int"   and n :: "nat" assumes "x \<in> {1..2 ^ n - 1}" shows "- x mod 2 ^ n = 2 ^ n - x"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "domain_semigroup.ds_ord (antidomain_left_monoid.am_d a) (\<cdot>) x y"
fflip_vsv: fixes f :: "V" shows "vsv (fflip f)"
in_set_subseqs: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs \<in> set (subseqs ys)) = subseq xs ys"
openin_diff: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "openin U S"   and "closedin U T" shows "openin U (S - T)"
iT_Minus_card: fixes k :: "nat"   and I :: "nat set" shows "card (k \<ominus> I) = card (I \<down>\<le> k)"
simps_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Var x1 = Var y1) = (x1 = y1)"
assms: fixes k :: "'a" shows "f k (0::'c) (0::'d) = (0::'b)"
hseq_char: fixes \<nu> :: "'d"   and \<mu> :: "'d" shows "D.hseq \<nu> \<mu> = (D.arr \<mu> \<and> D.arr \<nu> \<and> src\<^sub>D \<nu> = trg\<^sub>D \<mu>)"
simps_1: fixes I :: "('e, 'f, 'd) interp"   and x :: "'d"   and v :: "(real, 'd) vec" shows "frechet I (Var x) v = (\<lambda>v'. v' \<bullet> axis x 1)"
hfref_weaken_pre: fixes P :: "'e \<Rightarrow> bool"   and P' :: "'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g Heap"   and h :: "'e \<Rightarrow> 'h nres"   and R :: "('e \<Rightarrow> 'f \<Rightarrow> assn) \<times> ('e \<Rightarrow> 'f \<Rightarrow> assn)"   and S :: "'h \<Rightarrow> 'g \<Rightarrow> assn" assumes "\<And>x. P x \<longrightarrow> P' x"   and "(f, h) \<in> [P']\<^sub>a R \<rightarrow> S" shows "(f, h) \<in> [P]\<^sub>a R \<rightarrow> S"
induct: fixes P :: "('b, 'c) arr \<Rightarrow> bool"   and arr :: "('b, 'c) arr" assumes "P Null"   and "\<And>x1 x2 x3. P (MkArr x1 x2 x3)" shows "P arr"
Kcknows_1: shows "subjective a \<^bold>C\<^bsub>as\<^esub> \<psi>"
conc_trans_additional_3: fixes A :: "'a nres"   and R :: "('a \<times> 'b) set"   and B :: "'b nres"   and C :: "'b nres" assumes "A \<le> \<Down> R B"   and "B \<le> \<Down> Id C" shows "A \<le> \<Down> R C"
subst_sends_dom_to_img: fixes v :: "'a"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<in> subst_domain \<theta>" shows "fv (Var v \<cdot> \<theta>) \<subseteq> range_vars \<theta>"
\<chi>1: shows "\<chi>1 \<in> fmla"