hcju/isabelleps huggingface dataset:
forest_subE: fixes E' :: "(nat \<times> int \<times> nat) set" assumes "forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>" shows "E' \<subseteq> E"
Class_revI: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E" shows "b \<in> Class a"
intros_2: fixes dims :: "nat list"   and P :: "complex mat"   and U :: "complex mat" assumes "state_sig.is_quantum_predicate dims P" shows "state_sig.hoare_partial dims (adjoint U * P * U) (Utrans U) P"
A1b: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<^bold>\<not>\<P> x \<^bold>\<rightarrow> \<P> (\<^bold>\<rightharpoondown>x)) w)\<rfloor>"
dirichlet_prod'_one_left: fixes f :: "real \<Rightarrow> 'a"   and x :: "real" shows "dirichlet_prod' (\<lambda>n. if n = 1 then 1::'a else (0::'a)) f x = (if 1 \<le> x then f x else (0::'a))"
c_fas_def: shows "c_fas \<equiv> [floatarith.Minus (floatarith.Add (floatarith.Minus (floatarith.Var 1)) (floatarith.Mult (floatarith.Var 0) (floatarith.Add (floatarith.Add (floatarith.Num (Float 1 0)) (floatarith.Minus (floatarith.Var 0 ^\<^sub>e 2))) (floatarith.Minus (floatarith.Var 1 ^\<^sub>e 2))))), floatarith.Minus (floatarith.Add (floatarith.Var 0) (floatarith.Mult (floatarith.Var 1) (floatarith.Add (floatarith.Add (floatarith.Num (Float 1 0)) (floatarith.Minus (floatarith.Var 0 ^\<^sub>e 2))) (floatarith.Minus (floatarith.Var 1 ^\<^sub>e 2)))))]"
simps_2: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and v :: "'f"   and va :: "bool" shows "redT_updT ts (ThreadExists v va) = ts"
unitary_unit_col: fixes U :: "complex mat"   and n :: "nat"   and i :: "nat" assumes "unitary U"   and "dim_col U = n"   and "i < n" shows "\<parallel>col U i\<parallel> = 1"
hn_refineI: fixes m :: "'a nres"   and \<Gamma> :: "assn"   and c :: "'b Heap"   and \<Gamma>' :: "assn"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "nofail m \<Longrightarrow> <\<Gamma>> c <\<lambda>r. \<Gamma>' * (\<exists>\<^sub>Ax. R x r * \<up> (RETURN x \<le> m))>\<^sub>t" shows "hn_refine \<Gamma> c \<Gamma>' R m"
disj_conj_distrib: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x OR y AND z = (x OR y) AND (x OR z)"
inter_spec_def: fixes sys :: "('j, 'k set, 'l) sa_rec_scheme"   and ba :: "('m, 'k set, 'n) igba_rec_scheme" shows "inter_spec sys ba \<equiv> ASSERT (sa sys) \<bind> (\<lambda>_. ASSERT (finite ((g_E sys)\<^sup>* `` g_V0 sys)) \<bind> (\<lambda>_. ASSERT (igba ba) \<bind> (\<lambda>_. ASSERT (finite ((g_E ba)\<^sup>* `` g_V0 ba)) \<bind> (\<lambda>_. SPEC (\<lambda>(G, project). igb_graph G \<and> finite ((g_E G)\<^sup>* `` g_V0 G) \<and> (\<forall>r. (\<exists>r'. igb_graph.is_acc_run G r' \<and> r = project \<circ> r') = (graph_defs.is_run sys r \<and> sa_L sys \<circ> r \<in> igba.lang ba)))))))"
subst_atm_mset_list_Cons: fixes A :: "'a multiset"   and As :: "'a multiset list"   and \<sigma> :: "'s" shows "(A # As) \<cdot>aml \<sigma> = A \<cdot>am \<sigma> # As \<cdot>aml \<sigma>"
assms_2: fixes x :: "'a" assumes "nofail S'"   and "inres S x" shows "inres S' x"
isomorphicE: fixes a :: "('a S.E.term, 'a) S.arr"   and a' :: "('a S.E.term, 'a) S.arr" assumes "S.isomorphic a a'" obtains f :: "('a S.E.term, 'a) S.arr" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>S a'\<guillemotright> \<and> S.iso f"
cB: shows "convex_hull B \<subseteq> carrier_vec n"
domain: shows "Domainp pcr_unit = (\<lambda>x. \<exists>y. x = y \<and> y \<in> {0..1})"
While: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and b :: "bexp"   and C :: "com" assumes "\<turnstile>\<^sub>3\<^sub>' {\<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b True s} C { (\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1}" shows "\<turnstile>\<^sub>3\<^sub>' {(\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1} WHILE b DO C { \<lambda>(s, n). P (s, n) \<and> lmaps_to_axpr b False s}"
subst_psubst_idEnv: fixes ys :: "'varSort"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and y :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "X #[Y / y]_ys = X #[idEnv [y \<leftarrow> Y]_ys]"
elims_2: fixes x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "Derivation x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = ab__" and "xa = []" and "xb = b__" and "ab__ = b__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<exists>x. Derives1 a (fst d) (snd d) x \<and> Derivation x D b"
sams_lea2_suma2__lea123: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a"   and G' :: "'a"   and H' :: "'a"   and I' :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong D' E' F' D E F"   and "Tarski_neutral_dimensionless.LeA Bet Cong G H I G' H' I'"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A' B' C' D' E' F' G' H' I'" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C A' B' C'"
filter_by_key_merge_all_sequences: fixes k :: "'k"   and xs :: "'a list" shows "filter (\<lambda>x. key x = k) (local.merge_all (local.sequences xs)) = filter (\<lambda>x. key x = k) xs"
Jmp: fixes G :: "prog"   and s :: "st"   and j :: "jump"   and n :: "nat" shows "G\<turnstile>Norm s \<midarrow>Jmp j\<midarrow>n\<rightarrow> (Some (Jump j), s)"
qVar_alpha_iff: fixes xs :: "'c"   and x :: "'d"   and X' :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "(qVar xs x #= X') = (X' = qVar xs x)"
b: shows "xs @ x # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ 0 # anything = xs @ x # 0 \<up> (max (n + 3) (max fft gft) - n) @ anything"
asym: fixes b :: "'c multiset"   and a :: "'c multiset" assumes "b \<subset># a"   and "a \<subset># b" shows "False"
gfp_unfold: fixes f :: "'c \<Rightarrow> 'c" assumes "mono f" shows "gfp f = f (gfp f)"
numadd: fixes bs :: "int list"   and t :: "Cooper.num"   and s :: "Cooper.num" shows "Inum bs (numadd t s) = Inum bs (Add t s)"
remove_comp_val_process: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and y :: "'c" assumes "portfolio pf1" shows "\<forall>n w. val_process Mkt (qty_rem_comp pf1 y) n w = val_process Mkt pf1 n w - prices Mkt y n w * pf1 y (Suc n) w"
llist_all2_LCons2: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and y :: "'f"   and ys :: "'f llist" shows "llist_all2 P xs (LCons y ys) = (\<exists>x xs'. xs = LCons x xs' \<and> P x y \<and> llist_all2 P xs' ys)"
list_sum_distrib_aux: fixes xs :: "'a list"   and n :: "'a" shows "sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs"
const_name_simps_2: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and t :: "'a" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "pre_term.is_const unconst t" shows "const (pre_term.const_name unconst t) = t"
state_to_strips_state_dom_element_iff: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c"   and a :: "'d"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>" shows "((v, a) \<in> dom (\<phi>\<^sub>S \<Psi> s)) = (v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> s v \<noteq> None \<and> a \<in> \<R>\<^sub>+ \<Psi> v)"
indep_events_def: fixes A :: "'c \<Rightarrow> 'a set"   and I :: "'c set" shows "indep_events A I = (A ` I \<subseteq> events \<and> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))))"
Dom_Diagonalize_rep: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "Dom \<^bold>\<lfloor>\<F>C.rep f\<^bold>\<rfloor> = \<^bold>\<lfloor>\<F>C.rep (\<F>C.dom f)\<^bold>\<rfloor>"
PlusE: fixes u :: "'a + 'b"   and A :: "'a set"   and B :: "'b set" assumes "u \<in> A <+> B" obtains x :: "'a" where "x \<in> A" and "u = Inl x"   | y :: "'b" where "y \<in> B" and "u = Inr y"
separatedin_open_sets: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S"   and "openin X T" shows "separatedin X S T = disjnt S T"
l_get_shadow_root_axioms: shows "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
intro: fixes get_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "get_component = l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_component get_root_node to_tree_order"   and "is_strongly_dom_component_safe = l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe get_root_node to_tree_order"   and "is_weakly_dom_component_safe = l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe get_root_node to_tree_order" shows "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node"
perron_frobenius_both: fixes A :: "((complex, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "real_non_neg_mat A" shows "\<exists>v. eigen_vector A v (complex_of_real (HMA_Connect.spectral_radius A)) \<and> real_non_neg_vec v"
lv_ivl_def: fixes xrs :: "'b list"   and yrs :: "'b list" shows "lv_ivl xrs yrs = {zrs. list_all2 (\<le>) xrs zrs \<and> list_all2 (\<le>) zrs yrs}"
hyps: shows "mat\<^sub>h (A ^\<^sub>m k) = mat\<^sub>h A ^\<^sub>m k"
trg_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.hseq \<mu> \<nu>" shows "trg\<^sub>B (\<mu> \<star>\<^sub>B \<nu>) = trg\<^sub>B \<mu>"
AbGroup_inner_dirsum_el_decomp_in_nth: fixes Gs :: "'a set list"   and n :: "nat"   and x :: "'a" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "n < length Gs"   and "x \<in> Gs ! n" shows "\<Oplus>Gs\<leftarrow> x = (replicate (length Gs) (0::'a))[n := x]"
abs: shows "abs.strong_low_bisim_mm \<R>\<^sub>A"
usubstappt_const_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and f :: "char" assumes "usubstappt \<sigma> U (Const f) \<noteq> undeft" shows "SConst \<sigma> f = undeft \<or> (\<exists>r. SConst \<sigma> f = Aterm r \<and> FVT r \<inter> U = {})"
intros: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma> {c} \<Gamma>'"   and "mds_consistent mds \<Gamma>"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'\<^esub> \<langle>c, mds, mem\<^sub>2\<rangle>"
bisims1_refl: fixes P :: "'a prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and xs :: "'addr val list" shows "P,es,h \<turnstile> (es, xs) [\<leftrightarrow>] ([], xs, 0, None)"
rel: fixes xa :: "('r option \<times> 's) option \<times> ('r option \<times> 's) option"   and s1 :: "'s"   and s2 :: "'s"   and x :: "'c" assumes "xa \<in> set_pmf (joint s1 s2 x)"   and "S s1 s2"   and "\<not> stop s2" shows "case xa of (x, xa) \<Rightarrow> ord_option (\<lambda>(r1, s1') (r2, s2'). le_option r2 r1 \<and> S s1' s2' \<and> (r2 = None \<and> r1 \<noteq> None) = stop s2') x xa"
length_inorder: fixes t :: "('b \<times> 'c) tree" shows "length (Tree2.inorder t) = size t"
constrains_strongest_rhs: fixes F :: "'c program"   and A :: "'c set" shows "F \<in> A co strongest_rhs F A"
induct: fixes P :: "('b common_primitive negation_type \<Rightarrow> bool) \<Rightarrow> 'b common_primitive match_expr \<Rightarrow> bool"   and a0 :: "'b common_primitive negation_type \<Rightarrow> bool"   and a1 :: "'b common_primitive match_expr" assumes "\<And>uu_. P uu_ MatchAny"   and "\<And>disc a. P disc (Match a)"   and "\<And>disc a. P disc (MatchNot (Match a))"   and "\<And>disc v. P disc (MatchNot v) \<Longrightarrow> P disc (MatchNot (MatchNot v))"   and "\<And>disc v va. P disc (MatchAnd v va) \<Longrightarrow> P disc (MatchNot (MatchAnd v va))"   and "\<And>disc. P disc MatchAny \<Longrightarrow> P disc (MatchNot MatchAny)"   and "\<And>disc m1 m2. \<lbrakk>P disc m1; P disc m2\<rbrakk> \<Longrightarrow> P disc (MatchAnd m1 m2)" shows "P a0 a1"
min_gallery_least_length: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "chamber D"   and "C \<noteq> D" shows "min_gallery (C # (ARG_MIN length Cs. gallery (C # Cs @ [D])) @ [D])"
ipresIGWlsAbs_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'd, 'e, 'j, 'k, 'l, 'g, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGWlsAbs hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresWlsAbs Delta hA MOD"
trans_joinable: fixes R :: "('b \<times> 'b) set" assumes "trans R"   and "confluent R" shows "trans (joinable R)"
inject: fixes x :: "'a \<Rightarrow> 'b \<times> 'a"   and ya :: "'a \<Rightarrow> 'b \<times> 'a" shows "(State x = State ya) = (x = ya)"
square_free_m_factor_2: fixes f :: "int poly"   and g :: "int poly" assumes "p.square_free_m (f * g)" shows "p.square_free_m g"
uniform_discrete_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.metric_space dist uniformity open" shows "metric_space.uniform_discrete dist S = (\<exists>e>0. \<forall>x\<in>S. \<forall>y\<in>S. dist x y < e \<longrightarrow> x = y)"
evaln_elim_cases_16: fixes G :: "prog"   and s :: "st"   and c1 :: "stmt"   and c2 :: "stmt"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (c1;; c2)\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains s1 :: "state" where "x = \<diamondsuit>" and "G\<turnstile>Norm s \<midarrow>c1\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>c2\<midarrow>n\<rightarrow> s'"
block_set_nempty_imp_block_ex: assumes "\<B>\<^sup>+ \<noteq> {#}" shows "\<exists>bl. bl \<in># \<B>\<^sup>+"
prj0_simps_arr: fixes f :: "'a"   and g :: "'a" shows "arr (prj0 f g) = cospan f g"
map: fixes f :: "'b \<Rightarrow> 'a"   and x :: "unit \<Rightarrow> 'b" shows "map_future f (future.fork x) = future.fork (f \<circ> x)"
istopology_def: fixes L :: "'d set \<Rightarrow> bool" shows "istopology L \<equiv> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>\<K>. (\<forall>K\<in>\<K>. L K) \<longrightarrow> L (\<Union> \<K>))"
inv_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "category C"   and "category.iso C f"   and "partial_magma.in_hom C f a b" shows "partial_magma.in_hom C (category.inv C f) b a"
prob_space_exponential_escape_rate: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'d" assumes "transition_rates R" shows "prob_space (exponential (transition_rates.escape_rate R x))"
msb_word_iff_bit: fixes w :: "'b word" shows "msb w = bit w (LENGTH('b) - Suc 0)"
restrict_ext: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "restrict f A = restrict g A"
exhaust: fixes y :: "ValD" obtains "y = \<bottom>"   | cfun :: "ValD \<rightarrow> ValD" where "y = ValF\<cdot>cfun"   | "y = ValTT"   | "y = ValFF"   | nat :: "nat" where "y = ValN\<cdot>nat"
span_is_subset: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and S :: "'g set"   and N :: "'g set" assumes "module R M"   and "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "module.span R M S \<subseteq> N"
inj_on_xor: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "inj_on ((\<oplus>\<^bsub>L\<^esub>) x) (carrier L)"
le_fun_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
reachable_closed_dfs: fixes xs :: "'a list" assumes "list_all isNode xs" shows "nodeAbs ` reachable (set xs) \<subseteq> nodeAbs ` set_of (dfs empt xs)"
perm_3: fixes pi :: "(name \<times> name) list"   and x1 :: "pi" shows "pi \<bullet> \<tau>.x1 = \<tau>.(pi \<bullet> x1)"
summable_geometric: fixes c :: "'b" assumes "norm c < 1" shows "summable ((^) c)"
forall_M_small_big: fixes h :: "'a"   and f :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and xs :: "'c list"   and h' :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "h \<turnstile> forall_M f xs \<rightarrow>\<^sub>h h'"   and "P h"   and "\<And>h h' x. \<lbrakk>x \<in> set xs; h \<turnstile> f x \<rightarrow>\<^sub>h h'; P h\<rbrakk> \<Longrightarrow> P h'" shows "P h'"
gle_e: shows "\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball (0::'a) B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar> < e"
cring_def: fixes R :: "('a, 'b) ring_scheme" shows "cring R \<equiv> ring R \<and> Group.comm_monoid R"
max_wp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "maximal (wp (Apply f))"
permutes_imp_bij: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "bij_betw p S S"
tllist_all2_reflI: fixes xs :: "('g, 'h) tllist"   and P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>x. x \<in> tset xs \<Longrightarrow> P x x"   and "tfinite xs \<Longrightarrow> Q (terminal xs) (terminal xs)" shows "tllist_all2 P Q xs xs"
T_TAbs: fixes T\<^sub>1 :: "POPLmarkRecord.type"   and \<Gamma> :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "TVarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2" shows "\<Gamma> \<turnstile> (\<lambda><:T\<^sub>1. t\<^sub>2) : (\<forall><:T\<^sub>1. T\<^sub>2)"
path_image_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "path_image (reversepath g) = path_image g"
\<X>_is_prefix: fixes t :: "symbol \<times> character list"   and k :: "nat" assumes "t \<in> \<X> k" shows "is_prefix (snd t) (drop k Doc)"
sscan_scons_snth: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b stream"   and i :: "nat" shows "(a ## sscan f xs a) !! i = fold f (stake i xs) a"
subspace_def: fixes S :: "'a set" shows "ideal.subspace S = ((0::'a) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c * x \<in> S))"
telescope: fixes n :: "nat" shows "F (Suc n) \<subseteq> F n"
extensional_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "\<gamma>\<^sub>T\<^sub>R K \<phi> = \<gamma>\<^sub>T\<^sub>R K \<psi>"
is_interval_ball_real: fixes a :: "real"   and b :: "real" shows "is_interval (ball a b)"
subgroup_card_dvd_group_ord: fixes H :: "'a set" assumes "subgroup H G" shows "card H dvd order G"
sigma_sets_mono': fixes A :: "'d set set"   and B :: "'d set set"   and X :: "'d set" assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
assms_2: shows "zs \<noteq> xs \<or> z \<noteq> x"
rebalance_last_tree_height: fixes t :: "'a btree"   and sub :: "'a btree"   and ts :: "('a btree \<times> 'a) list"   and list :: "('a btree \<times> 'a) list"   and sep :: "'a"   and k :: "nat" assumes "height t = height sub"   and "ts = list @ [(sub, sep)]" shows "height (rebalance_last_tree k ts t) = height (Node ts t)"
false_eqvt: fixes pi :: "('d \<times> 'd) list" shows "pi \<bullet> False = False"
gcd_0_nat: fixes x :: "nat" shows "gcd x 0 = x"
subst_cls_add_mset: fixes L :: "'a literal"   and C :: "'a literal multiset"   and \<sigma> :: "'s" shows "add_mset L C \<cdot> \<sigma> = add_mset (L \<cdot>l \<sigma>) (C \<cdot> \<sigma>)"
measure_uniform_count_measure: fixes A :: "'a set"   and X :: "'a set" assumes "finite A"   and "X \<subseteq> A" shows "Sigma_Algebra.measure (uniform_count_measure A) X = real (card X) / real (card A)"
foldl_mono: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and y :: "'b"   and y' :: "'b"   and xs :: "'a list" assumes "\<And>x y y'. y < y' \<Longrightarrow> f y x < f y' x"   and "y < y'" shows "foldl f y xs < foldl f y' xs"
witOpS_def: fixes s :: "'sort"   and delta :: "'opSym"   and R :: "('sort \<times> 'sort) set" shows "witOpS s delta R \<equiv> wlsOpS delta \<and> stOf delta = s \<and> liftAll (\<lambda>s'. (s', s) \<in> R) (arOf delta) \<and> liftAll (\<lambda>(us, s'). (s', s) \<in> R) (barOf delta)"
bex_UN: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>z\<in>\<Union> (B ` A). P z) = (\<exists>x\<in>A. \<exists>z\<in>B x. P z)"
square_diff_one_factored: fixes x :: "'a" shows "x * x - (1::'a) = (x + (1::'a)) * (x - (1::'a))"
simps_1: fixes \<xi> :: "nat \<Rightarrow> 'b set" shows "\<xi> \<Turnstile>\<^sub>c true\<^sub>c = True"
disj_wf_def: fixes r :: "('b \<times> 'b) set" shows "disj_wf r = (\<exists>T n. (\<forall>i<n. wf (T i)) \<and> r = \<Union> (T ` {..<n}))"
grid_single_level: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and d :: "nat" assumes "p \<in> grid b ds"   and "d < length b" shows "lv b d \<le> lv p d"
mod_eq_dvd_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "(modulo a c = modulo b c) = dvd.dvd times c (minus a b)"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "int \<times> int" assumes "bezw x xa = y" obtains xb :: "nat" and ya :: "nat" where "x = xb" and "xa = ya" and "y = (if ya = 0 then (1, 0) else (snd (bezw ya (xb mod ya)), fst (bezw ya (xb mod ya)) - snd (bezw ya (xb mod ya)) * int (xb div ya)))"
assms_2: shows "\<mu> \<star> a \<noteq> null"
assms: shows "\<C> \<turnstile> es : tn _> tm"