hcju/isabelleps huggingface dataset:
ring_hom_restrict: fixes f :: "'a \<Rightarrow> 'c"   and S :: "('c, 'd) ring_scheme"   and g :: "'a \<Rightarrow> 'c" assumes "f \<in> ring_hom R S"   and "\<And>r. r \<in> carrier R \<Longrightarrow> f r = g r" shows "g \<in> ring_hom R S"
intros_4: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n"   and m' :: "nat"   and n :: "nat"   and v :: "'var" assumes "(\<pi>, m) \<in> scp"   and "(\<pi>', m') \<in> scp"   and "n dcd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m via \<pi>' m'" shows "(\<pi>, n) \<in> scp"
b: shows "ID CENUM('b) = Some (enum_b, enum_all_b, enum_ex_b)"
pr_conv_3_to_1_lm: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec3" shows "pr_conv_3_to_1 f \<in> PrimRec1"
mem_val_alt_10_unchanged1: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 ((pa AND 68719476732) + 1) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 ((pa AND 68719476732) + 1) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
continuous_eval_fds: fixes f :: "'a fds"   and s :: "'a"   and A :: "'a set" assumes "conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "continuous (at s within A) (eval_fds f)"
someI: fixes x :: "char list \<Rightarrow> nat" assumes "bij x" shows "bij (SOME \<phi>. bij \<phi>)"
comp_in_homI': fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr f"   and "arr g"   and "local.dom f = a"   and "cod g = c"   and "local.dom g = cod f" shows "\<guillemotleft>C g f : a \<rightarrow> c\<guillemotright>"
fdemodalisation2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "( |x\<rangle> y \<le> ad (ad z)) = (x \<cdot> ad (ad y) \<le> ad (ad z) \<cdot> x)"
prems: shows "\<nexists>a l. a \<noteq> 0 \<and> l = 0 \<and> pCons c cs = pCons a l"
finfun_apply_inverse: fixes x :: "'d \<Rightarrow>f 'e" shows "Abs_finfun (($) x) = x"
gen_llength_cons_def: fixes n :: "enat"   and s :: "'s" shows "local.gen_llength_cons n s = n + llength (lunstream g s)"
mono_lex1: fixes s :: "('c \<times> 'c) set" shows "mono (\<lambda>r. r <*lex*> s)"
reduced_GB_subset_monic_dgrad_p_set: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m"   and "is_monomial_set F" shows "reduced_GB F \<subseteq> monic ` F"
steps_reaches: fixes xs :: "'a list" assumes "G'.steps xs" shows "G'.reaches (hd xs) (last xs)"
zero: fixes i :: "nat" assumes "i < nat n" shows "(deriv ^^ i) f z = 0"
lens_put_comm: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'f"   and u :: "'d" assumes "X \<bowtie> Y" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"
I_2: shows "fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! (snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! (fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! dfa_steps M 0 w)) = fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! dfa_steps M 0 w"
singleton_conv: fixes a :: "'c" shows "_Collect x (x = a) = {a}"
unit_factor_gcd: fixes a :: "'a"   and b :: "'a" shows "unit_factor (gcd a b) = (if a = (0::'a) \<and> b = (0::'a) then 0::'a else (1::'a))"
that: fixes v :: "'a" assumes "verts G = {v}" shows "thesis"
next_subseqs2_set_def: fixes v :: "'b"   and js :: "nat list" shows "next_subseqs2_set v js \<equiv> {(j, f (tail !! j) v) |j. j \<in> set js}"
size_hp_def: fixes h :: "'b hp" shows "Pairing_Heap_List2_Analysis.size_hp h = size_hps (hps h) + 1"
path_weight_def: fixes p :: "('c \<times> 'b \<times> 'c) list" shows "path_weight p \<equiv> sum_list (map (fst \<circ> snd) p)"
Res_def: fixes x1 :: "name"   and x2 :: "ccs" shows "\<lparr>\<nu>x1\<rparr>x2 \<equiv> Abs_ccs (Res_Rep ([x1].Rep_ccs x2))"
ma_plus: fixes r1 :: "mini_alg"   and r2 :: "mini_alg" shows "real_of r1 + real_of r2 = (if ma_compatible r1 r2 then real_of (ma_plus r1 r2) else Code.abort STR ''different base'' (\<lambda>_. real_of r1 + real_of r2))"
dynlookup_access_prop: fixes emh :: "ref_ty \<times> mhead"   and G :: "prog"   and accC :: "qtname"   and statT :: "ref_ty"   and sig :: "sig"   and dynC :: "qtname"   and dynM :: "qtname \<times> methd" assumes "emh \<in> mheads G accC statT sig"   and "dynlookup G statT dynC sig = Some dynM"   and "G,statT \<turnstile> dynC valid_lookup_cls_for is_static emh"   and "isrtype G statT"   and "wf_prog G" shows "G \<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
assms_1: shows "\<bar>x\<bar> \<le> pi"
select_convs_2: fixes access :: "acc_modi"   and static :: "bool"   and pars :: "vname list"   and resT :: "ty"   and more :: "'b" shows "resT \<lparr>access = access, static = static, pars = pars, resT = resT, \<dots> = more\<rparr> = resT"
A\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "SkolemizedAxiomsSetMcL" shows "x \<cdot> (y \<cdot> z) \<cong> (x \<cdot> y) \<cdot> z"
list_comb_snoc: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "app (term.list_comb app f xs) y = term.list_comb app f (xs @ [y])"
eq_\<I>_gpv_reflI: fixes \<I> :: "('b, 'c) \<I>"   and gpv :: "('a, 'b, 'c) gpv"   and A :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. x \<in> results_gpv \<I> gpv \<Longrightarrow> A x x"   and "\<I> \<turnstile>g gpv \<surd>" shows "eq_\<I>_gpv A \<I> gpv gpv"
simps_2: fixes xss :: "'b list list" shows "transpose ([] # xss) = transpose xss"
awalk_verts_drop_conv: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and u' :: "'a"   and n :: "nat" assumes "cas u p v" shows "awalk_verts u' (drop n p) = (if n < length p then drop n (awalk_verts u p) else [u'])"
div_minus1_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "divide a (uminus one) = uminus a"
preserves_cod_def: fixes G :: "('a, 'b, 'a, 'b) functor" shows "preserves_cod G \<equiv> \<forall>f\<in>Ar. G \<^bsub>\<o>\<^esub> (Cod f) = Cod\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
group_type_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set" assumes "group_design point_set block_collection groups" shows "group_design.group_type groups \<equiv> {(g, group_design.groups_of_size groups g) |g. g \<in> group_design.group_sizes groups}"
IH1: fixes C :: "int poly"   and Gs :: "int poly list" assumes "C =m \<Prod>\<^sub># (factors_of_factor_tree l)"   and "\<And>F. F \<in># factors_of_factor_tree l \<Longrightarrow> Mp F = F \<and> monic F"   and "hensel_main C l = Gs"   and "monic C"   and "pn.Mp C = C"   and "square_free_m C"   and "\<And>t f. \<lbrakk>t \<in> sub_trees l; factor_node_info t = f\<rbrakk> \<Longrightarrow> f = Mp (\<Prod>\<^sub># (factors_of_factor_tree t))" shows "pn.eq_m C (prod_list Gs) \<and> factors_of_factor_tree l = mset (map Mp Gs) \<and> (\<forall>G. G \<in> set Gs \<longrightarrow> monic G \<and> pn.Mp G = G)"
relabel_precond_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" shows "relabel_precond f l u \<equiv> u \<noteq> t \<and> (0::'capacity) < excess f u \<and> (\<forall>v. (u, v) \<in> cfE_of f \<longrightarrow> l u \<noteq> l v + 1)"
sym_group_carrier': fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p \<in> carrier (sym_group n)" shows "permutation p"
Entry_reachesE: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)"   and "invar g" obtains ns :: "'node list" where "g \<turnstile> Entry g-ns\<rightarrow>n" and "old.EntryPath g ns"
quotientI2: fixes x :: "'b"   and A :: "'b set"   and X :: "'b set"   and r :: "('b \<times> 'b) set" assumes "x \<in> A"   and "X = r `` {x}" shows "X \<in> A // r"
assms: shows "\<exists>B>0. \<exists>c. \<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2"
one_final_but_antimono: fixes g :: "graph"   and E :: "(nat \<times> nat) set"   and E' :: "(nat \<times> nat) set" assumes "one_final_but g E"   and "E \<subseteq> E'" shows "one_final_but g E'"
conditional_entropy_generic_eq: fixes S :: "'e measure"   and T :: "'f measure"   and Y :: "'a \<Rightarrow> 'f"   and Py :: "'f \<Rightarrow> real"   and X :: "'a \<Rightarrow> 'e"   and Pxy :: "'e \<times> 'f \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space T \<Longrightarrow> 0 \<le> Py x"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)" shows "conditional_entropy b S T X Y = - (LINT (x, y)|(S \<Otimes>\<^sub>M T). Pxy (x, y) * log b (Pxy (x, y) / Py y))"
hm_hashcode_eq: fixes j :: "nat"   and l :: "('a \<times> 'b) list list"   and i :: "nat"   and ht :: "('a, 'b) hashtable"   and h :: "heap \<times> nat set" assumes "j < length (l ! i)"   and "i < length l"   and "h \<Turnstile> is_hashtable l ht" shows "bounded_hashcode_nat (length l) (fst (l ! i ! j)) = i"
transpose_mat_def: fixes A :: "'d mat" shows "A\<^sup>T \<equiv> Matrix.mat (dim_col A) (dim_row A) (\<lambda>(i, j). A $$ (j, i))"
disc_15: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "\<not> isRecv (RecvMarker x51 x52 x53)"
quicksort_trivial1: fixes r :: "nat"   and l :: "nat"   and xs :: "'b list" assumes "r \<le> l" shows "quicksort xs l r = xs"
Map_cod: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.arr f" shows "Cop_S.Map (Cop_S.cod f) = Cop_S.Cod f"
i_in: shows "\<forall>i\<in>{0..<dim_vec y}. y $ i = v (i + dim_vec x)"
vds_map_length: fixes P :: "cld list"   and ctx :: "ctx"   and vds :: "vd list" shows "length (map (case_vd (\<lambda>clk vark. find_type_f P ctx clk)) vds) = length vds"
adv: assumes "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g \<A> \<surd>" shows "advantage \<A> (obsf_resource (sim_outer |\<^sub>= 1\<^sub>C \<rhd> middle)) (obsf_resource real) \<le> adv2"
h_def: shows "h \<equiv> concatMapStep\<cdot>f\<cdot>ha"
Pi_coprime_suf: fixes i :: "nat"   and j :: "nat"   and ps :: "nat list" assumes "i < j" shows "coprime (Pi i) (godel_code' ps j)"
inner_Pair: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and d :: "'b" shows "(a, b) \<bullet> (c, d) = a \<bullet> c + b \<bullet> d"
inject: fixes x1 :: "'b"   and y1 :: "'b" shows "(extT.Val x1 = extT.Val y1) = (x1 = y1)"
right_idem: fixes a :: "'d multiset"   and b :: "'d multiset" shows "a \<union># b \<union># b = a \<union># b"
rel_fset_def: shows "rel_fset \<equiv> map_fun id (map_fun fset (map_fun fset id)) rel_set"
l_one: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.monoid G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
append_is_Nil_conv: fixes xs :: "'d list"   and ys :: "'d list" shows "(xs @ ys = []) = (xs = [] \<and> ys = [])"
a_d_closed: fixes x :: "'a" shows "apd_d (ad x) = ad x"
tm_dghm_ObjMap_in_Vset: shows "\<GG>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
add_numeral_left: fixes v :: "num"   and w :: "num"   and z :: "'b" shows "numeral v + (numeral w + z) = numeral (v + w) + z"
Lev1_Suc: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and x :: "'d" shows "Lev1 s1 s2 (Suc n) x = {Inl (tobd_F12 s1 x b1) # kl |b1 kl. b1 \<in> F1set2 (s1 x) \<and> kl \<in> Lev1 s1 s2 n b1} \<union> {Inr (tobd_F13 s1 x b2) # kl |b2 kl. b2 \<in> F1set3 (s1 x) \<and> kl \<in> Lev2 s1 s2 n b2}"
Ord_sup: fixes x :: "V"   and y :: "V" assumes "Ord x"   and "Ord y" shows "Ord (x \<union>\<^sub>\<circ> y)"
homeomorphic_map_openness: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set" assumes "homeomorphic_map X Y f"   and "U \<subseteq> topspace X" shows "openin Y (f ` U) = openin X U"
error_free_Error: fixes e :: "error"   and s :: "st" shows "error_free (Some (Error e), s) = False"
push_effect_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a" shows "push_effect f \<equiv> \<lambda>(u, v). augment_edge f (u, v) (min (excess f u) (cf_of f (u, v)))"
hyps_10: fixes b :: "(name \<times> ty) list"   and T :: "ty" assumes "b \<turnstile> e[x::=e\<^sub>2'] : T" shows "b \<turnstile> e' : T"
filter_union_zmset: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b zmultiset"   and N :: "'b zmultiset" shows "filter_zmset P (M + N) = filter_zmset P M + filter_zmset P N"
finite_distinct: fixes A :: "'c set" assumes "finite A" shows "finite {p. distinct p \<and> set p \<subseteq> A}"
dpd2: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> d y) = d (x \<cdot> y)"
cring0_is_zariski_open: shows "cring0.is_zariski_open = (\<lambda>U. U = {})"
nb\<^sub>1: shows "cnf ((\<Phi>\<^sub>G \<Pi>) t) = \<Union> {cnf (encode_state_variable t (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>G) v)) |v. v \<in> set (\<Pi>\<^sub>\<V>) \<and> (\<Pi>\<^sub>G) v \<noteq> None}"
assms_2: shows "Q t\<^sub>1 t\<^sub>2"
i_join_union: fixes A :: "nat set"   and B :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "finite A"   and "Max A < iMin B"   and "B \<noteq> {}" shows "f \<Join>\<^sub> (A \<union> B) = f \<Down> Suc (Max A) \<Join>\<^sub> A \<frown> f \<Join>\<^sub> B"
parent_child_rel_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "(child \<in> set children) = ((ptr, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<in> parent_child_rel h)"
filtrated_prob_space_axioms: shows "filtrated_prob_space M G"
folding: fixes C :: "'a set" assumes "folding_g.chamber C"   and "C \<in> g \<turnstile> X" shows "\<exists>!D. folding_g.chamber D \<and> D \<notin> g \<turnstile> X \<and> g ` D = C"
simps_2: fixes \<phi> :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and x :: "'d"   and \<Pi> :: "('c, 'd) chain" shows "chain_all \<phi> (\<lbrace> \<sigma> \<rbrace> \<cdot> x \<cdot> \<Pi>) = (\<phi> (\<sigma>, x, pre \<Pi>) \<and> chain_all \<phi> \<Pi>)"
subst_lemma: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and a :: "('b, 'c) form"   and t :: "'b FOL_Fitting.term"   and i :: "nat" shows "eval e f g (a[t/i]) = eval (e\<langle>i:evalt e f t\<rangle>) f g a"
list_all2_ConsR: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and y :: "'d"   and ys' :: "'d list" assumes "list_all2 R xs (y # ys')" shows "\<exists>x xs'. xs = x # xs' \<and> R x y \<and> list_all2 R xs' ys'"
mbs: fixes h :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "h \<in> BAD P" shows "\<exists>f\<in>BAD P. \<forall>g. (f, g) \<in> gseq \<longrightarrow> good P g"
openin_product_topology_alt: fixes X :: "'a \<Rightarrow> 'b topology"   and I :: "'a set"   and S :: "('a \<Rightarrow> 'b) set" shows "openin (product_topology X I) S = (\<forall>x\<in>S. \<exists>U. finite {i \<in> I. U i \<noteq> topspace (X i)} \<and> (\<forall>i\<in>I. openin (X i) (U i)) \<and> x \<in> Pi\<^sub>E I U \<and> Pi\<^sub>E I U \<subseteq> S)"
assms: fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>i. (f (Suc i), f i) \<in> r" shows "False"
SUP_identity_eq: fixes Sup :: "'b set \<Rightarrow> 'b"   and A :: "'b set" shows "Sup ((\<lambda>x. x) ` A) = Sup A"
weight_le_mono_one: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and f :: "'f"   and ss1 :: "('f, 'b) Term.term list"   and ss2 :: "('f, 'b) Term.term list" assumes "weight_le s t" shows "weight_le (Fun f (ss1 @ s # ss2)) (Fun f (ss1 @ t # ss2))"
cases: fixes a1 :: "'a cfg"   and a2 :: "'a cfg stream" assumes "MDP.MC.enabled a1 a2" obtains \<omega> :: "'a cfg stream" and s :: "'a cfg" where "a1 = s" and "a2 = \<omega>" and "MDP.MC.enabled (shd \<omega>) (stl \<omega>)" and "shd \<omega> \<in> set_pmf (K_cfg s)"
Rep_resumption_cases: fixes y :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "y \<in> resumption" obtains x :: "('h, 'g, 'f, 'i) resumption" where "y = Rep_resumption x"
assms_2: shows "\<guillemotleft>\<mu> : trg f \<star> f \<Rightarrow> f\<guillemotright>"
inject_3: fixes x51 :: "'a fm"   and x52 :: "'a fm"   and y51 :: "'a fm"   and y52 :: "'a fm" shows "(Or x51 x52 = Or y51 y52) = (x51 = y51 \<and> x52 = y52)"
least_fixpoint_unique: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_fixpoint f" shows "\<exists>!x. is_least_fixpoint f x"
asIMOD_igVarIPresIGWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igVarIPresIGWls (asIMOD SEM)"
h3a: shows "\<forall>x. List.member (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) x \<longrightarrow> x < length signs"
not_zero_divisor_mult_1: fixes a :: "'b"   and b :: "'b" assumes "\<not> zero_divisor (a * b)" shows "\<not> zero_divisor a"
lim: shows "AE x in M. ((\<lambda>i. s i x) \<longlongrightarrow> f x) (at_left c)"
eq_strict_1: fixes x :: "'a" shows "eq\<cdot>x\<cdot>\<bottom> = \<bottom>"
mbisim_simulation1': fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and tl1 :: "'b \<times> ('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s1' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 s2"   and "\<not> \<tau>multithreaded.m\<tau>move \<tau>move1 s1 tl1 s1'"   and "multithreaded_base.redT final1 r1 convert_RA s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. (\<tau>trsys.silent_move (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2))\<^sup>*\<^sup>* s2 s2' \<and> multithreaded_base.redT final2 r2 convert_RA s2' tl2 s2'' \<and> \<not> \<tau>multithreaded.m\<tau>move \<tau>move2 s2' tl2 s2'' \<and> FWbisimulation_base.mbisim bisim bisim_wait s1' s2'' \<and> FWbisimulation_base.mta_bisim bisim tl1 tl2"
pseudo_proj_True_suc_img_pref: fixes n :: "nat" shows "range (pseudo_proj_True (Suc n)) = {y. \<exists>w\<in>range (pseudo_proj_True n). y = True ## w} \<union> {y. \<exists>w\<in>range (pseudo_proj_True n). y = False ## w}"
fresh_fun_app': fixes h :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp h)"   and "a \<sharp> h"   and "a \<sharp> h a" shows "fresh_fun h = h a"
disc_2: fixes x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "\<not> Resumption.resumption.is_Done (Resumption.resumption.Pause x21 x22)"
foreach_impl_correct_3: fixes Qi :: "'q"   and \<pi>i :: "'m"   and u :: "'v"   and Qi' :: "'q"   and \<pi>i' :: "'m" assumes "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')"   and "prim_invar_impl Qi \<pi>i" shows "Q_\<alpha> Qi' = Qinter (Q_\<alpha> Qi) (M_lookup \<pi>i) u"
assms_3: shows "proj_list_3 xs' (rs, w, ws) = (rs', w', ws')"
finite_ran: fixes p :: "'a \<Rightarrow> 'b option" assumes "finite (dom p)" shows "finite (ran p)"
D_approx: fixes fs :: "int vec list" assumes "LLL_invariant_weak fs"   and "g_bound fs" shows "D fs \<le> N ^ (m * m)"
inject_5: fixes x5 :: "'a Regex.regex"   and y5 :: "'a Regex.regex" shows "(regex.Star x5 = regex.Star y5) = (x5 = y5)"
type_definition_finfun: shows "type_definition finfun_apply Abs_finfun finfun"
conforms_to_another_strategy: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.path_conforms_with_strategy G p P \<sigma>'" shows "vmc_path G P v0 p \<sigma>'"
assms_1: shows "action_dom (fst a) (snd a) \<subseteq> vs1"
h: fixes f :: "'a \<Rightarrow> 'b stream" assumes "F f" shows "(\<lambda>x. shd (f x)) \<in> N \<rightarrow>\<^sub>M M"
Cpd1_def: fixes x3 :: "char list"   and x1 :: "NominalSequents.var"   and x2 :: "form" shows "x3 \<nabla> [x1].x2 \<equiv> Abs_form (Cpd1_Rep x3 ([x1].Rep_form x2))"
emeasure_lborel_cbox_eq': fixes a :: "'a"   and b :: "'a" shows "emeasure lborel (cbox a b) = ennreal (\<Prod>e\<in>Basis. max 0 ((b - a) \<bullet> e))"
matrix_to_iarray_mult_row: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and q :: "'b" shows "matrix_to_iarray (mult_row A i q) = mult_row_iarray (matrix_to_iarray A) (mod_type_class.to_nat i) q"
g_growth1: shows "\<exists>C c1. 0 < c1 \<and> C < Min (set bs) \<and> (\<forall>\<^sub>F x in at_top. \<forall>u\<in>{C * x..x}. c1 * g' x \<le> g' u)"
modes_eq: shows "snd cms\<^sub>1 = snd cms\<^sub>2"
eval_ec_subseteq: fixes ec :: "'s odlist"   and \<phi> :: "('b, 'p) Kform" shows "toSet (eval envVal ec \<phi>) \<subseteq> toSet ec"
while_bot_def: fixes b :: "'b upred"   and P :: "'b hrel" shows "while\<^sub>\<bottom> b do P od = (\<mu> X \<bullet> P ;; X \<triangleleft> b \<triangleright>\<^sub>r II)"
valid: fixes x :: "'a" assumes "x \<in> X" shows "valid_intv (k x) (I x)"
convex_on_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real" shows "convex_on S f = (\<forall>x\<in>S. \<forall>y\<in>S. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)"
nth_xs: shows "xs ! j = (if j < i then take i xs ! j else (xs ! i # drop (Suc i) xs) ! (j - i))"
fresh_1: fixes a :: "name" shows "a \<sharp> \<zero> = True"
multiplicity_aprimedivisor_gt_0_nat: fixes d :: "nat" assumes "Suc 0 < d" shows "0 < multiplicity (aprimedivisor d) d"
index_mat_multrow_mat_2: fixes n :: "nat"   and k :: "nat"   and a :: "'c" shows "dim_row (multrow_mat n k a) = n"
ZFTFst: fixes a :: "ZF"   and b :: "ZF"   and c :: "ZF" shows "ZFTFst (ZFTriple a b c) = a"