hcju/isabelleps huggingface dataset:
red_external_wt_hconf_hext: fixes wf_md :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> String.literal \<Rightarrow> (nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) mdecl \<Rightarrow> bool"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap"   and h'' :: "'heap"   and U :: "ty" assumes "wf_prog wf_md P"   and "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "h'' \<unlhd> h"   and "P,h'' \<turnstile> a\<bullet>M(vs) : U"   and "P,h'' \<turnstile> t \<surd>t"   and "hconf h''" shows "\<exists>ta' va' h'''. P,t \<turnstile> \<langle>a\<bullet>M(vs),h''\<rangle> -ta'\<rightarrow>ext \<langle>va',h'''\<rangle> \<and> collect_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> = collect_locks \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<and> collect_cond_actions \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> = collect_cond_actions \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<and> collect_interrupts \<lbrace>ta\<rbrace>\<^bsub>i\<^esub> = collect_interrupts \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub>"
assms_1: shows "a' = \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
gcs_adds: fixes s :: "'a"   and t :: "'a" shows "gcs s t adds s"
islimptI: fixes x :: "'a"   and S :: "'a set" assumes "\<And>T. \<lbrakk>x \<in> T; open T\<rbrakk> \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x" shows "x islimpt S"
chain_def: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph" shows "chain S \<equiv> \<forall>i. subgraph (S i) (S (i + 1))"
shead_smap_distr: fixes f :: "'s \<rightarrow> 'r"   and xs :: "[:'s:]" assumes "f\<cdot>\<bottom> = \<bottom>"   and "\<And>x. x \<in> sset xs \<Longrightarrow> f\<cdot>x \<noteq> \<bottom>" shows "shead\<cdot>(smap\<cdot>f\<cdot>xs) = f\<cdot>(shead\<cdot>xs)"
nhds_2: fixes i :: "nat" shows "x \<in> A i"
jipsen4r: fixes x :: "'a" shows "x \<cdot> (x \<rightarrow> x) = x"
restricted_subformulas_inner_subset: fixes \<phi> :: "'b ltln" shows "restricted_subformulas_inner \<phi> \<subseteq> subformulas\<^sub>\<nu> \<phi> \<union> subformulas\<^sub>\<mu> \<phi>"
simps_4: fixes v :: "bool" shows "is_Addr (Bool v) = False"
sup_id_star2: fixes x :: "'a" assumes "reflexive x" shows "x\<^sup>\<star> ; x = x\<^sup>\<star>"
def_hashmap_size_channel_def: fixes uu :: "channel itself" shows "def_hashmap_size uu \<equiv> 10"
assms_1: shows "f \<in> borel_measurable (F t)"
left_neutral: fixes a :: "'b hrel" shows "II ;; a = a"
monic_degree_m: fixes f :: "int poly" assumes "monic f" shows "q.degree_m f = degree f"
keySetup_def: shows "keySetup \<equiv> {(ltK K, A) |K A. (K, A) \<in> ltkeySetup}"
sqns_increase: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). \<forall>ip. sqn (rt \<xi>) ip \<le> sqn (rt \<xi>') ip)"
ipresWlsAll_defs_2: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'j, 'e, 'k, 'l, 'g, 'm, 'n) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresWls Delta h MOD \<equiv> \<forall>s X. FixSyn.wls Delta s X \<longrightarrow> igWls MOD s (h X)"
simps_1: shows "local.linear_hensel_main (Suc 0) = (D1, H1)"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "x \<le> xa" obtains xb :: "'a" and y :: "'a" where "x = Some xb" and "xa = Some y__" and "xb \<le> y__"   | y :: "'a option" where "x = None" and "xa = y"
list_augment_as_update: fixes k :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "k < length xs" shows "list_augment xs k x = xs[k := x]"
map_pmf_compose: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
in_degree_def: fixes G :: "('e, 'f) pre_digraph"   and v :: "'e" shows "in_degree G v \<equiv> card (in_arcs G v)"
effectE: fixes c :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b" assumes "effect c h h' r" obtains "r = fst (the (execute c h))" and "h' = snd (the (execute c h))" and "success c h"
dvd_add_left_iff: fixes 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 a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one"   and "dvd.dvd times a c" shows "dvd.dvd times a (plus b c) = dvd.dvd times a b"
delete_shadow_root_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and getter :: "('ab, 'z, 'aa, (\<lparr>RShadowRoot.nothing :: unit, mode :: shadow_root_mode, child_nodes :: (_) node_ptr list, \<dots> :: 'ag option\<rparr> + 'ah) option) RDocument_scheme \<Rightarrow> 'aj" assumes "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<noteq> ptr"   and "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
term_templateInduct: fixes rel :: "(('c, 'd, 'varSort, 'var, 'e) qTerm set \<times> ('c, 'd, 'varSort, 'var, 'e) qTerm set) set"   and phi :: "('c, 'd, 'varSort, 'var, 'e) qTerm set \<Rightarrow> bool"   and phiAbs :: "('c, 'd, 'varSort, 'var, 'e) qAbs set \<Rightarrow> bool"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and A :: "('c, 'd, 'varSort, 'var, 'e) qAbs set" assumes "\<And>X Y. \<lbrakk>good X; (X, Y) \<in> rel\<rbrakk> \<Longrightarrow> good Y \<and> skel Y = skel X"   and "\<And>xs x. phi (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>goodInp inp; goodBinp binp; liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (Op delta inp binp)"   and "\<And>xs x X. \<lbrakk>good X; \<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (Abs xs x X)" shows "(good X \<longrightarrow> phi X) \<and> (goodAbs A \<longrightarrow> phiAbs A)"
Super: fixes E :: "env"   and C :: "qtname"   and c :: "class"   and dt :: "loc \<Rightarrow> ty option" assumes "lcl E This = Some (Class C)"   and "C \<noteq> Object"   and "class (prg E) C = Some c" shows "E,dt\<Turnstile>Super\<Colon>-Class (super c)"
GodExistenceImpliesNecExistence: shows "\<lfloor>(\<lambda>w. (\<exists>x. \<forall>xa. \<P> xa w = (\<^bold>\<box>xa x) w) \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v = (\<^bold>\<box>xa x) v))) w)\<rfloor>"
run_fail_env: fixes fail :: "'c"   and r :: "'d" shows "run_env (fail_env fail) r = fail"
pochhammer_poly_Suc: fixes n :: "nat" shows "pochhammer_poly (Suc n) = [:of_nat n, 1::'b:] * pochhammer_poly n"
Gcd_insert: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd (insert a A) = gcd a (Gcd A)"
cong_sym_poly: fixes a :: "'c poly"   and b :: "'c poly"   and m :: "'c poly" assumes "[a = b] (mod m)" shows "[b = a] (mod m)"
assms_6: shows "ptr \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 |h \<turnstile> create_element document_ptr tag|\<^sub>r"
hyps_Diff: fixes p :: "'b pl"   and t :: "'b set"   and v :: "'b" shows "hyps p (t - {v}) \<subseteq> insert (#v -> false) (hyps p t - {#v})"
assms: assumes "\<Phi>" shows "mi \<le> \<Down>\<^sub>E E R m"
Set_filter_fold: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "finite A" shows "Set.filter P A = Finite_Set.fold (\<lambda>x A'. if P x then insert x A' else A') {} A"
coinduct: fixes X :: "ccs \<Rightarrow> ccs \<Rightarrow> bool"   and x1 :: "ccs"   and x2 :: "ccs" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>P Q. x1 = P \<and> x2 = Q \<and> P \<leadsto>\<^sup>^<{(xa, x). X xa x \<or> xa \<approx> x}> Q \<and> (X Q P \<or> Q \<approx> P)" shows "x1 \<approx> x2"
simps: fixes g :: "'g"   and v :: "'var"   and unvisited :: "'node list"   and n :: "'node" shows "phiDefNodes_aux g v unvisited n = (if n \<notin> set unvisited \<or> v \<in> defs g n then {} else fold (\<union>) (map (phiDefNodes_aux g v (removeAll n unvisited)) (predecessors g n)) (if length (predecessors g n) \<noteq> 1 then {n} else {}))"
E: fixes n :: "nat" shows "\<gamma> - 1 / 2 ^ n < d (E n)"
map_rbt_less: fixes k :: "'a"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'd"   and t :: "('a, 'e) rbt" shows "(RBT_Impl.map f t |\<guillemotleft> k) = (t |\<guillemotleft> k)"
spp_inv_alt: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "spp_inv sp = (vec_of sp \<noteq> 0)"
inc_predict_support_trading_strat: fixes pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and B :: "'d set" assumes "trading_strategy pf1" shows "\<forall>asset\<in>support_set pf1 \<union> B. borel_predict_stoch_proc F (pf1 asset)"
assms_1: shows "RModuleHom R (\<cdot>) M (\<star>) S"
length_zip: fixes xs :: "'c list"   and ys :: "'d list" shows "length (zip xs ys) = min (length xs) (length ys)"
intros: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emb I J X \<in> generator"
dist_eqs_1: fixes cfun :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue"   and discr' :: "bool discr" shows "CFn\<cdot>cfun \<noteq> CB\<cdot>discr'"
no_initial_process_snapshot: shows "\<forall>p. \<not> ps init p \<noteq> None"
alw_InvGlobPointstampsEq: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (holds InvGlobPointstampsEq) s"
rep_eq: fixes x :: "'a dual" shows "undual (- x) = - undual x"
wajsberg1: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) r\<rightarrow> b = (b l\<rightarrow> a) r\<rightarrow> a"
familyUnionFiniteEverySetFinite: fixes XX :: "'a set set" assumes "finite (\<Union> XX)" shows "\<forall>X\<in>XX. finite X"
Inf_between_alt: fixes N :: "'f set"   and M :: "'f set" shows "Inf_between N M = {\<iota> \<in> Inf. \<iota> \<in> Inf_from (N \<union> M) \<and> set (prems_of \<iota>) \<inter> M \<noteq> {}}"
fold_simps_2: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'f"   and xs :: "'f list"   and s :: "'e" shows "fold f (x # xs) s = fold f xs (f x s)"
prefixToLevelPrefixToLevelHigherLevel: fixes level :: "nat"   and level' :: "nat"   and M :: "('a \<times> bool) list" assumes "level \<le> level'" shows "prefixToLevel level M = prefixToLevel level (prefixToLevel level' M)"
lemma4_prev: fixes a :: "'a"   and a1 :: "'a"   and d :: "'a"   and b :: "'a"   and b1 :: "'a" assumes "a = a1 * d"   and "b = b1 * d"   and "ideal_generated {a1, b1} = ideal_generated {1::'a}" shows "ideal_generated {a, b} = ideal_generated {d}"
ty_cases: fixes T :: "ty" shows "(\<exists>s. T = PR s) \<or> (\<exists>T'. T = NOT T') \<or> (\<exists>S U. T = S OR U) \<or> (\<exists>S U. T = S AND U) \<or> (\<exists>S U. T = S IMP U)"
Gromov_completion_rel_const: fixes x :: "'b" shows "Gromov_completion_rel (\<lambda>n. x) (\<lambda>n. x)"
res: shows "LLL_Impl.reduce_basis \<alpha> fs_init = fs"
aff_dim_parallel_eq: fixes S :: "'a set"   and T :: "'a set" assumes "affine_parallel (affine hull S) (affine hull T)" shows "aff_dim S = aff_dim T"
simps_4: fixes f1 :: "'b"   and f2 :: "'c set \<Rightarrow> 'b" shows "(case FAILi of FAILi \<Rightarrow> f1 | RES x \<Rightarrow> f2 x) = f1"
greatest_postfixpoint_char: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_greatest_postfixpoint f x = (has_greatest_postfixpoint f \<and> x = p\<nu> f)"
dickson_leD1: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "gd_powerprod.dickson_le ord d m s t" shows "d s \<le> m"
cgenideal_pirreducible: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "pirreducible K p"   and "pirreducible K q"   and "q \<in> PIdl\<^bsub>K [X]\<^esub> p" shows "p \<sim>\<^bsub>K [X]\<^esub> q"
isolated_verts_eq: shows "G.isolated_verts = H.isolated_verts"
is_zero_row_upt_k_def: fixes i :: "'d"   and k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec" shows "is_zero_row_upt_k i k A = (\<forall>j. mod_type_class.to_nat j < k \<longrightarrow> A $ i $ j = (0::'e))"
dim_row_echelon_2: fixes A :: "'c mat" shows "dim_col (row_echelon A) = dim_col A"
basis_subspace_exists: fixes S :: "('a, 'b) vec set" assumes "vec.subspace S" obtains B :: "('a, 'b) vec set" where "finite B" and "B \<subseteq> S" and "vec.independent B" and "vec.span B = S" and "card B = vec.dim S"
mem_Rep_star_iff: fixes X :: "nat \<Rightarrow> 'b"   and x :: "'b star" shows "(X \<in> Rep_star x) = (x = star_n X)"
runit_char_3: fixes a :: "'c" assumes "C.ide a" shows "\<exists>!f. \<guillemotleft>f : a \<otimes>\<^sub>C \<I>\<^sub>C \<rightarrow>\<^sub>C a\<guillemotright> \<and> f \<otimes>\<^sub>C \<I>\<^sub>C = (a \<otimes>\<^sub>C \<iota>\<^sub>C) \<cdot>\<^sub>C \<a>\<^sub>C[a, \<I>\<^sub>C, \<I>\<^sub>C]"
echelon_form_of_invertible: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'd) vec, 'c) vec" assumes "is_bezout_ext bezout" shows "\<exists>P. invertible P \<and> P ** A = echelon_form_of A bezout \<and> echelon_form (echelon_form_of A bezout)"
Afix_above_arg: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "ae \<sqsubseteq> Afix \<Gamma>\<cdot>ae"
rel_map_1: fixes Sb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e \<Rightarrow> 'c"   and x :: "'e option"   and y :: "'d option" shows "rel_option Sb (map_option i x) y = rel_option (\<lambda>x. Sb (i x)) x y"
finite_Diff_insert: fixes A :: "'d set"   and a :: "'d"   and B :: "'d set" shows "finite (A - insert a B) = finite (A - B)"
split: fixes P :: "'d \<Rightarrow> bool"   and f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and stateT :: "('e, 'f) stateT" shows "P (case stateT of StateT x \<Rightarrow> f x) = (\<forall>x. stateT = StateT x \<longrightarrow> P (f x))"
as: shows "usappconst \<sigma> U f \<noteq> undeft"
diff_eq2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "x + - y = x - y"
tutm_uhalt': fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'a" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (1, Bk \<up> l, <lm>) tp stp)" shows "\<forall>stp. \<not> is_final (steps0 (1, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)"
eq_tllist_def: fixes p :: "'c llist \<times> 'd"   and q :: "'c llist \<times> 'd" shows "eq_tllist p q = (fst p = fst q \<and> (if lfinite (fst p) then snd p = snd q else True))"
insert_remove: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "abel_semigroup_set f z"   and "finite A" shows "abel_semigroup_set.F f z g (insert x A) = f (g x) (abel_semigroup_set.F f z g (A - {x}))"
simps_2: fixes g :: "'b ugraph"   and u :: "'b"   and e :: "'b \<times> 'b"   and ps :: "('b \<times> 'b) list"   and w :: "'b" shows "path g u (e # ps) w = (\<exists>v. e = (u, v) \<and> e \<in> edges g \<and> path g v ps w)"
elt_set_times_def: fixes a :: "'a"   and B :: "'a set" shows "a *o B = {c. \<exists>b\<in>B. c = a * b}"
strict_iff_order: fixes a :: "nat list"   and b :: "nat list" shows "(a <\<^sub>v b) = (a \<le>\<^sub>v b \<and> a \<noteq> b)"
multiple_is_finite: fixes n :: "nat" shows "finite_incidence_system \<V> (fis.multiple_blocks n)"
that_1: shows "y \<in> aform_err e (c, d')"
extendConc_def: fixes S :: "('a, 'b) sequent"   and r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" shows "extendConc S r \<equiv> (fst r, extend S (snd r))"
f_closed: shows "f ` U\<^sub>M\<^sub>_\<^sub>1 \<subseteq> U\<^sub>M\<^sub>_\<^sub>2"
subterm_of_iff_subtermeq: fixes t :: "('c, 'd) Term.term"   and t' :: "('c, 'd) Term.term" shows "(t \<sqsubseteq> t') = subterm_of t t'"
of_nat_lt_hmset_of_enat_iff: fixes m :: "nat"   and n :: "enat" shows "(of_nat m < hmset_of_enat n) = (enat m < n)"
min_int_poly_real_alg: fixes x :: "real_alg" shows "min_int_poly_real_alg x = min_int_poly (real_of x)"
cone_\<chi>: shows "D.mkCone e \<in> D.cones (C.dom e)"
singleton_ball: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>{a}. P x) = P a"
typ_of1_imp_has_typ1: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ" assumes "typ_of1 Ts t = Some ty" shows "Ts \<turnstile>\<^sub>\<tau> t : ty"
measurable_expr_sem_rf: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and V :: "nat set" assumes "\<Gamma> \<turnstile> e : t"   and "randomfree e"   and "free_vars e \<subseteq> V" shows "(\<lambda>\<sigma>. expr_sem_rf \<sigma> e) \<in> state_measure V \<Gamma> \<rightarrow>\<^sub>M stock_measure t"
Rolle_pderiv: fixes x1 :: "real"   and x2 :: "real"   and q :: "real poly" shows "x1 < x2 \<and> poly q x1 = 0 \<and> poly q x2 = 0 \<longrightarrow> (\<exists>w>x1. w < x2 \<and> poly (pderiv q) w = 0)"
ExecAssignEvalError: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and e :: "expr"   and v :: "char list" assumes "eval \<sigma> e = None" shows "Assign v e \<rhd> \<sigma> \<leadsto> None"
irrefl: fixes x :: "'a" assumes "x \<in> A" shows "\<not> x \<sqsubset> x"
inv_mult_group: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "\<ominus> (x \<oplus> y) = \<ominus> y \<oplus> \<ominus> x"
eventually_mono: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter"   and Q :: "'c \<Rightarrow> bool" assumes "eventually P F"   and "\<And>x. P x \<Longrightarrow> Q x" shows "eventually Q F"
subState: fixes x :: "'a \<Rightarrow> 'b option"   and y :: "'a \<Rightarrow> 'b option"   and v :: "'a" assumes "x \<preceq> y"   and "v \<in> dom x" shows "x v = y v"
simps_1: fixes x :: "'d" shows "ground (Var x) = False"
hseq_char': fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "H.hseq \<nu> \<mu> = (\<nu> \<cdot> \<mu> \<noteq> V.null)"
disc_83: fixes x6 :: "ipt_l4_ports" shows "\<not> is_MultiportPorts (Src_Ports x6)"
rtrancl_apply_insert: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and S :: "'a set" shows "R\<^sup>* `` insert x S = insert x (R\<^sup>* `` (S \<union> R `` {x}))"
P2L_to_L2P: fixes p :: "proj2"   and l :: "proj2_line" shows "(P2L p = l) = (p = L2P l)"
field_vector_diff_chain_at: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and g :: "'a \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_vector_derivative f') (at x)"   and "(g has_field_derivative g') (at (f x))" shows "(g \<circ> f has_vector_derivative f' * g') (at x)"
group_size_implies_group_existance: fixes x :: "nat" assumes "x \<in> group_sizes" shows "\<exists>G. G \<in> \<G> \<and> card G = x"
simps_1: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'b" assumes "propositional_atoms atom_ordering" shows "atom (Pos A) = A"
simps_4: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
simps_4: fixes x :: "'a"   and y :: "'a" shows "extB.Val x + extB.Val y = extB.Val (x + y)"
conjugation_pair_def: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "conjugation_pair f g \<equiv> \<forall>x y. (f x \<sqinter> y = \<bottom>) = (x \<sqinter> g y = \<bottom>)"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 \<nu> (\<pi> \<star> \<rho>)"
zero_less_mult_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (times a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
Broadcast_def: fixes n :: "nat"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list" shows "Broadcast n nSend recv \<equiv> \<forall>t. if \<exists>k<n. nSend k t \<noteq> [] then recv t = nSend (SOME k. k < n \<and> nSend k t \<noteq> []) t else recv t = []"
pickQFreshEnv_card_of: fixes V :: "'var set"   and XS :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "Ball XS qGood"   and "|Rho| <o |UNIV|"   and "Ball Rho qGoodEnv" shows "pickQFreshEnv xs V XS Rho \<notin> V \<and> (\<forall>X\<in>XS. qFresh xs (pickQFreshEnv xs V XS Rho) X) \<and> (\<forall>rho\<in>Rho. qFreshEnv xs (pickQFreshEnv xs V XS Rho) rho)"
exhaust: fixes y :: "exp_or_val" obtains x1 :: "exp0" where "y = Exp x1"   | x2 :: "v" where "y = Val x2"
Star_imp_cptn: fixes P :: "'b com option"   and s :: "'b"   and R :: "'b com option"   and t :: "'b" assumes "(P, s) -c*\<rightarrow> (R, t)" shows "\<exists>l\<in>cp P s. last l = (R, t) \<and> (\<forall>i. Suc i < length l \<longrightarrow> l ! i -c\<rightarrow> l ! Suc i)"
intpow_uminus: fixes x :: "'c"   and n :: "int" shows "x ^\<^sub>Z - n = inverse (x ^\<^sub>Z n)"
has_bochner_integral_lebesgue_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'a"   and i :: "'a"   and t :: "real" assumes "c \<noteq> 0" shows "has_bochner_integral lebesgue f i = has_bochner_integral lebesgue (\<lambda>x. f (t + c * x)) (i /\<^sub>R \<bar>c\<bar>)"
weakCongTransE: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" obtains P'' :: "ccs" and P''' :: "ccs" where "P \<Longrightarrow>\<^sub>\<tau> P''" and "P'' \<longmapsto>\<alpha> \<prec> P'''" and "P''' \<Longrightarrow>\<^sub>\<tau> P'"
drop_aux: fixes n :: "nat"   and l :: "'d list" assumes "Suc n \<le> length l" shows "drop (length l - Suc n) l = l ! (length l - Suc n) # drop (length l - n) l"