hcju/isabelleps huggingface dataset:
alg_MapOps_empty_1: fixes ec :: "'rep" assumes "k_isNode ec" shows "lookup aOps (MapOps.empty aOps) ec = None"
simps_5: fixes n :: "nat"   and r :: "'b rexp"   and s :: "'b rexp" shows "local.wf n (Plus r s) = (local.wf n r \<and> local.wf n s)"
col_smult: fixes i :: "nat"   and A :: "'a mat"   and k :: "'a" assumes "i < dim_col A" shows "col (k \<cdot>\<^sub>m A) i = k \<cdot>\<^sub>v col A i"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'l" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
substitutes_unique: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and B :: "'a trm"   and C :: "'a trm" assumes "substitutes A x M B"   and "substitutes A x M C" shows "B = C"
intro: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "morphism_sheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
mem_of_WWa: fixes W :: "'a set"   and a :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "W \<subseteq> carrier D"   and "Worder (Iod D W)"   and "a \<in> W"   and "\<forall>x\<in>W. a \<preceq> x"   and "\<forall>x\<in>W. if ExPre (Iod D W) x then g (Pre (Iod D W) x) = x else if a \<noteq> x then Algebra2.Sup D (segment (Iod D W) x) = x else a = a" shows "W \<in> WWa D g a"
subset_Un_eq: fixes A :: "'b set"   and B :: "'b set" shows "((A::'b set) \<subseteq> (B::'b set)) = (A \<union> B = B)"
g_sinks: fixes L :: "'a \<Rightarrow> g_level"   and xs :: "'a list" shows "sinks g_I L High xs \<subseteq> {High}"
induct: fixes P :: "dbfm \<Rightarrow> bool"   and dbfm :: "dbfm" assumes "\<And>x1 x2. P (DBMem x1 x2)"   and "\<And>x1 x2. P (DBEq x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (DBDisj x1 x2)"   and "\<And>x. P x \<Longrightarrow> P (DBNeg x)"   and "\<And>x. P x \<Longrightarrow> P (DBEx x)" shows "P dbfm"
single_eq_add_mset: fixes a :: "'a"   and b :: "'a"   and M :: "'a multiset" shows "(\<LM> a  \<RM> = add_mset b M) = (b = a \<and> M = \<Empt>)"
Red_I_\<G>_def: fixes N :: "('f \<times> 'l) set" shows "Red_I_\<G> N = (\<Inter>q\<in>Q. Red_I_\<G>_q q N)"
good_ruleset_tail: fixes r :: "'b rule"   and rs :: "'b rule list" assumes "good_ruleset (r # rs)" shows "good_ruleset rs"
pnormal_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a list" assumes "class.semiring_0 plus zero times" shows "semiring_0.pnormal zero p = (semiring_0.pnormalize zero p = p \<and> p \<noteq> [])"
RSD_pareto_eqclass_rsd_winners: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a list" assumes "pref_profile_wf agents alts R"   and "finite alts"   and "alts \<noteq> {}"   and "set agents' = agents" shows "RSD_pareto_eqclass agents alts R (rsd_winners R alts agents')"
awalk_empty_ends: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u [] v" shows "u = v"
tendsto_norm: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'e"   and F :: "'d filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
continuous_closed_preimage_constant: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b" assumes "continuous_on S f"   and "closed S" shows "closed {x \<in> S. f x = a}"
int_ideal: fixes I :: "'a set"   and J :: "'a set" assumes "ideal R I"   and "ideal R J" shows "ideal R (I \<inter> J)"
double_negation: fixes i :: "char list \<Rightarrow> tv"   and p :: "fm" shows "eval i p = eval i (Neg' (Neg' p))"
ide_char: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp" shows "partial_magma.ide (concrete_category.COMP Obj Hom Comp) f = (f \<noteq> Null \<and> concrete_category.Dom f \<in> Obj \<and> Cod f = concrete_category.Dom f \<and> Map f = Id (concrete_category.Dom f))"
getFrN_FvarsT_Fvars: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and n :: "nat"   and t :: "'trm"   and \<phi> :: "'fmla" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla" shows "set (getFrN xs ts \<phi>s n) \<subseteq> var \<and> set (getFrN xs ts \<phi>s n) \<inter> set xs = {} \<and> (t \<in> set ts \<longrightarrow> set (getFrN xs ts \<phi>s n) \<inter> FvarsT t = {}) \<and> (\<phi> \<in> set \<phi>s \<longrightarrow> set (getFrN xs ts \<phi>s n) \<inter> Fvars \<phi> = {}) \<and> length (getFrN xs ts \<phi>s n) = n \<and> distinct (getFrN xs ts \<phi>s n)"
span_finiteE: fixes B :: "('b \<Rightarrow>\<^sub>0 'c) set"   and p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "finite B"   and "p \<in> phull B" obtains q :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'c" where "p = (\<Sum>b\<in>B. q b \<cdot> b)"
conf_def: fixes G :: "prog"   and s :: "st"   and v :: "val"   and T :: "ty" shows "G,s\<turnstile>v\<Colon>\<preceq>T = (? T':typeof (\<lambda>a. map_option obj_ty (heap s a)) v: G\<turnstile>T'\<preceq>T)"
tauChainBisim: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "\<Psi> \<rhd> P \<sim> P''" obtains P''' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'''" and "\<Psi> \<rhd> P' \<sim> P'''"
prob_spaceI: fixes M :: "'a measure" assumes "emeasure M (space M) = 1" shows "prob_space M"
intros: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and y :: "real" assumes "u\<up>f"   and "\<And>n. x n \<in> sfis (u n) M"   and "x\<up>y" shows "y \<in> nnfis f M"
affine_hull_span2: fixes a :: "'a"   and s :: "'a set" assumes "a \<in> s" shows "affine hull s = (+) a ` span ((+) (- a) ` (s - {a}))"
W_3: shows "\<exists>Ta. FGModuleIso G (*) W (\<cdot>) Ta (T ` FG)"
find_path1_impl_def: fixes node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'b \<Rightarrow> nat"   and node_def_hash_size :: "nat" shows "find_path1_impl node_eq_impl node_hash_impl node_def_hash_size \<equiv> \<lambda>x xa xb. wset_find_path_impl node_hash_impl node_def_hash_size node_eq_impl x (x xa) xb \<bind> case_option (RETURN None) (\<lambda>(xc, xd). RETURN (Some (xa # xc, xd)))"
complexes: fixes Y :: "'a set set" assumes "Y \<in> \<A>" shows "ChamberComplex Y"
eq_key_imp_eq_value: fixes xs :: "('c \<times> 'd) list"   and k :: "'c"   and v1 :: "'d"   and v2 :: "'d" assumes "distinct (map fst xs)"   and "(k, v1) \<in> set xs"   and "(k, v2) \<in> set xs" shows "v1 = v2"
WHILE_annotRVI_def: fixes R :: "('b \<times> 'b) set"   and V :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'b"   and I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "WHILE_annotRVI R V I \<equiv> While"
alist_ext: fixes xs :: "('a, 'b) alist"   and ys :: "('a, 'b) alist" assumes "impl_of xs = impl_of ys" shows "xs = ys"
ord_isom_segment_segment: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D" shows "ord_isom (Iod D (segment D a)) (Iod E (segment E (f a))) (restrict f (carrier (Iod D (segment D a))))"
WhileT\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and c :: "nat exp"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>3 :: "nat exp"   and s\<^sub>3 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>c,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>2\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>2\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>3,s\<^sub>3\<rangle>"
list_all_zip_map1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and as :: "'c list"   and bs :: "'b list" shows "list_all_zip P (map f as) bs = list_all_zip (\<lambda>a. P (f a)) as bs"
ereal_inverse_not_MInfty: fixes a :: "ereal" shows "inverse a \<noteq> - \<infinity>"
append_self_conv2: fixes xs :: "'c list"   and ys :: "'c list" shows "(append xs ys = ys) = (xs = [])"
Order_Pow_def: fixes A :: "'c set" shows "po A = \<lparr>carrier = Pow A, rel = {(X, Y). X \<in> Pow A \<and> Y \<in> Pow A \<and> X \<subseteq> Y}\<rparr>"
simps_4: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and b :: "expr"   and c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd"   and ps :: "char list list"   and p :: "char list" shows "containsCall procs (if (b) c\<^sub>1 else c\<^sub>2) ps p = (containsCall procs c\<^sub>1 ps p \<or> containsCall procs c\<^sub>2 ps p)"
g_def: shows "g \<equiv> dual \<circ> f \<circ> undual"
check_AudioPlayer_CTRL: shows "(AudioPlayer_CTRL_States, AudioPlayer_CTRL_Init, AudioPlayer_CTRL_Labels, AudioPlayer_CTRL_Delta) \<in> seqauto"
converseD: fixes a :: "'e"   and b :: "'f"   and r :: "('f \<times> 'e) set" assumes "(a, b) \<in> r\<inverse>" shows "(b, a) \<in> r"
prv_all_congW: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var"   and "prv (imp \<phi> \<chi>)"   and "prv (all x \<phi>)" shows "prv (all x \<chi>)"
impl_of: fixes x :: "('c, 'd) trie" shows "impl_of x \<in> {t. invar_trie t}"
simps: fixes x :: "'c regular"   and y :: "'c dense Filters.filter"   and z :: "'c regular"   and w :: "'c dense Filters.filter" shows "stone_phi.pairs_less_eq (x, y) (z, w) = (x \<le> z \<and> w \<le> y)"
ideI: fixes a :: "'a" assumes "left b a"   and "ide a" shows "S.ide a"
IH: fixes ys :: "('k \<times> ('b, 'l, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef) list" assumes "Result.those (map (compile_env_entry \<A> \<O>) xs) = Ok ys" shows "rel_option (\<lambda>fd1 fd2. compile_fundef \<A> (\<O> f) fd1 = Ok fd2) (map_of xs f) (map_of ys f)"
delta_mono: fixes p :: "fm"   and bs :: "real list"   and d :: "int"   and d' :: "int" assumes "iszlfm p bs"   and "d dvd d'"   and "d_\<delta> p d" shows "d_\<delta> p d'"
simps_1: fixes z :: "complex" shows "Re (cnj z) = Re z"
rat_to_int_factor: fixes p :: "int poly"   and g :: "rat poly"   and h :: "rat poly" assumes "of_int_poly p = g * h" shows "\<exists>g' h'. p = g' * h' \<and> degree g' = degree g \<and> degree h' = degree h"
children_eq2_h: fixes ptr' :: "(_) object_ptr" shows "|h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r"
rep_eq: fixes x :: "integer" shows "int_of_integer (NOT x) = NOT (int_of_integer x)"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (b \<le> a \<and> a \<le> b)"
assms: shows "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)"
eval_plus: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a"   and G :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval (plus F G) = (\<lambda>x. eval F x + eval G x)"
assms_3: fixes A :: "'a set" assumes "A \<in> \<A>" shows "infinite A"
borel_measurable_Max: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. MAX i\<in>I. f i x) \<in> borel_measurable M"
subcocycle_avg_def: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle_avg u = real_of_ereal (subcocycle_avg_ereal u)"
image_compose_permutations_left: fixes q :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "q permutes S" shows "{q \<circ> p |p. p permutes S} = {p. p permutes S}"
pnet_never_newpkt: fixes s :: "'c net_state"   and a :: "'d node_action"   and s' :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and n :: "net_tree"   and i :: "nat"   and d :: "nat"   and di :: "nat" assumes "(s, a, s') \<in> automaton.trans (pnet np n)" shows "a \<noteq> i:newpkt(d, di)"
assms: shows "\<F>\<^sub>\<bullet> r \<inter>\<^sub>\<circ> \<F>\<^sub>\<bullet> s = []\<^sub>\<circ>"
exi_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and F :: "('a \<times> 'b set) stream \<Rightarrow> ('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.exi S s0 \<delta> AP L AP' F \<pi>l \<equiv> \<exists>\<pi>. Kripke.wfp S \<delta> AP L AP' \<pi> \<and> stateOf \<pi> = (if \<pi>l \<noteq> [] then stateOf (last \<pi>l) else s0) \<and> F \<pi> \<pi>l"
Dup_head_oob': fixes p :: "('b, 'c) fm"   and i :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and k :: "nat" assumes "Dup p i ((ps, a) # branch) xs" shows "(length branch, k + length ps) \<notin> xs"
ord_eq_2_iff: fixes n :: "nat"   and x :: "nat" shows "(ord n x = 2) = ([x \<noteq> 1] (mod n) \<and> [x\<^sup>2 = 1] (mod n))"
sl'_def: shows "sl' = pickEff (shd (trim rs s)) s"
\<pi>n: shows "\<pi>\<^sub>1 n = local.return"
2: shows "A n \<inter> \<Union> (A ` {0..<n}) = {}"
reach_def: fixes \<Sigma> :: "'d set"   and \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c" shows "reach \<Sigma> \<delta> q\<^sub>0 = {run \<delta> q\<^sub>0 w n |w n. range w \<subseteq> \<Sigma>}"
prod_chart_compat: fixes c1 :: "('a, 'b) chart"   and d1 :: "('a, 'b) chart"   and c2 :: "('c, 'd) chart"   and d2 :: "('c, 'd) chart" assumes "k-smooth_compat c1 d1"   and "k-smooth_compat c2 d2" shows "k-smooth_compat (prod_chart c1 c2) (prod_chart d1 d2)"
hom_sum: fixes f :: "'b \<Rightarrow> 'a poly"   and X :: "'b set"   and a :: "'a" shows "poly (sum f X) a = (\<Sum>x\<in>X. poly (f x) a)"
normal_poly_coeff_mult: fixes p :: "'a poly"   and i :: "nat" assumes "normal_poly p" shows "coeff p i * coeff p (i + 2) \<le> (coeff p (i + 1))\<^sup>2"
IdentityFunctor_def: fixes C :: "('a, 'b, 'c) Category_scheme" shows "FId C \<equiv> MakeFtor FId' C"
elims: fixes x :: "nat \<Rightarrow> 'n"   and xa :: "nat"   and y :: "'n list" assumes "cs\<^bsup>x\<^esup> xa = y" obtains \<pi> :: "nat \<Rightarrow> 'n" and n :: "nat" where "x = \<pi>" and "xa = n" and "y = (if Ex (is_icdi n \<pi>) then cs\<^bsup>\<pi>\<^esup> The (is_icdi n \<pi>) @ [\<pi> n] else [\<pi> n])"
list_all_simps_1: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" shows "list_all P (x # xs) = (P x \<and> list_all P xs)"
SpecAnno: fixes P :: "'d set"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and Q :: "'d set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('d, 'f, 'g) com" assumes "P \<subseteq> {s. \<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A}"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c Z (Q' Z),(A' Z)"   and "\<forall>Z. c Z = c undefined" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P specAnno P' c Q' A' Q,A"
f_approx_def: fixes x :: "real" shows "f_approx x = x powr p * (1 + integral (\<lambda>u. g u / u powr (p + 1)) x\<^sub>0 x)"
meridian_reflect_def: shows "meridian_reflect \<equiv> cltn2_abs rep_meridian_reflect"
distinct_adj_def: shows "distinct_adj = successively (\<noteq>)"
domain_eq: shows "Domainp (pcr_nd_fun (=)) = s2p UNIV"
run_altc_sample_option: fixes altc_sample :: "'f \<Rightarrow> ('g \<Rightarrow> 'd) \<Rightarrow> 'd"   and p :: "'f"   and f :: "'g \<Rightarrow> ('e, 'd) optionT" shows "run_option (altc_sample_option altc_sample p f) = altc_sample p (\<lambda>x. run_option (f x))"
smult_distrib_left: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and m :: "'b"   and n :: "'b" assumes "RModule R smult M"   and "r \<in> R"   and "m \<in> M"   and "n \<in> M" shows "smult r (m + n) = smult r m + smult r n"
B: shows "(tFirst t, tLast t) \<in> set (toList (snd ec))"
not_sym: fixes t :: "'h"   and s :: "'h" assumes "t \<noteq> s" shows "s \<noteq> t"
Bprop: fixes T :: "real" shows "\<bar>Si T\<bar> \<le> B"
cInf_le_cSup: fixes A :: "'a multiset set" assumes "A \<noteq> {}"   and "subset_mset.bdd_above A"   and "subset_mset.bdd_below A" shows "Inf A \<subseteq># Sup A"
apply_swap_simp_2: fixes a :: "'a"   and b :: "'a" shows "\<langle>a \<leftrightarrow> b\<rangle> \<langle>$\<rangle> b = a"
arrE: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VxVxV.arr f" obtains "B.arr (fst f) \<and> B.VxV.arr (snd f)"
sin_converges: fixes x :: "'b" shows "(\<lambda>n. sin_coeff n *\<^sub>R x ^ n) sums sin x"
T_insert_def: fixes x :: "'b"   and t :: "'b tree" shows "T_insert x t = 1 + (if t = \<langle>\<rangle> then 0 else T_splay x t)"
deriv_closed: fixes R :: "('a, 'b) ring_scheme"   and f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "a \<in> carrier R" shows "derivative R f a \<in> carrier R"
elims: fixes x :: "'b btree"   and y :: "nat" assumes "height x = y" obtains "x = Leaf" and "y = 0"   | ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = Node ts t" and "y = Suc (Max (height ` set (subtrees ts @ [t])))"
fresh_3: fixes a :: "name"   and x2 :: "trm"   and x1 :: "trm" shows "a \<sharp> App x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
map_spmf_idI: fixes p :: "'a spmf"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> f x = x" shows "map_spmf f p = p"
in_set_zip: fixes p :: "'c \<times> 'd"   and xs :: "'c list"   and ys :: "'d list" shows "(p \<in> set (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n < length xs \<and> n < length ys)"
eq_implies_ltl_equiv: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<sim>\<^sub>L \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<psi>"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> parts H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>X Y. a = X \<and> \<lbrace>X, Y\<rbrace> \<in> parts H) \<or> (\<exists>X Y. a = Y \<and> \<lbrace>X, Y\<rbrace> \<in> parts H) \<or> (\<exists>K X. a = X \<and> Crypt K X \<in> parts H) \<or> (\<exists>x y. a = Pri_AgrK x \<and> Auth_Data x y \<in> parts H) \<or> (\<exists>x y. a = Pri_AgrK y \<and> Auth_Data x y \<in> parts H))"
tagged_division_of_def: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" shows "(s tagged_division_of i) = (s tagged_partial_division_of i \<and> \<Union> {K. \<exists>x. (x, K) \<in> s} = i)"
invariantD: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and s :: "'e" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "s \<in> reachable A I" shows "P s"
bqtran_def: fixes x :: "'b"   and y :: "'b" shows "\<lfloor>x\<rfloor> y = - x \<squnion> y"
assms: fixes n :: "nat" assumes "n < length ias" shows "interrupt_action_ok' (redT_updIs is (take n ias)) (ias ! n)"
K2_def: shows "K2 \<equiv> {p. conic_sgn p < 0}"
multiplicative_dirichlet_inverse: fixes f :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f" shows "multiplicative_function (dirichlet_inverse f (1::'b))"
pos_bounded: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_linear f" shows "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
a_def: shows "a \<equiv> \<lambda>k. let (dk, ak, bk, M) = DF (Suc k) in ak"
ide_cod: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and f :: "'ag" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.cod C f)"
less_aexp_aux_antisym: fixes x :: "'b aexp"   and y :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux x y = (\<not> AExp_Lexorder.less_aexp_aux y x \<and> x \<noteq> y)"
newton_seq_is_Zp_cauchy: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "hensel p f a" shows "padic_integers.is_Zp_cauchy p (hensel.newton_seq p f a)"
product_composition_lemma: fixes c :: "'a fps"   and d :: "'a fps"   and a :: "'a fps"   and b :: "'a fps"   and n :: "nat" assumes "fps_nth c 0 = (0::'a)"   and "fps_nth d 0 = (0::'a)" shows "fps_nth ((a oo c) * (b oo d)) n = (\<Sum>(k, m)\<in>{(k, m). k + m \<le> n}. fps_nth a k * fps_nth b m * fps_nth (c ^ k * d ^ m) n)"
k_isNode_def: fixes ec :: "'rep" shows "k_isNode ec \<equiv> simAbs ec \<in> sim_equiv_class a ` jkbpC"
upper_asymptotic_density_lim: fixes A :: "nat set"   and l :: "real" assumes "(\<lambda>n. real (card (A \<inter> {..<n})) / real n) \<longlonglongrightarrow> l" shows "upper_asymptotic_density A = l"
stable_always: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool" shows "stable P = ((\<exists>\<sigma>. P \<sigma>) \<and> P = \<box>P)"