hcju/isabelleps huggingface dataset:
new_document_is_document_ptr: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "is_document_ptr\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr"
simps_6: fixes bs :: "int list"   and a :: "Cooper.num"   and b :: "Cooper.num" shows "Inum bs (Sub a b) = Inum bs a - Inum bs b"
eq_sym: fixes x :: "'d list"   and y :: "'d list" assumes "equivclp cancel1 x y" shows "equivclp cancel1 y x"
subsumes_list_modulo_Cons: fixes L :: "('a, 'b) Term.term literal"   and Ls :: "('a, 'b) Term.term literal list"   and Ks :: "('a, 'b) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term option" shows "subsumes_list_modulo (L # Ls) Ks \<sigma> = (\<exists>K\<in>set Ks. \<exists>\<tau>. extends_subst \<sigma> \<tau> \<and> dom \<tau> = vars_lit L \<union> dom \<sigma> \<and> L \<cdot>lit subst_of_map Var \<tau> = K \<and> subsumes_list_modulo Ls (remove1 K Ks) \<tau>)"
intro: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "po_on P A"   and "wfp_on P A" shows "minimal_element P A"
bchoice: fixes S :: "'c set"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<forall>x\<in>S. \<exists>y. Q x y" shows "\<exists>f. \<forall>x\<in>S. Q x (f x)"
cSUP_upper: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> A"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq (f x) (Sup (f ` A))"
m_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (homology_group p X)"   and "y \<in> carrier (homology_group p X)" shows "x \<otimes>\<^bsub>homology_group p X\<^esub> y \<in> carrier (homology_group p X)"
ccpo_Sup_singleton: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.ccpo Sup less_eq less" shows "Sup (\<eta> x) = x"
runit_simps_4: fixes f :: "'a" assumes "B.ide f" shows "B.dom (B.runit f) = f \<star>\<^sub>B src\<^sub>B f"
paths_le_idempotent: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "paths_le k (paths_le k P) = paths_le k P"
rep_eq: fixes x :: "nat"   and xa :: "float"   and xb :: "float" shows "real_of_float (float_divl x xa xb) = real_divl x (real_of_float xa) (real_of_float xb)"
set_5: fixes x51 :: "'b formula"   and x52 :: "'b formula" shows "atoms (x51 \<^bold>\<or> x52) = atoms x51 \<union> atoms x52"
inorder_inv_insert: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "rbt t \<and> Sorted_Less.sorted (Tree2.inorder t)" shows "rbt (RBT_Set.insert x t)"
alternating_order2_cancel_2left: fixes s :: "'a"   and t :: "'a"   and n :: "nat" assumes "s + s = (0::'a)"   and "t + t = (0::'a)" shows "sum_list (t # s # alternating_list (Suc (Suc n)) s t) = sum_list (alternating_list n s t)"
listt_update_in_list: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set"   and x :: "'b"   and i :: "nat" assumes "xs \<in> list n A"   and "x \<in> A" shows "xs[i := x] \<in> list n A"
list_se_match_2: fixes l2 :: "'a list"   and l1 :: "'a list"   and a :: "'a" assumes "l2 \<noteq> []" shows "(l1 @ l2 = [a]) = (l1 = [] \<and> l2 = [a])"
linear_scale_self: fixes c :: "'a" shows "Vector_Spaces.linear (*s) (*s) ((*s) c)"
while_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and e :: "'d"   and c :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((While e c, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = mds \<and> mem' = mem"
general_def: fixes t :: "triangle" shows "general t = (\<not> special t)"
cadjoint_eqI: fixes F :: "'a \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'a" assumes "\<And>x y. F x \<bullet>\<^sub>C y = x \<bullet>\<^sub>C G y" shows "G\<^sup>\<dagger> = F"
v0_v1_Digraph_def: fixes G :: "('a, 'b) Graph_scheme"   and v0 :: "'a"   and v1 :: "'a" shows "v0_v1_Digraph G v0 v1 \<equiv> Digraph G \<and> v0_v1_Digraph_axioms G v0 v1"
g_orbitalI: fixes X :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b"   and t :: "real"   and G :: "'b \<Rightarrow> bool" assumes "X \<in> Sols f U S t\<^sub>0 s"   and "t \<in> U s"   and "\<P> X (down (U s) t) \<subseteq> {s. G s}" shows "X t \<in> x\<acute>=f & G on U S @ t\<^sub>0 s"
le_refl: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a" assumes "weak_partial_order L"   and "x \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> x"
meld_correct_1: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and q' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar q"   and "SkewBinomialHeapStruc.invar q'" shows "SkewBinomialHeapStruc.invar (SkewBinomialHeapStruc.meld q q')"
dual_star_MonoTran_def: fixes x :: "'a MonoTran" shows "x ^ \<otimes> = ((x ^ o) ^ *) ^ o"
ucast_mask_drop: fixes n :: "nat"   and x :: "'c word" assumes "LENGTH('b) \<le> n" shows "UCAST('c \<rightarrow> 'b) (x AND mask n) = UCAST('c \<rightarrow> 'b) x"
find_closest_bf_code_dist_eq: fixes ps :: "(int \<times> int) list"   and \<delta> :: "int"   and c :: "int \<times> int"   and p :: "int \<times> int" assumes "0 < length ps"   and "(\<delta>, c) = find_closest_bf_code p ps" shows "\<delta> = dist_code p c"
carrier_0: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "(x \<in> carrier \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>) = (x \<noteq> zero)"
distinct_74: fixes pi1' :: "pi"   and pi2' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "pi1' \<parallel> pi2' \<noteq> [name1\<noteq>name2]pi"
has_derivative_in_compose2: fixes t :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b"   and s :: "'d set"   and x :: "'d"   and f' :: "'d \<Rightarrow> 'b" assumes "\<And>x. x \<in> t \<Longrightarrow> (g has_derivative g' x) (at x within t)"   and "f ` s \<subseteq> t"   and "x \<in> s"   and "(f has_derivative f') (at x within s)" shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>y. g' (f x) (f' y))) (at x within s)"
monad_fmap: fixes f :: "'g \<rightarrow> 'e"   and xs :: "'g\<cdot>'f" shows "fmap\<cdot>f\<cdot>xs = xs \<bind> (\<Lambda> x. return\<cdot>(f\<cdot>x))"
hom_component_minus: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "hom_component (p - q) n = hom_component p n - hom_component q n"
k2: shows "k < ncols (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n))"
fold_simps_2: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list"   and s :: "'b" shows "fold f (x # xs) s = fold f xs (f x s)"
is_Prog\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Prog\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = False"
simps_1: fixes uu :: "'b"   and uv :: "'b"   and F :: "'a" shows "\<chi>.\<Phi>.Ya.Cop_S.Map (\<chi>.\<Phi>.Ya.Cop_S.MkArr uu uv F) = F"
resl_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> inv_resl z y) = (x \<cdot> y \<le> z)"
intro: fixes is_map :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'c \<Rightarrow> assn"   and lookup :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option Heap" assumes "imp_map_lookup is_map lookup" shows "gen_contains_key_by_lookup is_map lookup"
IH: fixes xs :: "'a stream" assumes "(holds P until holds Q) xs"   and "\<forall>i\<le>j. \<not> Q (xs !! i)" shows "P (xs !! j)"
arr_inv: fixes f :: "'a" assumes "B.iso f" shows "B.arr (B.inv f)"
ext_inject: fixes vars :: "(String.literal, variable) assoc_list"   and channels :: "channel list"   and timeout :: "bool"   and procs :: "pState list"   and more :: "'a"   and vars' :: "(String.literal, variable) assoc_list"   and channels' :: "channel list"   and timeout' :: "bool"   and procs' :: "pState list"   and more' :: "'a" shows "(\<lparr>gState.vars = vars, channels = channels, timeout = timeout, procs = procs, \<dots> = more\<rparr> = \<lparr>gState.vars = vars', channels = channels', timeout = timeout', procs = procs', \<dots> = more'\<rparr>) = (vars = vars' \<and> channels = channels' \<and> timeout = timeout' \<and> procs = procs' \<and> more = more')"
dghm_id_components_4: fixes \<CC> :: "V" shows "dghm_id \<CC>\<lparr>HomCod\<rparr> = \<CC>"
norm_cblinfun: fixes f :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and x :: "'b" shows "norm (f *\<^sub>V x) \<le> norm f * norm x"
sep_conj_impl1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and I :: "'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> bool"   and h :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "\<And>h. P h \<Longrightarrow> I h"   and "sep_algebra.sep_conj plus sep_disj P R h" shows "sep_algebra.sep_conj plus sep_disj I R h"
P_lnull_ltl_deadend_v0: assumes "lnull (ltl P')" shows "deadend (P $ n)"
map_of_zip_nth: fixes xs :: "'b list"   and ys :: "'c list"   and i :: "nat" assumes "length xs = length ys"   and "distinct xs"   and "i < length ys" shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)"
set_child_nodes_get_child_nodes_different_pointers: fixes ptr :: "(_) object_ptr"   and ptr' :: "(_) object_ptr"   and w :: "((_) heap, exception, unit) prog"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr \<noteq> ptr'"   and "w \<in> Shadow_DOM.set_child_nodes_locs ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> Shadow_DOM.get_child_nodes_locs ptr'" shows "r h h'"
rExt: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> {uu_. \<exists>\<Psi> P P' Q' Q. uu_ = (\<Psi>, P, Q) \<and> \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> {uu_. \<exists>\<Psi> P P' Q' Q. uu_ = (\<Psi>, P, Q) \<and> \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}"
matches_iff_apply_f: fixes \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and p :: "'b"   and f :: "'a match_expr \<Rightarrow> 'a match_expr"   and m :: "'a match_expr"   and \<alpha> :: "action \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "action" assumes "ternary_ternary_eval (map_match_tac \<beta> p (f m)) = ternary_ternary_eval (map_match_tac \<beta> p m)" shows "matches (\<beta>, \<alpha>) (f m) a p = matches (\<beta>, \<alpha>) m a p"
abs_e3: shows "\<bar>e3\<bar> \<le> snd (trunc_bound_pdevs p (scaleR_pdevs (fst (fst X)) (snd (fst Y))))"
c_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.c p q = card (QR.C p q)"
assms: shows "(s, a, s') \<in> automaton.trans (pnet np p)"
finite_UB: fixes xs :: "'a list"   and f :: "atom fm" shows "finite (UB f xs)"
strip_while_Cons: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "\<not> P x" shows "strip_while P (x # xs) = x # strip_while P xs"
monomials_liftPoly: fixes b :: "nat"   and i :: "nat"   and mp :: "'a mpoly" shows "monomials (liftPoly b i mp) = lowerPowers b i ` monomials mp"
vertices_def: fixes g :: "('a, 'b) rgraph_scheme" shows "vertices g = {root g} \<union> src ` edges g \<union> tgt ` edges g"
induct_hyp: shows "\<forall>f. (\<forall>n. f n = n + i) \<longrightarrow> f \<in> CofGroups.Ex1"
ts_tuple_rel_ext: fixes tas :: "nat \<times> 'a option list"   and nt :: "nat"   and X :: "'a option list set"   and Y :: "'a option list set"   and tass :: "(nat \<times> 'a option list set) list" assumes "tas \<in> ts_tuple_rel {(nt, X)}" shows "tas \<in> ts_tuple_rel (set ((nt, Y \<union> X) # tass))"
not_squarefree_primepow: fixes n :: "'a" assumes "primepow n" shows "squarefree n = prime n"
choose_good: fixes v :: "'a" assumes "v \<in> S - W" shows "choose v \<in> good v"
atMost_Int_atLeast: fixes n :: "'c" shows "{..n} \<inter> {n..} = {n}"
continuous_comp: fixes f :: "'b set \<Rightarrow> 'c set"   and g :: "'c set \<Rightarrow> 'd set" assumes "continuous f"   and "continuous g" shows "continuous (g \<circ> f)"
eOp_simp4: fixes einp :: "'index \<Rightarrow> 'c withERR option"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and ebinp :: "'bindex \<Rightarrow> 'd withERR option" assumes "\<not> liftAll (\<lambda>eX. eX \<noteq> ERR) einp" shows "eOp MOD delta einp ebinp = ERR"
CSP2_idem: shows "CSP2 = CSP2 \<circ> CSP2"
valid_data_dependency_nth_update: fixes i :: "nat"   and \<theta> :: "nat \<Rightarrow> nat option"   and "is" :: "instr list"   and sb :: "'a memref list"   and p :: "'a"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "data_dependency_consistent_instrs (dom \<theta>) is"   and "load_tmps is \<inter> \<Union> (fst ` write_sops sb) = {}" shows "valid_data_dependency (ts\<^sub>s\<^sub>b[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
nsets_def: fixes A :: "'b set"   and n :: "nat" shows "[A]\<^bsup>n\<^esup> \<equiv> {N. N \<subseteq> A \<and> finite N \<and> card N = n}"
ts_tp_lt'_def: fixes ts :: "nat"   and tp :: "nat"   and tstp :: "nat + nat" shows "ts_tp_lt' ts tp tstp = (case tstp of Inl ts' \<Rightarrow> ts < ts' | Inr tp' \<Rightarrow> tp \<le> tp')"
converse_Subobjs_Rep: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "P \<turnstile> last Cs \<prec>\<^sub>R C'"   and "wf_prog wf_md P" shows "Subobjs P C (Cs @ [C'])"
assigns_good_approx: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "normal s1" shows "assigns t \<subseteq> dom (locals (snd s1))"
cnf2bhv_lActive: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" shows "dynamic_component.cnf2bhv tCMP active c t (dynamic_component.lActive active c t) = the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1"
cmp_def: fixes g :: "'g \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'g" shows "cmp g f = (\<lambda>x. g (f x))"
chambersubcomplex_opp_fold_A_im: shows "ChamberComplex.ChamberSubcomplex A (opp_fold_A \<turnstile> A)"
has_integral_const_real: fixes c :: "'b"   and a :: "real"   and b :: "real" shows "((\<lambda>x. c) has_integral content {a..b} *\<^sub>R c) {a..b}"
stalks_are_local: fixes x :: "'a"   and U :: "'a set" assumes "x \<in> U"   and "is_open U" shows "stalk.is_local is_open \<FF> \<rho> add_str mult_str zero_str one_str (neighborhoods x) x U"
greedy_approach_leads_to_minBasis: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and F :: "'c set"   and weight :: "'c \<Rightarrow> 'd" assumes "weighted_matroid carrier indep"   and "indep F"   and "\<forall>e\<in>carrier - F. \<not> indep (insert e F)"   and "weighted_matroid.no_smallest_element_skipped carrier indep weight {} F" shows "weighted_matroid.minBasis carrier indep weight F"
Y_surjective_on_homs: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a"   and a' :: "'a"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "yoneda_lemma_for_hom C S \<phi> a a'"   and "natural_transformation (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) (yoneda_functor.Y C S \<phi> a') \<tau>" shows "yoneda_functor.Y C S \<phi> (hom_functor.\<psi> C \<phi> (a, a') (yoneda_lemma.\<E> S \<phi> a \<tau>)) = \<tau>"
simps_19: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and e :: "('d, 'e, 'b) exp"   and e' :: "('d, 'e, 'b) exp" shows "J1_heap_base.\<tau>move1 typeof_addr P h (e;; e') = (J1_heap_base.\<tau>move1 typeof_addr P h e \<or> final e)"
distinct_1: fixes x1 :: "real" shows "ereal x1 \<noteq> PInfty"
nn_integral_pmf_of_set: fixes S :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "S \<noteq> {}"   and "finite S" shows "integral\<^sup>N (measure_pmf (pmf_of_set S)) f = sum f S / of_nat (card S)"
prems_5: shows "\<forall>x\<in>snd ` set (a # txs). \<forall>t\<in>fst ` set (a # txs). x \<notin> FvarsT t"
\<I>_full_le_plus_\<I>: fixes \<I>1 :: "('a, 'b) \<I>"   and \<I>2 :: "('c, 'd) \<I>" assumes "\<I>_full \<le> \<I>1"   and "\<I>_full \<le> \<I>2" shows "\<I>_full \<le> \<I>1 \<oplus>\<^sub>\<I> \<I>2"
sum_squares_le_zero_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 x :: "'a"   and y :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (plus (times x x) (times y y)) zero = (x = zero \<and> y = zero)"
exhaust: fixes y :: "cld" obtains x1 :: "char list" and x2 :: "cl" and x3 :: "fd list" and x4 :: "meth_def list" where "y = cld_def x1 x2 x3 x4"
finter_finsert_right_if1: fixes a :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "a |\<in>| A" shows "A |\<inter>| finsert a B = finsert a (A |\<inter>| B)"
non_overlapping_extension2: fixes P :: "'b set set"   and X :: "'b set"   and new_el :: "'b" assumes "is_non_overlapping P"   and "X \<in> P"   and "new_el \<notin> \<Union> P" shows "is_non_overlapping (insert_into_member new_el P X)"
assms_3: assumes "\<psi> \<pi>l" shows "\<chi>"
4: shows "card {x. regular x \<and> x \<le> - - h \<and> x \<le> - minarc h \<and> x \<le> - minarc h\<^sup>T} < card {x. regular x \<and> x \<le> - - h}"
chamber_in_fold_A_im: fixes F :: "'a set" assumes "chamber F"   and "F \<in> fold_A \<turnstile> A" shows "F \<in> fold_A \<turnstile> ChamberComplex.chamber_system A"
abelian_group_axioms: shows "abelian_group R (+) \<zero>"
integrable_bij_count_space: fixes g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "bij_betw g A B" shows "integrable (count_space A) (\<lambda>x. f (g x)) = integrable (count_space B) f"
max2_underS'_2: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded" assumes "x \<in> underS Bonk_Schramm_extension_unfolded_wo y" shows "max2 Bonk_Schramm_extension_unfolded_wo y x = y"
scale_left_distrib: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "(a + b) * x = a * x + b * x"
not_fulfilled_single: shows "\<forall>e\<in>edges G - set (filter_IFS_no_violations G M edgesList). \<not> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = {e} \<union> set (filter_IFS_no_violations G M edgesList)\<rparr>)"
lc_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "lc (monom_mult c t p) = c * lc p"
obs_neq_contradicts_term: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and n :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and n' :: "nat" assumes "path \<sigma> n = local.return"   and "path \<sigma>' n' = local.return"   and "obs \<sigma> \<noteq> obs \<sigma>'" shows "\<exists>k k'. (\<sigma>', k') \<cc> (\<sigma>, k) \<and> path \<sigma> k \<in> dom local.att \<or> (\<sigma>, k) \<cc> (\<sigma>', k') \<and> path \<sigma>' k' \<in> dom local.att"
eOp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'a option"   and binp :: "'bindex \<Rightarrow> 'b option" assumes "igWlsInp MOD delta inp"   and "igWlsBinp MOD delta binp" shows "eOp MOD delta (OKI inp) (OKI binp) = OK (igOp MOD delta inp binp)"
Pdrop_valid: fixes n :: "nat" shows "valid_path (ldropn n P)"
add_edge_ind: fixes a :: "nat"   and w :: "int"   and b :: "nat"   and F :: "(nat \<times> int \<times> nat) set" assumes "(a, w, b) \<in> E" shows "add_edge a w b \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = F\<rparr> = \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert (a, w, b) F\<rparr>"
case_eq_if: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g"   and option :: "'h option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
cring_lcf_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "p (deg R p) \<otimes> q (deg R q) \<noteq> \<zero>" shows "(p \<otimes>\<^bsub>P\<^esub> q) (deg R (p \<otimes>\<^bsub>P\<^esub> q)) = p (deg R p) \<otimes> q (deg R q)"
eq_iff: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a = b) = (b \<le>s a \<and> a \<le>s b)"
InvGlobNonposImpRecordsNonpos_def: fixes c :: "('c, 'loc \<times> 't) Exchange.configuration" shows "cri.InvGlobNonposImpRecordsNonpos c = (\<forall>t q. cri.GlobNonposUpto c q t \<longrightarrow> cri.nonpos_upto (cri.records c) t)"
pmf_neq_exists_less: fixes M :: "'b pmf"   and N :: "'b pmf" assumes "M \<noteq> N" shows "\<exists>x. pmf M x < pmf N x"