hcju/isabelleps huggingface dataset:
valid_data_dependency_prog_step: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')"   and "valid ts" shows "data_dependency_consistent_instrs (dom \<theta> \<union> load_tmps is) is' \<and> load_tmps is' \<inter> \<Union> (fst ` store_sops is) = {} \<and> load_tmps is' \<inter> \<Union> (fst ` write_sops sb) = {}"
src_dom: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "src (partial_magma.dom V \<mu>) = src \<mu>"
nodes_def: shows "DGCA.nodes \<equiv> automaton.nodes' dgca.alphabet dgca.initial dgca.transition"
hyps_2: assumes "as \<noteq> bs" shows "as \<parallel> bs"
not_empty_means_seq: shows "\<forall>i\<in>{S. \<exists>m n. {m..n} = S}. i \<noteq> {} \<longrightarrow> (\<exists>m n. m \<le> n \<and> {m..n} = i)"
GuardFault: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'f"   and c :: "('d, 'e, 'f) com"   and n :: "nat" assumes "s \<notin> g" shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> =n\<Rightarrow> Fault f"
S_add_order2: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "add_order s = 2"
addr_of_sys_xcpt_def: fixes C :: "String.literal" shows "addr_of_sys_xcpt C = the (map_of (zip initialization_list start_addrs) C)"
span_explicit': fixes b :: "'b set" shows "cspan b = {\<Sum>v | f v \<noteq> 0. f v *\<^sub>C v |f. finite {v. f v \<noteq> 0} \<and> (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> b)}"
that: fixes x :: "'a" assumes "x \<noteq> (0::'a)"   and "x \<in> span T"   and "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y" shows "thesis"
ccomp_ccomp: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "ccomp c\<^sub>1 @ ccomp c\<^sub>2 \<Turnstile> cfs\<box>"   and "\<And>cfs. ccomp c\<^sub>1 \<Turnstile> cfs\<box> \<Longrightarrow> cpred c\<^sub>1 (cfs ! 0) (cfs ! (length cfs - 1))"   and "\<And>cfs. ccomp c\<^sub>2 \<Turnstile> cfs\<box> \<Longrightarrow> cpred c\<^sub>2 (cfs ! 0) (cfs ! (length cfs - 1))" shows "case cfs ! 0 of (pc, s, stk) \<Rightarrow> pc = 0 \<and> (\<exists>k<length cfs. \<exists>t. case cfs ! k of (pc', s', stk') \<Rightarrow> pc' = size (ccomp c\<^sub>1 @ ccomp c\<^sub>2) \<and> (c\<^sub>1, s) \<Rightarrow> t \<and> (c\<^sub>2, t) \<Rightarrow> s' \<and> stk' = stk)"
br_def: fixes \<alpha> :: "'g \<Rightarrow> 'h"   and I :: "'g \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
implode_explode_eq: fixes s :: "String.literal" shows "String.implode (literal.explode s) = s"
product_nn_integral_component: fixes f :: "real \<Rightarrow> ennreal"   and i :: "nat" assumes "??.local.random_variable i borel f"   and "i \<in> UNIV" shows "\<integral>\<^sup>+ x. f (x i) \<partial>Pi\<^sub>M UNIV (\<lambda>n. exponential (R n)) = integral\<^sup>N (exponential (R i)) f"
eq_iff: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(x = y) = (ord (PP x) (PP y) \<and> ord (PP y) (PP x))"
recurse_rev_def: fixes f :: "'c \<Rightarrow> ('d, 'e) state dual" shows "recurse_rev f = un_B \<circ> traverse_rev_rev (B \<circ> f)"
subterms_subst_subset: fixes \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term"   and t :: "('d, 'f) Term.term" shows "subterms t \<cdot>\<^sub>s\<^sub>e\<^sub>t \<sigma> \<subseteq> subterms (t \<cdot> \<sigma>)"
ainv_def: shows "ainv \<equiv> dirichlet_inverse a y"
simps_7: fixes var :: "'e"   and p :: "('d, 'e, 'f) formula" shows "SIGF (InContext var p) = {Inr (Inl var)} \<union> SIGF p"
option_of_exception_\<I>: fixes \<I> :: "('a, 'b) \<I>" shows "map_\<I> id option_of_exception (exception_\<I> \<I>) = stop_\<I> \<I>"
sat_eq: fixes c :: "('a, 'b) conf" shows "Conf.sat c = (states c \<noteq> {})"
filtercomap_neq_bot: fixes F :: "'a filter"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>P. eventually P F \<Longrightarrow> \<exists>x. P (f x)" shows "filtercomap f F \<noteq> bot"
intros_10: fixes X :: "cmsg"   and Y :: "cmsg" assumes "X \<in> cpayload"   and "Y \<in> cpayload" shows "cExp X Y \<in> cpayload"
face_cycle_sets_def: shows "face_cycle_sets = face_cycle_set ` arcs G"
Zero: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_mset' R {#} {#}"
zero_literal_def: shows "STR '''' \<equiv> literal.Abs_literal []"
induct: fixes x1 :: "'a ltln"   and x2 :: "'a ltln"   and P :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" assumes "x1 \<turnstile>\<^sub>s x2"   and "\<And>uu_. P uu_ true\<^sub>n"   and "\<And>uv_. P false\<^sub>n uv_"   and "\<And>\<phi>. \<phi> = \<phi> \<Longrightarrow> P \<phi> \<phi>"   and "\<And>\<phi> \<chi> \<psi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>\<rbrakk> \<Longrightarrow> P (\<phi> and\<^sub>n \<psi>) \<chi>"   and "\<And>\<psi> \<chi> \<phi>. \<lbrakk>\<psi> \<turnstile>\<^sub>s \<chi>; P \<psi> \<chi>\<rbrakk> \<Longrightarrow> P (\<phi> and\<^sub>n \<psi>) \<chi>"   and "\<And>\<phi> \<psi> \<chi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<psi>; P \<phi> \<psi>; \<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>\<rbrakk> \<Longrightarrow> P \<phi> (\<psi> and\<^sub>n \<chi>)"   and "\<And>\<phi> \<psi> \<chi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<psi>; P \<phi> \<psi>\<rbrakk> \<Longrightarrow> P \<phi> (\<psi> or\<^sub>n \<chi>)"   and "\<And>\<phi> \<chi> \<psi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>\<rbrakk> \<Longrightarrow> P \<phi> (\<psi> or\<^sub>n \<chi>)"   and "\<And>\<phi> \<chi> \<psi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>; \<psi> \<turnstile>\<^sub>s \<chi>; P \<psi> \<chi>\<rbrakk> \<Longrightarrow> P (\<phi> or\<^sub>n \<psi>) \<chi>"   and "\<And>\<phi> \<chi> \<psi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>\<rbrakk> \<Longrightarrow> P \<phi> (\<psi> U\<^sub>n \<chi>)"   and "\<And>\<phi> \<chi> \<psi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>; \<psi> \<turnstile>\<^sub>s \<chi>; P \<psi> \<chi>\<rbrakk> \<Longrightarrow> P (\<phi> U\<^sub>n \<psi>) \<chi>"   and "\<And>\<phi> \<chi> \<psi> \<nu>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>; \<psi> \<turnstile>\<^sub>s \<nu>; P \<psi> \<nu>\<rbrakk> \<Longrightarrow> P (\<phi> U\<^sub>n \<psi>) (\<chi> U\<^sub>n \<nu>)"   and "\<And>\<chi> \<phi> \<psi>. \<lbrakk>\<chi> \<turnstile>\<^sub>s \<phi>; P \<chi> \<phi>\<rbrakk> \<Longrightarrow> P (\<psi> R\<^sub>n \<chi>) \<phi>"   and "\<And>\<chi> \<phi> \<psi>. \<lbrakk>\<chi> \<turnstile>\<^sub>s \<phi>; P \<chi> \<phi>; \<chi> \<turnstile>\<^sub>s \<psi>; P \<chi> \<psi>\<rbrakk> \<Longrightarrow> P \<chi> (\<phi> R\<^sub>n \<psi>)"   and "\<And>\<phi> \<chi> \<psi> \<nu>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<chi>; P \<phi> \<chi>; \<psi> \<turnstile>\<^sub>s \<nu>; P \<psi> \<nu>\<rbrakk> \<Longrightarrow> P (\<phi> R\<^sub>n \<psi>) (\<chi> R\<^sub>n \<nu>)"   and "\<And>\<phi> \<psi>. \<lbrakk>G\<^sub>n \<phi> \<turnstile>\<^sub>s \<psi>; P (G\<^sub>n \<phi>) \<psi>\<rbrakk> \<Longrightarrow> P (G\<^sub>n \<phi>) (X\<^sub>n \<psi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s F\<^sub>n \<psi>; P \<phi> (F\<^sub>n \<psi>)\<rbrakk> \<Longrightarrow> P (X\<^sub>n \<phi>) (F\<^sub>n \<psi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>\<phi> \<turnstile>\<^sub>s \<psi>; P \<phi> \<psi>\<rbrakk> \<Longrightarrow> P (X\<^sub>n \<phi>) (X\<^sub>n \<psi>)" shows "P x1 x2"
scaleC_scaleC: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "complex"   and b :: "complex"   and x :: "'d" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC a (scaleC b x) = scaleC (a * b) x"
simps_47: fixes \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" shows "mmonitorable_exec (formula.Until \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> right I \<noteq> \<infinity> \<and> (mmonitorable_exec \<phi> \<or> (case \<phi> of formula.Neg x \<Rightarrow> mmonitorable_exec x | _ \<Rightarrow> False)) \<and> mmonitorable_exec \<psi>)"
find_None_iff: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(find P xs = None) = (\<nexists>x. x \<in> set xs \<and> P x)"
cinsert_cDiff_single: fixes a :: "'f"   and A :: "'f cset" shows "cinsert a (cDiff A (csingle a)) = cinsert a A"
VcompNml_Can_Inv: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Can V src trg t"   and "bicategorical_language.Nml V src trg t" shows "bicategorical_language.VcompNml V t (bicategorical_language.Inv V t) = bicategorical_language.Cod V src trg t"
nth_nths: fixes i :: "nat"   and xs :: "'a list"   and I :: "nat set" assumes "i < card {i. i < length xs \<and> i \<in> I}" shows "nths xs I ! i = xs ! pick I i"
trim_def': shows "trim rs s = sdrop (minWait rs s) rs"
assms: shows "coincide_on \<sigma> \<eta> (vars_of_eq e)"
normalize_left_idem: fixes a :: "'b"   and b :: "'b" shows "gcd (normalize a) b = gcd a b"
simps: fixes uu :: "name"   and rhs :: "pterm" shows "prule (uu, rhs) = (pre_strong_term_class.wellformed rhs \<and> closed rhs \<and> is_abs rhs)"
prod_mset_replicate_mset: fixes times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and n :: "nat"   and a :: "'k" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one (replicate_mset n a) = power.power one times a n"
hrp_imp_def: fixes RR :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'b \<Rightarrow> assn)"   and RR' :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'b \<Rightarrow> assn)" shows "hrp_imp RR RR' \<equiv> \<forall>a b. (fst RR' a b \<Longrightarrow>\<^sub>t fst RR a b) \<and> (snd RR a b \<Longrightarrow>\<^sub>t snd RR' a b)"
get_child_nodes_locs_def: shows "get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = (\<lambda>ptr. (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) RElement.child_nodes)} else {}) \<union> (if is_document_ptr_kind ptr then {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^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 ptr)) document_element)} else {}) \<union> {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
Abs_finfun_inject_finite: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "finite UNIV" shows "(Abs_finfun x = Abs_finfun y) = (x = y)"
path_AppendE: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l3 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and l2 :: "'a"   and s :: "'b"   and l2' :: "'a" assumes "graph weights"   and "graph.path weights l1 l3 (xs @ [(l2, s, l2')])" obtains "l3 = l2'" and "graph.path weights l1 l2 xs" and "s \<in>\<^sub>A weights l2 l2'"
strict_implies_order: fixes b :: "'t"   and a :: "'t" assumes "b \<prec>\<^sub>t a" shows "b \<preceq>\<^sub>t a"
redundant_def: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'b Literal set"   and S :: "'b Literal set set" assumes "propositional_atoms atom_ordering" shows "redundant C S = (tautology C \<or> (\<exists>D. D \<in> S \<and> subsumes D C))"
sorted_iff_nth_mono: fixes xs :: "'d Word.word list" shows "signed.sorted xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> xs ! i \<le>s xs ! j)"
best_def: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>" shows "best outcomes \<R> = {x \<in> \<P> outcomes. \<forall>y\<in>\<P> outcomes. x \<succeq>[\<R>] y}"
unique_x_axis_poincare_distance_negative: fixes d :: "real" assumes "0 \<le> d" shows "\<exists>!z. is_real z \<and> Re z \<le> 0 \<and> - 1 < Re z \<and> poincare_distance 0\<^sub>h (of_complex z) = d"
cases: fixes x :: "example_peers \<times> nat" obtains "x = (PeerA, 0)"   | "x = (PeerB, 0)"   | "x = (PeerA, Suc 0)"   | "x = (PeerB, Suc 0)"   | "x = (PeerC, 0)"   | "x = (PeerC, Suc 0)"   | "x = (PeerC, Suc (Suc 0))"   | "x = (PeerC, Suc (Suc (Suc 0)))"   | "x = (PeerA, Suc (Suc 0))"   | "x = (PeerA, Suc (Suc (Suc 0)))"   | "x = (PeerB, Suc (Suc 0))"   | "x = (PeerB, Suc (Suc (Suc 0)))"   | n :: "nat" where "x = (PeerA, Suc (Suc (Suc (Suc n__))))"   | n :: "nat" where "x = (PeerB, Suc (Suc (Suc (Suc na__))))"   | n :: "nat" where "x = (PeerC, Suc (Suc (Suc (Suc n))))"
nth_subst: fixes n :: "nat"   and A :: "'a list"   and S :: "nat \<Rightarrow> typ" assumes "n < length A" shows "$ S A ! n = $ S (A ! n)"
iSPEC_eq: fixes \<Phi> :: "'b \<Rightarrow> bool" shows "SPEC \<Phi> = Sup {return x |x. \<Phi> x}"
e_def: shows "e = min (e0 / 2) (e1 / 2)"
sheaf_morphisms_sheaf_spec: fixes s :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"   and U :: "('a \<times> 'a) set set set" assumes "s \<in> local.local.sheaf_spec U" shows "local.local.sheaf_spec_morphisms U U s = s"
compl_sup_top: fixes x :: "'b upred" shows "(\<not> x \<or> x) = true"
spectral_radius_ev: shows "\<exists>ev v. eigen_vector A v ev \<and> cmod ev = HMA_Connect.spectral_radius A"
HSigma_iff: fixes a :: "hf"   and b :: "hf"   and A :: "hf"   and B :: "hf \<Rightarrow> hf" shows "(\<langle>a, b\<rangle> \<^bold>\<in> HSigma A B) = (a \<^bold>\<in> A \<and> b \<^bold>\<in> B a)"
hyps_3: assumes "f \<in> U \<rightarrow> carrier_vec n" shows "conjugate (finsum V f U) = finsum V (conjugate \<circ> f) U"
shiftl_rev: fixes w :: "'a word"   and n :: "nat" shows "w << n = word_reverse (word_reverse w >> n)"
composition_series_prefix_closed: fixes i :: "nat" assumes "i \<le> length \<GG>"   and "0 < i" shows "composition_series (G\<lparr>carrier := \<GG> ! (i - 1)\<rparr>) (take i \<GG>)"
lift_SKIP: fixes i :: "nat" shows "lift i \<bottom> = \<bottom>"
grey_reachable_def: fixes y :: "'h" shows "grey_reachable y = (\<lambda>s. \<exists>x. grey x s \<and> (x reaches y) s)"
elims: fixes x :: "three"   and xa :: "three"   and y :: "three" assumes "f2 x xa = y" obtains xb :: "three" where "x = xb__" and "xa = Zero" and "y = xb__"   | xb :: "three" where "x = xba__" and "xa = One" and "y = One"   | xb :: "three" where "x = xb" and "xa = Two" and "y = Two"
expectation_gpv_Fail: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'c \<Rightarrow> ennreal" shows "expectation_gpv fail \<I> f Fail = fail"
intro: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" assumes "iso_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 "iso_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"
coprime_0_right_iff: 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 a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a zero = dvd.dvd times a one"
inject: fixes x21 :: "'a"   and x22 :: "'a list"   and y21 :: "'a"   and y22 :: "'a list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
mult_left_le_one_le: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times y x) x"
dist_euclidean_space_def: fixes x :: "'a euclidean_space"   and y :: "'a euclidean_space" shows "dist x y = norm (x - y)"
e_prim_intro: fixes n1 :: "nat"   and e1 :: "exp"   and \<rho> :: "(nat \<times> val) list"   and n2 :: "nat"   and e2 :: "exp"   and v :: "val"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "VNat n1 \<in> E e1 \<rho>"   and "VNat n2 \<in> E e2 \<rho>"   and "v = VNat (f n1 n2)" shows "v \<in> E (EPrim f e1 e2) \<rho>"
Rep_ordinal_inverse: fixes x :: "ordinal" shows "Abs_ordinal (Rep_ordinal x) = x"
llength_LNil: shows "LList2.llength LNil = 0"
rep_eq: fixes x :: "'c"   and xa :: "'d"   and xb :: "('c, 'd) fmap" shows "($$) (xb(x $$:= xa)) = map_upd x xa (($$) xb)"
assms_2: shows "(Q, P) \<in> R \<rightarrow> bool_rel"
d_reachable_reachable: fixes M :: "('b, 'c, 'a) FSM"   and p :: "'a" shows "d_reachable M p \<subseteq> reachable M p"
power: fixes n :: "nat"   and m :: "nat" shows "fds_nth f (n ^ m) = fds_nth f n ^ m"
le_fun_any_inv: fixes t :: "(val \<times> val) list"   and v :: "val" assumes "VFun t \<sqsubseteq> v" obtains t2 :: "(val \<times> val) list" where "v = VFun t2" and "t \<lesssim> t2"
measure_spmf_of_set: fixes S :: "'b set"   and A :: "'b set" shows "Sigma_Algebra.measure (measure_spmf (spmf_of_set S)) A = real (card (S \<inter> A)) / real (card S)"
fold_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set" shows "Finite_Set.fold f z A = (if finite A then THE y. fold_graph f z A y else z)"
simps: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a1 :: "'b SDG_node"   and a2 :: "'b SDG_node list"   and a3 :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.call_cdep_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses a1 a2 a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n) \<or> (\<exists>n ns n'' n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> SDG.call_cdep_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'' \<and> SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None None n') \<or> (\<exists>n ns n'' p n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> SDG.call_cdep_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'' \<and> SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None (Some (p, True)) n'))"
cases: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "('a, 'c) com"   and a2 :: "'b" assumes "PL.mayDiverge tval aval a1 a2" obtains c :: "('a, 'c) com" and s :: "'b" and c' :: "('a, 'c) com" and s' :: "'b" where "a1 = c" and "a2 = s" and "PL.transC tval aval (c, s) (c', s') \<and> PL.mayDiverge tval aval c' s'"
intros_2: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and d :: "nat"   and dir :: "dir" assumes "p \<in> grid b ds"   and "d \<in> ds" shows "child p dir d \<in> grid b ds"
a_get_attribute_def: fixes ptr :: "(_) element_ptr"   and k :: "char list" shows "a_get_attribute ptr k = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. return (fmlookup m k))"
uminus_ms_aux_MSLCons_eval: fixes c :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist" shows "uminus_ms_aux (MSLCons (c, e) xs) = MSLCons (- c, e) (uminus_ms_aux xs)"
relcompp_assoc: fixes r :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and s :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "(r OO s) OO t = r OO s OO t"
arr_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "arr f = (f \<noteq> Null \<and> Dom f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> Cod f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom f) (Cod f)))"
simps_4: fixes a :: "'b"   and b :: "'b" shows "inf (dRETURN a) (dRETURN b) = (if a = b then dRETURN b else dSUCCEEDi)"
mult_right_le_imp_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less_eq (times a c) (times b c)"   and "less zero c" shows "less_eq a b"
not_final_thread_conv: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "not_final_thread s t = (thr s t \<noteq> None \<and> \<not> final_thread s t)"