hcju/isabelleps huggingface dataset:
Oops_range_spies1: fixes A :: "agent"   and KeyA :: "nat"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A (Crypt KeyA \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "authK \<notin> range shrK \<and> authK \<in> symKeys"
ID_distr_completion_1: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (\<E> [e]) = RID\<^sub>C \<E> \<union> RID\<^sub>E e"
oexp_def: shows "st.oexp = {(f, g). f \<in> st.FINFUNC \<and> g \<in> st.FINFUNC \<and> ((let m = t.max_fun_diff f g in (f m, g m) \<in> s) \<or> f = g)}"
assms_1: shows "\<rho> \<equiv> run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w"
cmp_rule_eq: fixes s :: "'a"   and ni :: "'b"   and i :: "'c ifex"   and ni' :: "'b"   and i' :: "'c ifex" assumes "I s"   and "(ni, i) \<in> R s"   and "(ni', i') \<in> R s" shows "cmp ni ni' = (i = i')"
jipsen1l: fixes x :: "'a"   and y :: "'a" shows "x \<le> inv_resl y (inv_resr x y)"
simps_3: fixes f :: "'d"   and \<phi> :: "'b" shows "L_transform_Tree (FL_Formula.Tree.tPred f \<phi>) = Formula.Act\<^sub>\<alpha> (Eff f) (Formula.Pred\<^sub>\<alpha> \<phi>)"
simps_28: fixes va :: "'b formula"   and vb :: "\<I>"   and vc :: "'b formula" shows "safe_formula (Neg (Until va vb vc)) = False"
of_int_le_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w \<le> z)"
integral_eq: fixes A :: "'a set"   and i :: "nat" assumes "A \<in> sets M" shows "emeasure N (Q i \<inter> A) = \<integral>\<^sup>+ x. f i x * indicator (Q i \<inter> A) x \<partial>M"
add_mset_eq_singleton_iff: fixes x :: "'c"   and M :: "'c multiset"   and y :: "'c" shows "(add_mset x M = {#y#}) = (M = {#} \<and> x = y)"
0: shows "\<forall>p\<in>num. prv (neg (PPf p \<langle>\<phi>G\<rangle>))"
connectedD: fixes g :: "'a ugraph"   and u :: "'a"   and v :: "'a" assumes "Undirected_Graph.connected g"   and "u \<in> nodes g"   and "v \<in> nodes g" shows "(u, v) \<in> (edges g)\<^sup>*"
max_node_is_Max_V: fixes la :: "(nat \<times> int \<times> nat) list" assumes "E = \<alpha> ` set la" shows "max_node la = Max (insert 0 V)"
PivotUpdateMinVars_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state" shows "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update \<equiv> PivotAndUpdate eq_idx_for_lvar pivot_and_update \<and> MinVars min_lvar_not_in_bounds eq_idx_for_lvar min_rvar_incdec_eq"
simps_2: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and ops :: "'a strips_operator list" shows "execute_serial_plan s (op # ops) = (if is_operator_applicable_in s op then execute_serial_plan (s \<then> op) ops else s)"
PUTList_None: fixes x :: "'c"   and list :: "('c \<Rightarrow> 'd decision option) list"   and PUT :: "'c \<Rightarrow> 'd decision option" assumes "x \<notin> dom (list2policy list)" shows "PUTList PUT x list"
ospecD: fixes x :: "'a option"   and P :: "'a \<Rightarrow> bool"   and y :: "'a" assumes "ospec x P"   and "x = Some y" shows "P y"
assms: shows "f ` (A \<union> B) = g ` C"
sets_P: fixes J :: "'i set" assumes "finite J"   and "J \<subseteq> I" shows "sets (P J) = sets (Pi\<^sub>M J M)"
distrib_left: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * (b \<squnion> c) = a * b \<squnion> a * c"
prems_2: shows "last (markedElements (a # M')) \<notin> set (elements (a # M'))"
diff_strict_left_mono: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less b a" shows "less (minus c a) (minus c b)"
in_verts_sccsD_sccs: fixes S :: "'a set" assumes "S \<in> sccs_verts" shows "G \<restriction> S \<in> sccs"
ring_simprules_14: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" shows "x \<ominus>\<^bsub>P\<^esub> y = x \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> y"
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and a1 :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and a2 :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "exec_1p P a1 a2" obtains \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" where "a1 = \<sigma>" and "a2 = \<sigma>'" and "exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>"
a_def_equiv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "a_aux a b c = (a \<squnion> d_aux a b c) \<sqinter> e_aux a b c"
hfun_sigma_iff: fixes r :: "hf" shows "hfun_sigma r = (hfunction r \<and> hrelation r)"
get_disconnected_document_pure: fixes ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "pure (get_disconnected_document ptr) h"
bounds_consistent_geq_lb: fixes s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and c :: "'b" assumes "\<diamond> s"   and "\<B>\<^sub>u s x\<^sub>i = Some c" shows "c \<ge>\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i"
TBBC: shows "T \<inter> B' \<union> (T \<inter> B \<union> T \<inter> wp act M) \<in> C"
pigeonhole_infinite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "infinite A"   and "finite (f ` A)" shows "\<exists>a0\<in>A. infinite {a \<in> A. f a = f a0}"
not_ex_in_conv: fixes A :: "'a set" shows "(\<nexists>x. x \<in> A) = (A = {})"
edge'_simps_2: fixes y :: "'v option"   and x :: "'v option ??.MFMC_Flow_Attainability.flow_attainability.node" shows "??.local.\<Psi>.edge' (??.MFMC_Flow_Attainability.flow_attainability.node.Inner y) x = (\<exists>ya. x = ??.MFMC_Flow_Attainability.flow_attainability.node.Inner ya \<and> edge \<Psi> y ya)"
iIN_iFROM_union: fixes n' :: "nat"   and n :: "nat"   and d :: "nat" assumes "n' \<le> Suc (n + d)" shows "[n\<dots>,d] \<union> [n'\<dots>] = [min n n'\<dots>]"
bounded_iff: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd" shows "(b \<oplus> c \<preceq> a) = (b \<preceq> a \<and> c \<preceq> a)"
in_sets_SUP: fixes i :: "'b"   and I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and Y :: "'c set"   and X :: "'c set" assumes "i \<in> I"   and "\<And>i. i \<in> I \<Longrightarrow> space (M i) = Y"   and "X \<in> sets (M i)" shows "X \<in> sets (\<Squnion> (M ` I))"
undo_local_step: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option"   and i :: "nat"   and u_ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and u_m :: "nat \<Rightarrow> nat"   and u_shared :: "nat \<Rightarrow> bool option"   and u_p :: "'a"   and u_is :: "instr list"   and u_tmps :: "nat \<Rightarrow> nat option"   and u_x :: "unit"   and u_dirty :: "bool"   and u_owns :: "nat set"   and u_rels :: "nat \<Rightarrow> bool option" assumes "direct_computation.concurrent_step (ts, m, \<S>) (ts', m', \<S>')"   and "i < length ts"   and "ts ! i = ts' ! i"   and "safe_delayed (u_ts, u_m, u_shared)"   and "length u_ts = length ts"   and "\<forall>j<length ts. j \<noteq> i \<longrightarrow> u_ts ! j = ts ! j"   and "u_ts ! i = (u_p, u_is, u_tmps, u_x, u_dirty, u_owns, u_rels)"   and "\<forall>a. a \<notin> u_owns \<longrightarrow> u_m a = m a"   and "\<forall>a. a \<in> u_owns \<longrightarrow> a \<in> dom u_shared \<longrightarrow> u_m a = m a"   and "\<forall>a. a \<notin> u_owns \<longrightarrow> a \<notin> owned (ts ! i) \<longrightarrow> u_shared a = \<S> a"   and "simple_ownership_distinct u_ts"   and "simple_ownership_distinct ts" shows "\<exists>u_ts' u_shared' u_m'. direct_computation.concurrent_step (u_ts, u_m, u_shared) (u_ts', u_m', u_shared') \<and> u_ts' ! i = u_ts ! i \<and> (\<forall>a\<in>u_owns. u_shared' a = u_shared a) \<and> (\<forall>a\<in>u_owns. \<S>' a = \<S> a) \<and> (\<forall>a\<in>u_owns. u_m' a = u_m a) \<and> (\<forall>a\<in>u_owns. m' a = m a) \<and> (\<forall>j<length ts. j \<noteq> i \<longrightarrow> u_ts' ! j = ts' ! j) \<and> (\<forall>a. a \<notin> u_owns \<longrightarrow> a \<notin> owned (ts ! i) \<longrightarrow> u_shared' a = \<S>' a) \<and> (\<forall>a. a \<notin> u_owns \<longrightarrow> u_m' a = m' a)"
single_is_union: fixes a :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" shows "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
Cd_range': fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Bossy''.Cd d X" shows "x \<in> X \<inter> Field (BPd' d)"
\<psi>_G_char: fixes f :: "'c"   and x :: "'c"   and x' :: "'c" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>C x'\<guillemotright>" shows "\<psi> x' (G f) = f \<cdot>\<^sub>C \<epsilon>o x"
update_convs_9: fixes getWatch2' :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option"   and getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getWatch2_update getWatch2' \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2' getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr>"
bigomega_def: fixes F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" shows "\<Omega>[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)}"
foldr_fun_upd: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and x :: "'b"   and g :: "'b \<Rightarrow> 'd"   and \<A> :: "'c \<Rightarrow> 'd" assumes "inj_on f (set xs)"   and "x \<in> set xs" shows "foldr (\<lambda>x \<A>. \<A>(f x := g x)) xs \<A> (f x) = g x"
segment_open_subset_closed: fixes a :: "'b"   and b :: "'b" shows "open_segment a b \<subseteq> closed_segment a b"
gcd_complex_def: shows "gcd = normalization_euclidean_semiring_class.gcd"
unique_NA: fixes B :: "agent"   and NA :: "nat"   and A :: "agent"   and evs :: "event list"   and B' :: "agent"   and A' :: "agent" assumes "Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (pubK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts (knows Spy evs)"   and "Nonce NA \<notin> analz (knows Spy evs)"   and "evs \<in> ns_public" shows "A = A' \<and> B = B'"
Per_perm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C" shows "Tarski_neutral_dimensionless.Per Bet Cong A B C \<and> Tarski_neutral_dimensionless.Per Bet Cong C B A"
infinite: fixes A :: "'b set" assumes "infinite A" shows "Min A = the None"
maxne0_le_Max: fixes x :: "nat list"   and a :: "nat list" shows "maxne0 x a \<le> Max (set a)"
run_one_step_basic_call_indirect_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x13 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Call_indirect x13) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
prems_1: fixes t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t'"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t" shows "thesis"
ArrayCpy_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and y :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (x[] ::= y, s) \<Rightarrow> t = (t = s(x := s y))"
prems: shows "Suc i < card {a. a < m \<and> a \<in> I}"
relax_precise_id: fixes w :: "'b \<times> 'b \<Rightarrow> enat"   and s :: "'b"   and D :: "'b \<Rightarrow> enat"   and S :: "'b set"   and v :: "'b"   and u :: "'b" assumes "Dijkstra_Invar w s D S"   and "D v = WGraph.\<delta> w s v" shows "Dijkstra.relax_outgoing w u D v = WGraph.\<delta> w s v"
simps_2: fixes x1 :: "real" shows "ereal x1 \<noteq> PInfty"
FRAME: shows "Ps \<Longrightarrow>\<^sub>A P * F"
Sel_upper_bound: fixes A :: "(symbol \<times> character list) set"   and B :: "(symbol \<times> character list) set" assumes "A \<subseteq> B" shows "Sel A B \<subseteq> B"
assms_1: shows "I = (\<lambda>l s. 0 \<le> s a)"
ovr_idem: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "idem_overrider F" shows "F x x = x"
ordinal_utility_def: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and u :: "'a \<Rightarrow> real" shows "ordinal_utility carrier relation u \<equiv> (\<forall>x y. x \<in> carrier \<longrightarrow> y \<in> carrier \<longrightarrow> x \<succeq>[relation] y = (u y \<le> u x)) \<and> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> x \<in> carrier) \<and> (\<forall>x y. x \<succeq>[relation] y \<longrightarrow> y \<in> carrier)"
unique_connecting_path_decomp_2: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c"   and ps :: "'c list"   and u :: "'c"   and ps' :: "'c list" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>"   and "v \<leadsto>\<^bsub>G\<^esub> w = ps @ u # ps'" shows "u # ps' = u \<leadsto>\<^bsub>G\<^esub> w"
Gromov_product_at_diff2: fixes e :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" shows "\<bar>Gromov_product_at e x z - Gromov_product_at e y z\<bar> \<le> dist x y"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes> y = y \<otimes> x"
child_out: fixes p :: "(nat \<times> int) list"   and d :: "nat"   and dir :: "dir" assumes "length p \<le> d" shows "child p dir d = p"
\<Phi>\<Psi>: fixes f :: "'s" assumes "S.arr f" shows "\<Psi> (\<Phi> f) = f"
pred_of_set_set_fold_sup: fixes xs :: "'b list" shows "pred_of_set (set xs) = fold sup (map Predicate.single xs) bot"
group_hom_isoI: assumes "\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>"   and "h ` carrier G = carrier H" shows "h \<in> Group.iso G H"
naturality: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.\<rho>' (\<F>C.cod f) \<cdot> \<F>C.map f = \<F>C.tensor f \<F>C.unity \<cdot> \<F>C.\<rho>' (\<F>C.dom f)"
filter_sup_def: fixes F :: "'a set"   and G :: "'a set" shows "filter_sup F G \<equiv> {z. \<exists>x\<in>F. \<exists>y\<in>G. x \<sqinter> y \<le> z}"
complete: fixes size :: "'b \<Rightarrow> nat"   and exhaustive :: "('b \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"   and n :: "nat"   and f :: "'b \<Rightarrow> (bool \<times> term list) option" assumes "class.complete size exhaustive" shows "(\<exists>v. size v \<le> n \<and> is_some (f v)) = is_some (exhaustive f (natural_of_nat n))"
simps_1: fixes x :: "'f \<Rightarrow> 'g decision option"   and y :: "'h \<Rightarrow> 'i decision option"   and ys :: "('h \<Rightarrow> 'i decision option) list"   and par_comb :: "('f \<Rightarrow> 'g decision option) \<Rightarrow> ('h \<Rightarrow> 'i decision option) \<Rightarrow> 'f \<times> 'h \<Rightarrow> ('g \<times> 'i) decision option"   and ran_adapt :: "'g \<times> 'i \<Rightarrow> 'e"   and dom_adapt :: "'d \<Rightarrow> 'f \<times> 'h" shows "(x \<Otimes>\<^sub>L y # ys) par_comb ran_adapt dom_adapt = (ran_adapt o\<^sub>f (par_comb x y \<circ> dom_adapt)) # (x \<Otimes>\<^sub>L ys) par_comb ran_adapt dom_adapt"
may_lock_unlock_lock_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "has_lock l t" shows "may_lock (unlock_lock l) t = may_lock l t"
hom_char: fixes a :: "'c \<times> 'c \<times> 'c"   and b :: "'c \<times> 'c \<times> 'c" shows "C.VVV.hom a b = C.VxVxV.hom a b \<inter> {\<tau>\<mu>\<nu>. C.arr (fst \<tau>\<mu>\<nu>) \<and> C.VV.arr (snd \<tau>\<mu>\<nu>) \<and> src\<^sub>C (fst \<tau>\<mu>\<nu>) = trg\<^sub>C (fst (snd \<tau>\<mu>\<nu>))}"
i2: shows "k \<le> mod_type_class.to_nat i"
K: shows "r < K \<and> ereal K < conv_radius (fps_nth f)"
intro_elim_6_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]"   and "[\<psi> in v]" shows "[\<phi> in v]"
raw_match'_assoc_match: fixes T :: "typ"   and U :: "typ"   and subs :: "((variable \<times> String.literal set) \<times> typ) list" shows "raw_match' T U (map_of subs) = map_option map_of (assoc_match T U subs)"
setmult_rcos_assoc: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier H'"   and "K \<subseteq> carrier H'"   and "x \<in> carrier H'" shows "H <#>\<^bsub>H'\<^esub> (K #>\<^bsub>H'\<^esub> x) = H <#>\<^bsub>H'\<^esub> K #>\<^bsub>H'\<^esub> x"
un_Fun1_def: fixes "term" :: "('e, 'f) Term.term" shows "the_Fun term = (case term of Fun x21 x22 \<Rightarrow> x21)"
iwlsFSwSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "iwlsFSwSb MOD \<equiv> iwlsFSw MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igSubstCls MOD"
compSupp: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "supp (\<Psi> \<otimes> \<Psi>') \<subseteq> supp \<Psi> \<union> supp \<Psi>'"
simple_closed_path_winding_number_cases: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "simple_path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "z \<notin> path_image \<gamma>" shows "winding_number \<gamma> z \<in> {- 1, 0, 1}"
types_preserved_get_local: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and i :: "nat"   and ts :: "t list"   and ts' :: "t list"   and vi :: "v list"   and v :: "v"   and vs :: "v list" assumes "\<S>\<bullet>\<C> \<turnstile> [$Get_local i] : ts _> ts'"   and "length vi = i"   and "local \<C> = map typeof (vi @ [v] @ vs)" shows "\<S>\<bullet>\<C> \<turnstile> [$C v] : ts _> ts'"
run_get_option: fixes get :: "('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> ('b, 'a) optionT" shows "run_option (get_option get f) = get (\<lambda>s. run_option (f s))"
bind_eq_None_conv: fixes a :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "(Option.bind a f = None) = (a = None \<or> f (the a) = None)"
simps_9: shows "eval_ternary_Imp TernaryUnknown TernaryUnknown = TernaryUnknown"
iso_to_GIrrRep_rep: fixes G :: "'b set"   and smult :: "('c, 'b) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and V :: "'d set" assumes "IrrFinGroupRepresentation G smult V" shows "\<exists>U\<in>Group.GIrrRep_repset G. \<exists>T. FGModuleIso G smult V (*) T U"
nnf_simps_7: fixes x :: "'a"   and y :: "'a" shows "(\<not> x < y) = (y \<le> x)"
mult_indicator_subset: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<subseteq> B" shows "indicator A x * indicator B x = indicator A x"
emeasure_CI: fixes J :: "nat set"   and n :: "nat"   and X :: "(nat \<Rightarrow> 'a) set" assumes "J \<subseteq> {0..<n}"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure (CI J) X = emeasure (C 0 n (\<lambda>_. undefined)) (PF.emb {0..<n} J X)"
intras_same_level_path: fixes as :: "'edge list" assumes "\<forall>a\<in>set as. intra_kind (kind a)" shows "same_level_path as"
homeomorphic_open_interval_UNIV: fixes a :: "real"   and b :: "real" assumes "a < b" shows "{a<..<b} homeomorphic UNIV"
sorted_mid_iff: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "sorted (xs @ y # ys) = (sorted (xs @ [y]) \<and> sorted (y # ys))"
le_not_less_replicate: fixes xs :: "nat list"   and b :: "nat" assumes "\<forall>x\<in>set xs. x \<le> b"   and "\<not> xs <\<^sub>v replicate (length xs) b" shows "xs = replicate (length xs) b"
adj_Proj: fixes M :: "'a ccsubspace" shows "Proj M* = Proj M"
succ_0_plus_eq: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>" shows "ZFC_in_HOL.succ 0 + \<alpha> = ZFC_in_HOL.succ \<alpha>"
reachable_def: fixes ioa :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and s :: "'e" shows "reachable ioa s \<equiv> \<exists>ex\<in>executions ioa. \<exists>n. snd ex n = s"
fresh_atm: fixes a :: "name"   and b :: "name" shows "a \<sharp> b = (a \<noteq> b)"
rel_S_def: fixes L1 :: "'a \<Rightarrow> 'i \<Rightarrow> bool"   and L2 :: "'b \<Rightarrow> 'j \<Rightarrow> bool"   and Co1 :: "'c \<Rightarrow> 'k \<Rightarrow> bool"   and Co2 :: "'d \<Rightarrow> 'l \<Rightarrow> bool"   and Contra1 :: "'e \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2 :: "'g \<Rightarrow> 'n \<Rightarrow> bool" shows "rel_S L1 L2 Co1 Co2 Contra1 Contra2 = vimage2p Rep_S Rep_S (rel_G L1 L2 Co1 Co2 Contra1 Contra2)"
order_trans: fixes x :: "'d list"   and y :: "'d list"   and z :: "'d list" assumes "prefix x y"   and "prefix y z" shows "prefix x z"
linepath_image_01: fixes a :: "'a"   and b :: "'a" shows "linepath a b ` {0..1} = closed_segment a b"
Em: fixes n :: "nat" shows "emeasure lebesgue (E n) < ennreal (1 / 2 ^ n)"