hcju/isabelleps huggingface dataset:
drop_suffix': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and csxs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs'xs :: "('e, 'd, 'f) com list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and cs :: "('e, 'd, 'f) com list"   and xs :: "('e, 'd, 'f) com list"   and cs' :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile> (csxs, css, s) \<rightarrow> (cs'xs, css', t)"   and "css = css'"   and "csxs = cs @ xs"   and "cs'xs = cs' @ xs"   and "cs \<noteq> []" shows "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow> (cs', css, t)"
hpair_neq_one: fixes x :: "hf"   and y :: "hf" shows "\<langle>x, y\<rangle> \<noteq> 1"
plus_iarray_def: fixes A :: "'c iarray"   and B :: "'c iarray" shows "A + B = (let length_A = IArray.length A; length_B = IArray.length B; n = max length_A length_B; A' = IArray.of_fun (\<lambda>a. if a < length_A then A !! a else (0::'c)) n; B' = IArray.of_fun (\<lambda>a. if a < length_B then B !! a else (0::'c)) n in IArray.of_fun (\<lambda>a. A' !! a + B' !! a) n)"
carrier_both: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "carrier (t \<otimes>\<otimes> t') = carrier t \<union> carrier t'"
image2pI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "R x y" shows "BNF_Greatest_Fixpoint.image2p f g R (f x) (g y)"
su1_3: shows "su1 \<in> existence_ivl0 x"
eq_f: shows "functions_of P A B = range_permutation A B `` {f}"
I_l_closed: fixes a :: "'a"   and x :: "'a" assumes "a \<in> I"   and "x \<in> carrier R" shows "x \<otimes> a \<in> I"
in_mono: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
order_less_subst1: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and b :: "'d"   and c :: "'d" assumes "a < f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
list_member_conv_member: shows "equal_base.list_member (=) = List.member"
cases: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a1 :: "'e + 'g"   and a2 :: "'f + 'h" assumes "rel_sum R1 R2 a1 a2" obtains a :: "'e" and c :: "'f" where "a1 = Inl a" and "a2 = Inl c" and "R1 a c"   | b :: "'g" and d :: "'h" where "a1 = Inr b" and "a2 = Inr d" and "R2 b d"
simps_17: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and f4 :: "('c, 'd) Combinators \<Rightarrow> ('c, 'd) Combinators \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c" shows "(case DenyAllFromTo x21 x22 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f2 x21 x22"
mrw_value_cases: fixes x :: "('a, 'b) obs_event action" obtains ad :: "'a" and al :: "addr_loc" and v :: "'a val" where "x = NormalAction (WriteMem ad__ al__ v__)"   | ad :: "'a" and hT :: "htype" where "x = NormalAction (NewHeapElem ada__ hT)"   | ad :: "'a" and M :: "String.literal" and vs :: "'a val list" and v :: "'a val" where "x = NormalAction (ExternalCall adb__ M vs va__)"   | ad :: "'a" and al :: "addr_loc" and v :: "'a val" where "x = NormalAction (ReadMem adc__ al v)"   | t :: "'b" where "x = NormalAction (ThreadStart tb__)"   | t :: "'b" where "x = NormalAction (ThreadJoin tc__)"   | ad :: "'a" where "x = NormalAction (SyncLock add__)"   | ad :: "'a" where "x = NormalAction (SyncUnlock ad)"   | t :: "'b" where "x = NormalAction (ObsInterrupt td__)"   | t :: "'b" where "x = NormalAction (ObsInterrupted t)"   | "x = InitialThreadAction"   | "x = ThreadFinishAction"
axioms_2: fixes M :: "'b measure" assumes "prob_space M" shows "prob_space_axioms M"
result_bind_resumption: fixes x :: "('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done (x \<bind> f)" shows "Resumption.resumption.result (x \<bind> f) = Resumption.resumption.result x \<bind> Resumption.resumption.result \<circ> f"
HcompNml_term_Prim\<^sub>0: fixes t :: "'a local.term"   and \<mu> :: "'a" assumes "Src t = Trg \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = t"
sim_action_sym: fixes obs :: "('a, 'b) obs_event action"   and obs' :: "('a, 'b) obs_event action" assumes "obs \<approx> obs'" shows "obs' \<approx> obs"
Z_not_inj_def: shows "local.Z_not_inj = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> \<not> inj_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}}"
simps_3: fixes a :: "nat"   and b :: "nat"   and n :: "nat" shows "gen_fib a b (Suc (Suc n)) = gen_fib b (a + b) (Suc n)"
divide_cancel_right: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'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 :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a c = divide b c) = (c = zero \<or> a = b)"
set_concat: fixes xs :: "'g list list" shows "set (concat xs) = \<Union> (set ` set xs)"
RInitInitFail\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s :: "heap \<times> val list \<times> sheap"   and a :: "nat exp"   and h' :: "heap"   and l' :: "val list"   and sh' :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh'' :: "sheap"   and D :: "char list"   and Cs :: "char list list"   and e' :: "nat exp"   and e\<^sub>1 :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh')\<rangle>"   and "sh' C = \<lfloor>(sfs, i)\<rfloor>"   and "sh'' = sh'(C \<mapsto> (sfs, Error))"   and "P \<turnstile>\<^sub>1 \<langle>RI (D,throw a) ; Cs \<leftarrow> e',(h', l', sh'')\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>RI (C,e) ; D # Cs \<leftarrow> e',s\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>"
quasi_isometry_subset: fixes X :: "'b set"   and Y :: "'b set"   and C :: "real" assumes "X \<subseteq> Y"   and "\<And>y. y \<in> Y \<Longrightarrow> \<exists>x\<in>X. dist x y \<le> C"   and "0 \<le> C" shows "1 C -quasi_isometry_between X Y (\<lambda>x. x)"
multiple_incomplete: fixes n :: "nat" assumes "0 < n" shows "incomplete_design \<V> (multiple_blocks n) \<k>"
simps: fixes f\<^sub>1 :: "'c \<Rightarrow> 'b val"   and f\<^sub>2 :: "'d val \<Rightarrow> 'b val"   and v :: "'c" shows "F_Times2 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 v) (f\<^sub>2 Void)"
g: fixes y :: "'c" shows "mcont lub ord Sup (\<le>) (\<lambda>x. g x y)"
r_right_minus_eq: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)" shows "(a \<ominus>\<^bsub>poly_ring R\<^esub> b = \<zero>\<^bsub>poly_ring R\<^esub>) = (a = b)"
nucleus_alt_def3: fixes f :: "'b \<Rightarrow> 'b" assumes "\<forall>x y. f x \<rightarrow> f y = x \<rightarrow> f y"   and "\<forall>x y. f y \<leftarrow> f x = f y \<leftarrow> x" shows "nucleus f"
comparator_nat_term_compare_inv: fixes to :: "'a nat_term_order" shows "comparator (nat_term_compare_inv to)"
if_SE_D1: fixes P :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d"   and B\<^sub>1 :: "'d \<Rightarrow> (bool \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> (bool \<times> 'd) option" assumes "P \<sigma>" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2) = (\<sigma> \<Turnstile> B\<^sub>1)"
openin_Times_eq: fixes S :: "'a set"   and T :: "'b set"   and S' :: "'a set"   and T' :: "'b set" shows "openin (top_of_set (S \<times> T)) (S' \<times> T') = (S' = {} \<or> T' = {} \<or> openin (top_of_set S) S' \<and> openin (top_of_set T) T')"
minimal_antichain_path_weightp_member: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and s :: "'b"   and l1 :: "'a"   and l2 :: "'a" assumes "graph weights"   and "s \<in> {xs. graph.path_weightp weights l1 l2 xs}" shows "\<exists>u. u \<in> minimal_antichain {xs. graph.path_weightp weights l1 l2 xs} \<and> u \<le> s"
SigmaD1: fixes a :: "'h"   and b :: "'i"   and A :: "'h set"   and B :: "'h \<Rightarrow> 'i set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
monom_product_rule: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "pderiv (f \<otimes>\<^bsub>P\<^esub> monom P a n) = f \<otimes>\<^bsub>P\<^esub> pderiv (monom P a n) \<oplus>\<^bsub>P\<^esub> pderiv f \<otimes>\<^bsub>P\<^esub> monom P a n"
intros_3: fixes c1 :: "com"   and s1 :: "char list \<Rightarrow> int"   and x :: "nat"   and s2 :: "char list \<Rightarrow> int"   and c2 :: "com"   and y :: "nat"   and s3 :: "char list \<Rightarrow> int"   and z :: "nat" assumes "(c1, s1) \<Rightarrow> x \<Down> s2"   and "(c2, s2) \<Rightarrow> y \<Down> s3"   and "z = x + y" shows "(c1;; c2, s1) \<Rightarrow> z \<Down> s3"
left_invertible_iff_invertible: fixes M :: "((real, 'a) vec, 'a) vec" shows "(\<exists>N. N ** M = mat 1) = invertible M"
f_Exec_Stream_Acc_LocalState_eq_Acc_Trans_Fun_Step_LocalState: fixes k :: "nat"   and trans_fun :: "'d message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and localState :: "'e \<Rightarrow> 'f"   and pointwise_shrink :: "'e list \<Rightarrow> 'e"   and xs :: "'d message_af list"   and c :: "'e" assumes "0 < k"   and "Deterministic_Trans_Fun trans_fun localState"   and "is_correct_localState_Pointwise_Output_Shrink pointwise_shrink localState" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c = map localState (f_Exec_Comp_Stream (Acc_Trans_Fun_Step k trans_fun pointwise_shrink) xs c)"
simps_4: fixes x21 :: "'a"   and x22 :: "nat"   and x23 :: "'a interval"   and x11 :: "'a"   and x12 :: "nat"   and x13 :: "'a interval" shows "DEL x21 x22 x23 \<noteq> INS x11 x12 x13"
mult_le_cancel_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times a c) (times b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
allAllocationsFinite: fixes N :: "'d set"   and G :: "'e set" assumes "finite N"   and "finite G" shows "finite (allAllocations N G)"
is_pr_Iso_def: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list set"   and Fs\<^sub>2 :: "'b list set" shows "is_pr_Iso \<phi> Fs\<^sub>1 Fs\<^sub>2 \<equiv> is_pr_Hom \<phi> Fs\<^sub>1 Fs\<^sub>2 \<and> inj_on \<phi> (\<Union> (set ` Fs\<^sub>1))"
assms_2: fixes x :: "'a" shows "f x = \<langle>u, x\<rangle>"
listset_Cons: fixes x :: "'b set"   and xs :: "'b set list" shows "listset (x # xs) = (\<Union>y\<in>x. (#) y ` listset xs)"
eq_self_neq_FF: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and x :: "'b" assumes "class.Eq_equiv eq" shows "eq\<cdot>x\<cdot>x \<noteq> FF"
Fun_implies_PFun: fixes f :: "ZF"   and U :: "ZF"   and V :: "ZF" assumes "Elem f (Fun U V)" shows "Elem f (PFun U V)"
F_seq: fixes P :: "'b process"   and Q :: "'b process" shows "F (P `;` Q) = {(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> t2 \<in> D Q}"
sorted_list_of_set_nonempty: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}" shows "sorted_list_of_set A = Min A # sorted_list_of_set (A - {Min A})"
betw4_imp_neq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c d]]" shows "a \<noteq> b \<and> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<and> c \<noteq> d"
one_matrix_def: shows "1 \<equiv> Matrix (\<lambda>i j. if Rep_atMost i = Rep_atMost j then 1::'c else (0::'c))"
Inf_between_alt: fixes N :: "'f set"   and M :: "'f set" shows "no_labels.Inf_between N M = {\<iota> \<in> Inf_F. \<iota> \<in> no_labels.Inf_from (N \<union> M) \<and> set (prems_of \<iota>) \<inter> M \<noteq> {}}"
is_pole_def: fixes M :: "moebius"   and z :: "complex_homo" shows "is_pole M z = (moebius_pt M z = \<infinity>\<^sub>h)"
lift_step_into_qmsg: fixes A :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "nat"   and R :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'd) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'd) \<Rightarrow> bool" assumes "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala P"   and "\<And>\<xi>. U \<xi> \<xi>"   and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'"   and "\<And>\<sigma> \<sigma>' m. \<lbrakk>\<forall>j. U (\<sigma> j) (\<sigma>' j); R \<sigma> m\<rbrakk> \<Longrightarrow> R \<sigma>' m"   and "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). U (\<sigma> i) (\<sigma>' i))"   and "\<And>\<sigma> \<sigma>' m. \<lbrakk>\<forall>j. U (\<sigma> j) (\<sigma>' j); \<sigma>' i = \<sigma> i\<rbrakk> \<Longrightarrow> P (\<sigma>, receive m, \<sigma>')"   and "\<And>\<sigma> \<sigma>' m. P (\<sigma>, receive m, \<sigma>') \<Longrightarrow> P (\<sigma>, \<tau>, \<sigma>')" shows "A \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala P"
CallRedsThrowObj: fixes e :: "char list exp"   and s0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and M :: "char list"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1\<rangle>"
preserves_iso: fixes f :: "'b" assumes "local.iso f" shows "local.iso (\<ll>'.map f)"
intensional_simps_8: fixes y :: "'a \<Rightarrow> bool" shows "#True = y \<equiv> y"
star_right_induct_mult: fixes star :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and y :: "'c"   and x :: "'c" assumes "class.strong_left_kleene_algebra star one times sup less_eq less bot"   and "less_eq (times y x) y" shows "less_eq (times y (star x)) y"
mat_to_polys_nth: fixes i :: "nat"   and A :: "'c mat"   and ts :: "'d list" assumes "i < dim_row A" shows "mat_to_polys ts A ! i = row_to_poly ts (row A i)"
gdd_index_non_zero_iff: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2" shows "(int (\<B> index ps) = \<Lambda>) = (size (filter_mset ((\<subseteq>) ps) (mset_set \<G>)) = 0)"
antisym_conv: fixes y :: "'a"   and x :: "'a" assumes "y \<sqsubseteq> x" shows "(x \<sqsubseteq> y) = (x = y)"
vec_nth_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> UNIV" obtains x :: "('b, 'a) vec" where "y = ($) x"
sig_red_single_def: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" shows "sig_red_single sing_reg top_tail p q f t = (rep_list f \<noteq> 0 \<and> lookup (rep_list p) (t + punit.lt (rep_list f)) \<noteq> (0::'b) \<and> q = p - monom_mult (lookup (rep_list p) (t + punit.lt (rep_list f)) / punit.lc (rep_list f)) t f \<and> ord_term_lin.is_le_rel sing_reg \<and> ordered_powerprod_lin.is_le_rel top_tail \<and> sing_reg (t \<oplus> lt f) (lt p) \<and> top_tail (t + punit.lt (rep_list f)) (punit.lt (rep_list p)))"
assms: shows "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"
fin_join_ti_Suc: fixes s :: "'a list list"   and x :: "nat"   and i :: "nat" shows "fin_join_ti s x (Suc i) = fin_join_ti s x i @ s ! (x + Suc i)"
in_lang_INTERSECT: fixes n :: "nat"   and w :: "('a \<times> bool list) list"   and xs :: "'a atom rexp list" assumes "wf_word n w" shows "(w \<in> lang n (INTERSECT xs)) = (\<forall>r\<in>set xs. w \<in> lang n r)"
Abs_apps_eq_Abs_apps_conv: fixes r :: "dB"   and rs :: "dB list"   and s :: "dB"   and ss :: "dB list" shows "(Abs r \<degree>\<degree> rs = Abs s \<degree>\<degree> ss) = (r = s \<and> rs = ss)"
fields_mono_lemma: fixes x :: "fspec \<times> Decl.field"   and G :: "prog"   and C :: "qtname"   and D :: "qtname" assumes "x \<in> set (DeclConcepts.fields G C)"   and "G\<turnstile>D\<preceq>\<^sub>C C"   and "ws_prog G" shows "x \<in> set (DeclConcepts.fields G D)"
jordan_matrix_poly_bound2: fixes n_as :: "(nat \<times> 'a) list"   and N :: "nat" assumes "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; 0 < n\<rbrakk> \<Longrightarrow> norm a \<le> 1"   and "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; norm a = 1\<rbrakk> \<Longrightarrow> n \<le> N" shows "\<exists>c1. \<forall>k. \<forall>e\<in>elements_mat (jordan_matrix n_as ^\<^sub>m k). norm e \<le> c1 + real k ^ (N - 1)"
same_diag_refl: fixes A :: "'a mat" shows "local.same_diag A A"
max_step1_mono: fixes t :: "nat \<Rightarrow> real"   and t1 :: "real"   and t2 :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "t 0 \<le> t1"   and "t1 \<le> t2"   and "0 \<le> a"   and "0 \<le> b"   and "max_step1 t t2 a b c" shows "max_step1 t t1 a b c"
elims: fixes x :: "'b list"   and y :: "('b list \<times> 'b \<times> 'b list) list" assumes "list_splits x = y" obtains "x = []" and "y = []"   | xa :: "'b" and xs :: "'b list" where "x = xa # xs" and "y = ([], xa, xs) # map (\<lambda>(xs, x', xs'). (xa # xs, x', xs')) (list_splits xs)"
map_graph_fn_eqI: fixes G :: "('c, 'b) labeled_graph"   and f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "\<And>x. x \<in> vertices G \<Longrightarrow> f x = g x" shows "map_graph_fn G f = map_graph_fn G g"
Lxx_not_nullable: fixes x :: "nat"   and y :: "nat" shows "[] \<notin> Lxx x y"
Map_comp: fixes f :: "('a, 'a set) arr"   and g :: "('a, 'a set) arr" assumes "arr f"   and "arr g"   and "local.Dom g = local.Cod f" shows "Map (g \<odot> f) = Comp (Map g) (Map f)"
intro: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and cmp_term :: "'b nat_term_order" assumes "gd_term pair_of_term term_of_pair (\<lambda>s t. le_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term)"   and "gd_nat_term_axioms pair_of_term term_of_pair" shows "gd_nat_term pair_of_term term_of_pair cmp_term"
Abs_zet_inverse: fixes y :: "'b set" assumes "y \<in> zet" shows "Rep_zet (Abs_zet y) = y"
AbruptProp: fixes c :: "('a, 'b, 'c) com"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and f :: "'a" assumes "c \<noteq> Skip"   and "redex c = c" shows "\<Gamma>\<turnstile> (c, Abrupt f) \<rightarrow> (Skip, Abrupt f)"
heap_is_wellformed_disconnected_nodes_distinct: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "distinct disc_nodes"
used_subset: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set" assumes "s \<Turnstile> s'" shows "used s \<subseteq> used s'"
dup_naturality: fixes f :: "'a" assumes "arr f" shows "\<d>[cod f] \<cdot> f = (f \<otimes> f) \<cdot> \<d>[local.dom f]"
rep_eq: fixes x :: "('c, 'b) comp_fun_idem" shows "set_fold_cfi x = Finite_Set.fold (comp_fun_idem_apply x)"
simps_9: fixes vc :: "nat" shows "(extN.N vc \<le> extN.Top) = True"
INVe_def: shows "INVe \<equiv> hn_invalid (A +\<^sub>a B) e e'"
basech_halfchsys_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "C0 \<in> f \<turnstile> ChamberComplex.chamber_system X"
pos_convex_function: fixes C :: "real set"   and f' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "convex C"   and "\<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> f' x * (y - x) \<le> f y - f x" shows "convex_on C f"
coeff_0_reflect_poly_0_iff: fixes p :: "'a poly" shows "(poly.coeff (reflect_poly p) 0 = (0::'a)) = (p = 0)"
CARD_1: assumes "class.CARD_1 TYPE('a)" shows "CARD('a) = 1"
smult_diag: fixes x :: "'c"   and y :: "'c" shows "x *\<^sub>S diag y = diag (x * y)"
unit_system_class_def: shows "OFCLASS('a, unit_system_class) \<equiv> OFCLASS('a, unitary_class)"
i2: shows "i = mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1"
vec_simps_23: fixes aj :: "'b"   and bj :: "'b"   and ch :: "'b" shows "(aj - bj < ch) = (aj < ch + bj)"
2: shows "\<alpha> / pi - \<alpha>' = real_of_int (2 * k)"
nat_card_le_Max: fixes A :: "nat set" shows "card A \<le> Suc (Max A)"
induced_arrowI_1: fixes \<chi>' :: "(bool, bool list) concrete_category.arr \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "\<guillemotleft>induced_arrow a' \<chi>' : a' \<rightarrow> C.dom e\<guillemotright>"
WriteVolatile: fixes \<O>s :: "nat set list"   and i :: "nat"   and a :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j \<union> dom (\<R>s ! j)"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = _emptyset"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> (\<O>s ! j \<union> dom (\<R>s ! j)) = _emptyset"   and "a \<notin> read_only \<S>" shows "\<O>s,\<R>s,i\<turnstile> (Write True a (D, f) A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
assms: shows "parse_ipv6_address_compressed as = Some ipv6"