hcju/isabelleps huggingface dataset:
nat_cut_less_finite: fixes I :: "nat set"   and t :: "nat" shows "finite (I \<down>< t)"
hyps_5: shows "list_all (\<lambda>p. n \<notin> params p) G"
Q: fixes x :: "'a" assumes "Q x" shows "\<exists>y\<le>x. P y"
simps_2: fixes n :: "nat"   and p :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "repeat (Suc n) p = p ;; repeat n p"
g_nonneg: fixes y :: "real" assumes "a \<le> y" shows "0 \<le> g y"
f23: shows "t' \<otimes> t \<otimes> (snd x \<otimes> snd x' \<otimes> (r \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (fst x \<otimes> fst x')) = \<zero>"
termFSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme" shows "termFSbImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSubstAll h hA MOD"
carr_mat_1: shows "(mat_of_rows n G)\<^sup>T \<in> carrier_mat n n"
finite_power_dvd: fixes a :: "nat"   and y :: "nat" assumes "Suc 0 < a"   and "y \<noteq> 0" shows "finite {u. a ^ u dvd y}"
noread: shows "\<forall>\<sigma>. dom \<sigma> = X \<longrightarrow> (cms, subst \<sigma> mem) \<leadsto> (cms', subst \<sigma> mem')"
\<phi>_def: fixes y :: "'d"   and h :: "'c" shows "\<phi> y h = (\<psi>D (y, G (C.cod h)) \<circ> \<Phi>.FUN (y, C.cod h) \<circ> \<phi>C (F y, C.cod h)) h"
fn_norm_least: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real"   and f :: "'c \<Rightarrow> real"   and c :: "real" assumes "normed_vectorspace_with_fn_norm V norm"   and "Function_Norm.continuous V f norm"   and "\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * norm x"   and "0 \<le> c" shows "\<Squnion>({0} \<union> {\<bar>f x\<bar> / norm x |x. x \<noteq> (0::'c) \<and> x \<in> V}) \<le> c"
tObsC_prefix_closed: fixes t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState"   and t' :: "('a, 'es, 'as) BEState Trace"   and s' :: "('a, 'es, 'as) BEState" assumes "tObsC (t \<leadsto> s) = tObsC (t' \<leadsto> s')" shows "tObsC t = tObsC t'"
1: fixes i :: "nat" assumes "i \<in> s" shows "least s \<le> i"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "weakBisimp a1 a2" obtains P :: "pi" and Q :: "pi" where "a1 = P" and "a2 = Q" and "P \<leadsto>\<^sup>^<{(xa, x). weakBisimp xa x}> Q" and "weakBisimp Q P"
ndepth_Push_Node_Inl_aux: fixes n :: "'f"   and f :: "nat \<Rightarrow> 'f + 'g"   and k :: "nat" assumes "(case k of 0 \<Rightarrow> Inl n | Suc x \<Rightarrow> f x) = Inr (0::'g)" shows "Suc (LEAST x. f x = Inr (0::'g)) \<le> k"
unfree_free: fixes name :: "name" shows "unfree (free name) = Some name"
Barbara: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A Q B"   and "B Q C" shows "A Q C"
length_int_lt0: fixes i :: "int" assumes "i < 0" shows "length_int i = Suc (length_nat (nat (- i) - 1))"
n_zeros_def: fixes ts :: "t list" shows "n_zeros ts = map bitzero ts"
elims: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c list"   and y :: "nat \<Rightarrow> 'c" assumes "ring R"   and "ring.coeff R x = y" obtains "x = []" and "y = (\<lambda>_. \<zero>\<^bsub>R\<^esub>)"   | v :: "'c" and va :: "'c list" where "x = v # va" and "y = (\<lambda>i. if i = degree (v # va) then lead_coeff (v # va) else ring.coeff R (tl (v # va)) i)"
equiv_up_to_if_weak: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com"   and d :: "com"   and d' :: "com" assumes "P \<Turnstile> b <\<sim>> b'"   and "P \<Turnstile> c \<sim> c'"   and "P \<Turnstile> d \<sim> d'" shows "P \<Turnstile> IF b THEN c ELSE d \<sim> IF b' THEN c' ELSE d'"
defNode_cases: fixes v :: "'val"   and g :: "'g" assumes "v \<in> allVars g" obtains "v \<in> defs g (defNode g v)"   | "phi g v \<noteq> None"
get_dom_component_to_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and to :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> to_tree_order ptr' \<rightarrow>\<^sub>r to"   and "ptr \<in> set to" shows "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c"
index_row_1: fixes i :: "nat"   and A :: "'a Matrix.mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "Matrix.row A i $ j = A $$ (i, j)"
star_slide_var: fixes x :: "'a" shows "x\<^sup>\<star> ; x = x\<^sup>+"
log_eqI: fixes n :: "nat"   and k :: "nat" assumes "0 < n"   and "2 ^ k \<le> n"   and "n < 2 * 2 ^ k" shows "Discrete.log n = k"
distincts_lists_fcs: fixes as :: "('b \<times> 'b) list" assumes "distinct as" shows "distincts (lists_fcs as)"
inst_ok_code: fixes \<Theta> :: "exetheory"   and insts :: "(('a \<times> String.literal set) \<times> typ) list" assumes "exe_wf_theory' \<Theta>" shows "inst_ok (translate_theory \<Theta>) insts = exeinst_ok \<Theta> insts"
iszero_primfun_list_iff: fixes fs :: "('a \<times> 'b) list" shows "iszero_primfun_list fs = list_all ((=) (0::'b)) (map snd fs)"
k_def: shows "k = the (threshold \<phi> w \<G>)"
A3: shows "strong_reduction_simulation (Rel\<inverse>) (STCal Source Target)"
assms_2: shows "\<forall>x\<in>B. f x = 1"
Min_eqI: fixes A :: "'b word set"   and x :: "'b word" assumes "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> x \<le>s y"   and "x \<in> A" shows "signed.Min A = x"
pr_simps_3: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "cod \<p>\<^sub>0[a0, a1] = a0"
hnorm_triangle_ineq: fixes x :: "'a star"   and y :: "'a star" shows "hnorm (x + y) \<le> hnorm x + hnorm y"
DynCom: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "'b \<Rightarrow> ('b, 'a, 'c) com"   and s :: "'b"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> \<Rightarrow> t"
unit_factor_ratfps_def: fixes x :: "'a ratfps" shows "unit_factor x = ratfps_shift (ratfps_subdegree x) x"
simps_9: fixes e\<^sub>1 :: "char list exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" shows "fv (e\<^sub>1\<bullet>F{D} := e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
hcomp_in_hhom': fixes \<mu> :: "'a"   and \<nu> :: "'a"   and a :: "'a"   and c :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = a"   and "trg\<^sub>C \<nu> = c"   and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" shows "\<guillemotleft>\<nu> \<star>\<^sub>C \<mu> : a \<rightarrow>\<^sub>C c\<guillemotright>"
prv_nprv_emp: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv \<phi> = nprv {} \<phi>"
sum_mset_sum_list: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and xs :: "'c list" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum_mset plus zero (mset xs) = monoid_add.sum_list plus zero xs"
Norm_sum: fixes n :: "nat"   and u :: "nat \<Rightarrow> 'b"   and N :: "'b quasinorm" assumes "\<And>i. i < n \<Longrightarrow> u i \<in> space\<^sub>N N" shows "Norm N (sum u {..<n}) \<le> (\<Sum>i<n. defect N ^ Suc i * Norm N (u i))"
nonneg_matpow: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and n :: "nat"   and i :: "'b"   and j :: "'b" assumes "pf_nonneg_mat A" shows "(0::'a) \<le> matpow A n $h i $h j"
bounded_linear_blinfun_scaleR: fixes a :: "'e \<Rightarrow>\<^sub>L real" shows "bounded_linear (blinfun_scaleR a)"
l_append_child_wf_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" shows "l_append_child_wf type_wf known_ptr known_ptrs append_child heap_is_wellformed \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_append_child_wf_axioms type_wf known_ptrs append_child heap_is_wellformed"
small_def: fixes X :: "'a set" shows "small X \<equiv> \<exists>V_of. inj_on V_of X \<and> V_of ` X \<in> range elts"
symmetric: fixes x :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and y :: "'a state \<times> 'a state \<Rightarrow> Set('b)" shows "(x \<doteq> y) = (y \<doteq> x)"
assms: shows "[\<^bold>\<A>\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>) in v]"
jkbpC_represents_uniquely: fixes T :: "('a, 'es, 'as) BEState Trace set" assumes "SPR.represents T" shows "T = SPR.jkbpC"
classes_above_isfields: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "isfields P C = isfields P' C"
lincomb_def: fixes a :: "'a vec \<Rightarrow> 'a"   and A :: "'a vec set" shows "lincomb a A = (\<Oplus>\<^bsub>V\<^esub>v\<in>A. a v \<cdot>\<^sub>v v)"
max_in_chain_def: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "nat"   and C :: "nat \<Rightarrow> 'a" assumes "class.po below" shows "po.max_in_chain i C = (\<forall>j\<ge>i. C i = C j)"
nth_rotate1: fixes n :: "nat"   and xs :: "'c list" assumes "n < |xs|" shows "rotate1 xs ! n = xs ! (Suc n mod |xs|)"
eq: shows "steps0 (1, l, r) tcopy_loop stp = (s, l', r')"
old_bisim_def: fixes Rel :: "(pi \<times> pi) set" shows "old_bisim Rel \<equiv> \<forall>(P, Q)\<in>Rel. P \<leadsto>[Rel] Q \<and> (Q, P) \<in> Rel"
PiE_iff: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd set" shows "(f \<in> Pi\<^sub>E I X) = ((\<forall>i\<in>I. f i \<in> X i) \<and> f \<in> extensional I)"
inv\<^sub>2E_5: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and P\<^sub>1 :: "'b set set"   and P\<^sub>2 :: "'b set set"   and B\<^sub>1 :: "'b set"   and B\<^sub>2 :: "'b set"   and S :: "'b set"   and L :: "'b set" assumes "BinPacking_Complete U w c"   and "BinPacking_Complete.inv\<^sub>2 U w c P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L" shows "S \<subseteq> {u \<in> U. w u \<le> real c / 2} \<and> L \<subseteq> {u \<in> U. real c / 2 < w u}"
plus_V_def: fixes x :: "V" shows "(+) x \<equiv> transrec (\<lambda>f z. x \<squnion> set (f ` elts z))"
list_choice_zip: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list" shows "list_all (\<lambda>x. \<exists>y. P x y) xs = (\<exists>ys. length ys = length xs \<and> list_all (\<lambda>(x, y). P x y) (xs || ys))"
tube_lemma_gen: fixes S :: "'a set"   and T :: "'b set"   and T' :: "'b set"   and U :: "('a \<times> 'b) set" assumes "compact S"   and "S \<noteq> {}"   and "T \<subseteq> T'"   and "S \<times> T \<subseteq> U"   and "openin (top_of_set (S \<times> T')) U" obtains V :: "'b set" where "openin (top_of_set T') V" and "T \<subseteq> V" and "S \<times> V \<subseteq> U"
pickT_pred_pickT_unique: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and II :: "nat set"   and n :: "nat" assumes "PL_Indis cval indis"   and "\<And>n. n < length cl \<Longrightarrow> part {..<PL.brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "PL_Indis.pickT_pred cl P II n" shows "n = PL_Indis.pickT cl P II"
homotopy_equivalent_space_imp_isomorphic_homology_groups: fixes X :: "'b topology"   and Y :: "'c topology"   and p :: "int" assumes "X homotopy_equivalent_space Y" shows "homology_group p X \<cong> homology_group p Y"
null_sets_completion_subset: fixes B :: "'b set"   and A :: "'b set"   and M :: "'b measure" assumes "B \<subseteq> A"   and "A \<in> null_sets (completion M)" shows "B \<in> null_sets (completion M)"
vars_of_cnf_finite: fixes cnf :: "lit uprod set" assumes "finite cnf" shows "finite (vars_of_cnf cnf)"
Upper_def: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Upper L A = {u. \<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> le L x u} \<inter> carrier L"
subset_not_subset_eq: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subset> B) = (A \<subseteq> B \<and> \<not> B \<subseteq> A)"
pseudo_mod_monic_1: fixes g :: "'b poly"   and f :: "'b poly" assumes "monic g" shows "\<exists>q. f = g * q + pseudo_mod f g"
VPow_bottom: fixes B :: "V" shows "[]\<^sub>\<circ> \<in>\<^sub>\<circ> VPow B"
simps: fixes a :: "'b formula" shows "((a::'b formula) \<in> AX0) = ((\<exists>(F::'b formula) G::'b formula. a = F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F) \<or> (\<exists>(F::'b formula) (G::'b formula) H::'b formula. a = (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H))"
ftf_syn: fixes u :: "'a event list"   and t :: "'a event list"   and s :: "'a event list"   and A :: "'a set" assumes "front_tickFree u"   and "front_tickFree t"   and "s setinterleaves ((t, u), insert tick (ev ` A))" shows "front_tickFree s"
gderiv: shows "(sterm_sem I \<theta> has_derivative frechet I \<theta> (fst (mk_v I ODE (sol 0, b) (sol s)))) (at (fst (mk_v I ODE (sol 0, b) (sol s))) within (\<lambda>t. fst (mk_v I ODE (sol 0, b) (sol t))) ` {0..t})"
order_antisym_conv: fixes y :: "'d"   and x :: "'d" assumes "y \<le> x" shows "(x \<le> y) = (x = y)"
np_def: fixes \<G> :: "'a cyclic_group" shows "np \<G> \<equiv> np_base \<G> \<and> cyclic_group \<G>"
inject_5: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a"   and y51 :: "nat"   and y52 :: "'a"   and y53 :: "'a" shows "(RecvMarker x51 x52 x53 = RecvMarker y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
simps_4: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game"   and \<beta> :: "game" shows "usubstappp \<sigma> U (\<alpha> \<union>\<union> \<beta>) = (let V\<alpha> = usubstappp \<sigma> U \<alpha>; W\<beta> = usubstappp \<sigma> U \<beta> in (fst V\<alpha> \<union> fst W\<beta>, Choiceo (snd V\<alpha>) (snd W\<beta>)))"
step: shows "\<Gamma>\<turnstile> (c # cs @ xs, css, s) \<rightarrow> (cs' @ xs, css, t)"
reflp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
has_derivative_proj': fixes i :: "'a" shows "\<forall>x. ((\<lambda>x. x $ i) has_derivative (\<lambda>x. x $ i)) (at x)"
le_numeral_extra_3: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
CongruenceD: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Congruence" shows "\<exists>k\<in>K. a = b \<cdot> k"
lQuots_def: fixes xs :: "'c list"   and A :: "'c list set" shows "lQuots xs A = {ys. xs @ ys \<in> A}"
FW_neg_cycle_detect: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat" assumes "\<not> linordered_monoid.cycle_free m n" shows "\<exists>i\<le>n. linordered_monoid.fw m n n n n i i \<prec> Le (0::'c)"
more_than_one_mset_mset_diff: fixes a :: "'c"   and M :: "'c multiset" assumes "a \<in># remove1_mset a M" shows "set_mset (remove1_mset a M) = set_mset M"
apfst_apsnd: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and x :: "'c \<times> 'd" shows "apfst f (apsnd g x) = (f (fst x), g (snd x))"
execn_elim_cases_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and r :: "('e \<times> 'e) set"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Spec r,s\<rangle> =n\<Rightarrow> t" obtains f :: "'f" where "s = Fault f" and "t = Fault f"   | sa :: "'e" and ta :: "'e" where "s = Normal sa__" and "t = Normal ta" and "(sa__, ta) \<in> r"   | sa :: "'e" where "s = Normal saa__" and "t = Stuck" and "\<forall>t. (saa__, t) \<notin> r"   | "s = Stuck" and "t = Stuck"   | sa :: "'e" where "s = Abrupt sa" and "t = Abrupt sa"
P1: shows "Bet C A P \<and> A \<noteq> P"
Pow_iff: fixes A :: "'d set"   and B :: "'d set" shows "(A \<in> Pow B) = (A \<subseteq> B)"
ball_2: shows "(\<forall>y\<in>ball x e. f y \<bullet> k \<le> f x \<bullet> k) \<or> (\<forall>y\<in>ball x e. f x \<bullet> k \<le> f y \<bullet> k)"
correctCompositionDiffLevelsS7: shows "correctCompositionDiffLevels sS7"
elims: fixes x :: "('c \<times> 'd) tree"   and y :: "'c list" assumes "Tree2.inorder x = y" obtains "x = \<langle>\<rangle>" and "y = []"   | l :: "('c \<times> 'd) tree" and a :: "'c" and uu_ :: "'d" and r :: "('c \<times> 'd) tree" where "x = \<langle>l, (a, uu_), r\<rangle>" and "y = Tree2.inorder l @ a # Tree2.inorder r"
i_shrink_eq_NoMsg_iAll_start_event_conv: fixes k :: "nat"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af" assumes "0 < k"   and "\<And>t. event t = (t mod k = 0)"   and "t0 = t * k" shows "((s \<div> k) t = NoMsg) = (s t0 = NoMsg \<and> (\<circle> t' t0 [0\<dots>]. s t1 = NoMsg. t1 \<U> t2 [0\<dots>] \<oplus> t'. event t2))"
entt_disj_star_mono: fixes A :: "assn"   and C :: "assn"   and E :: "assn"   and B :: "assn"   and D :: "assn"   and F :: "assn" assumes "A \<or>\<^sub>A C \<Longrightarrow>\<^sub>t E"   and "B \<or>\<^sub>A D \<Longrightarrow>\<^sub>t F" shows "A * B \<or>\<^sub>A C * D \<Longrightarrow>\<^sub>t E * F"
msteps0_snoc: fixes \<pi> :: "((char list \<times> 'a list) set \<times> nat) list"   and tdb :: "(char list \<times> 'a list) set \<times> nat"   and st :: "'a mstate" shows "msteps0 (\<pi> @ [tdb]) st = (let (V', st') = msteps0 \<pi> st; (V'', st'') = mstep tdb st' in (V' \<union> V'', st''))"
set_disconnected_nodes_get_disconnected_nodes_different_pointers: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and ptr :: "(_) document_ptr"   and ptr' :: "(_) document_ptr"   and w :: "((_) heap, exception, unit) prog"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"   and "ptr \<noteq> ptr'"   and "w \<in> set_disconnected_nodes_locs ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
spanning_subset_independent: fixes B :: "'b set"   and A :: "'b set" assumes "B \<subseteq> A"   and "local.independent A"   and "A \<subseteq> local.span B" shows "A = B"
abs_convergent_prod_imp_summable: fixes f :: "nat \<Rightarrow> 'e" assumes "abs_convergent_prod f" shows "summable (\<lambda>i. norm (f i - (1::'e)))"
snapshot_state: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and p :: "'a"   and u :: "'b" assumes "trace init t final"   and "states (S t i) p = u"   and "\<not> ps (S t i) p \<noteq> None"   and "ps (S t (i + 1)) p \<noteq> None" shows "ps (S t (i + 1)) p = Some u"
contains_predI: fixes A :: "'b set"   and x :: "'b" assumes "Predicate_Compile.contains A x" shows "pred.eval (Predicate_Compile.contains_pred A x) ()"
lexordp_eq_pref: fixes u :: "'a list"   and v :: "'a list" shows "u \<le>lex u \<cdot> v"
assms_3: shows "angle a1 b1 c1 = angle a2 b2 c2"
rel_is_equivalence: shows "equivalence (Sigma {U. is_zariski_open U \<and> \<pp> \<in> U} sheaf_spec) {(x, y). st.rel x y}"
simps_2: fixes v :: "'a list"   and va :: "'a list list" shows "UP.eval (v # va) = (\<lambda>x. lead_coeff (v # va) \<otimes>\<^bsub>K [X]\<^esub> x [^]\<^bsub>K [X]\<^esub> degree (v # va) \<oplus>\<^bsub>K [X]\<^esub> UP.eval (tl (v # va)) x)"
WTrt_elim_cases_18: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and o' :: "'c expr"   and e :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (sync(o') e) T" obtains T :: "ty" where "J_heap_base.WTrt typeof_addr P h E o' T" and "is_refT T" and "J_heap_base.WTrt typeof_addr P h E e Ta__"
arcsin_ubound: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "arcsin y \<le> pi / 2"
assms_1: shows "vars F0 \<subseteq> decisionVars"
mk_conn_lower: fixes H1 :: "'b hrel \<Rightarrow> 'b hrel"   and \<H>\<^sub>1 :: "'a hrel \<Rightarrow> 'b hrel"   and \<H>\<^sub>2 :: "'b hrel \<Rightarrow> 'a hrel"   and H2 :: "'a hrel \<Rightarrow> 'a hrel" shows "\<pi>\<^sub>*\<^bsub>H1 \<Leftarrow>\<langle>\<H>\<^sub>1,\<H>\<^sub>2\<rangle>\<Rightarrow> H2\<^esub> = \<H>\<^sub>1"
B: fixes n :: "nat" assumes "N \<le> n" shows "p N * f N \<le> p n * f n"
apply_matrix_zero: fixes f :: "'c \<Rightarrow> 'b" assumes "f (0::'c) = (0::'b)" shows "apply_matrix f 0 = 0"
prems_2: shows "\<Gamma>\<turnstile> \<langle>sequence Seq ys,s'\<rangle> =n\<Rightarrow> t"
concrete_category_axioms: shows "concrete_category {False, True} (\<lambda>x y. Collect (G.path x y)) (\<lambda>_. []) (\<lambda>_ _ _ g f. f @ g)"
infsetsum_finite: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "infsetsum f A = sum f A"
get_scdom_component_ptrs_same_owner_document: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc"   and "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document" shows "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document"
rel_T_witness2: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "((('z, 'n, 'q, 't, 'w, 'aa) T \<times> (('ab, 'p, 's, 'v, 'y, 'aa) T \<Rightarrow> ('z \<times> 'ab, 'o, 'r, 'u, 'x, 'aa) T)) \<times> ((('z, 'n, 'q, 't, 'w, 'aa) T \<times> (('ab, 'p, 's, 'v, 'y, 'aa) T \<Rightarrow> ('z \<times> 'ab, 'o, 'r, 'u, 'x, 'aa) T)) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) T) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) T \<times> 'z \<times> ('z \<times> 'ab) \<times> 'ab \<times> 'aa) itself"   and L1 :: "'z \<Rightarrow> 'ab \<Rightarrow> bool"   and x :: "('z, 'n, 'q, 't, 'w, 'aa) T"   and y :: "('ab, 'p, 's, 'v, 'y, 'aa) T" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G"   and "rel_T L1 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_T (\<lambda>(x, y') y. y' = y \<and> L1 x y) Co1' Co2' Contra1' Contra2' (rel_T_witness L1 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' x y) y"
load_sub3_result: fixes instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and asi :: "word8"   and address :: "word32"   and s :: "'b sparc_state" shows "snd (load_sub3 instr curr_win rd asi address s) = False"
diff_measure_def: fixes M :: "'b measure"   and N :: "'b measure" shows "diff_measure M N = measure_of (space M) (sets M) (\<lambda>A. emeasure M A - emeasure N A)"