hcju/isabelleps huggingface dataset:
augment_Faults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and F' :: "'f set" assumes "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "F \<subseteq> F'" shows "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F'\<^esub> P c Q,A"
induces_elementary_cartesian_monoidal_category: shows "elementary_cartesian_monoidal_category (\<cdot>) (\<otimes>) \<I> lunit runit assoc \<tau> \<delta>"
d1: shows "\<forall>D. fine d1 (a, b) D \<longrightarrow> \<bar>rsum D f - k1\<bar> < \<bar>k1 - k2\<bar> / 2"
terminates_elim_cases_11: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile>Catch c1 c2 \<down> s" obtains f :: "'c" where "s = Fault f"   | "s = Stuck"   | sa :: "'b" where "s = Abrupt sa__"   | sa :: "'b" where "s = Normal sa" and "\<Gamma>\<turnstile>c1 \<down> Normal sa" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal sa\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s'"
x_def: shows "x \<equiv> getFr [y, z] [t] []"
c_tr_ipurge: fixes I :: "('g \<times> 'g) set"   and D :: "'e \<Rightarrow> 'g"   and u :: "'g"   and step :: "'h \<Rightarrow> 'e \<Rightarrow> 'h"   and out :: "'h \<Rightarrow> 'e \<Rightarrow> 'f"   and s :: "'h"   and xs :: "'e list" shows "c_ipurge I (c_dom D) u (c_tr step out s (c_ipurge I D u xs)) = c_tr step out s (c_ipurge I D u xs)"
zle_imp_zless_or_eq: fixes n :: "int"   and m :: "int" assumes "n \<le> m" shows "n < m \<or> n = m"
assms_2: shows "set \<phi>2s = set \<phi>1s"
policy_range_comp_def: fixes f :: "'g \<Rightarrow> 'f"   and p :: "'e \<Rightarrow> 'g decision option" shows "f o\<^sub>f p = (\<lambda>x. case p x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow y\<rfloor> \<Rightarrow> \<lfloor>allow (f y)\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> \<lfloor>deny (f y)\<rfloor>)"
normFace_neq: fixes a :: "nat"   and f :: "face"   and f' :: "face" assumes "a \<in> \<V> f"   and "a \<notin> \<V> f'"   and "vertices f' \<noteq> []" shows "normFace f \<noteq> normFace f'"
simps: fixes x :: "'d"   and xs :: "'d list" shows "min_list (x # xs) = (case xs of [] \<Rightarrow> x | a # list \<Rightarrow> min x (min_list xs))"
seqSubstAppend: fixes P :: "pi"   and \<sigma> :: "(name \<times> name) list"   and \<sigma>' :: "(name \<times> name) list" shows "P[<(\<sigma> @ \<sigma>')>] = P[<\<sigma>>][<\<sigma>'>]"
djf_def: fixes f :: "'b \<Rightarrow> fm"   and p :: "'b"   and q :: "fm" shows "djf f p q \<equiv> if q = T then T else if q = F then f p else let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q"
S: shows "sets (Pi\<^sub>M UNIV (\<lambda>_. borel)) = sets borel"
real_uminus_code: fixes x :: "rat" shows "- Ratreal x = Ratreal (- x)"
ndFalse2: fixes A :: "('b \<times> 'c set set) set"   and B :: "('d \<times> 'c set set) set"   and D :: "'c" assumes "\<forall>a b c d. (a, b) \<in> A \<and> (c, d) \<in> B \<longrightarrow> netsDistinct b d"   and "\<exists>(a, b)\<in>A. b \<in> subnetsOfAdr D"   and "\<exists>(a, b)\<in>B. b \<in> subnetsOfAdr D" shows "False"
cSup_eq: fixes X :: "'c set"   and a :: "'c" assumes "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"   and "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y" shows "Sup X = a"
assms: shows "M \<le> \<Down>\<^sub>E E R (f x)"
simple_distributed_joint: fixes X :: "'a \<Rightarrow> 'e"   and Y :: "'a \<Rightarrow> 'f"   and Px :: "'e \<times> 'f \<Rightarrow> real" assumes "simple_distributed M (\<lambda>x. (X x, Y x)) Px" shows "distributed M (count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M)) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Px x else 0))"
apex_def: shows "Cod\<mu>_Cod\<nu>.apex \<equiv> C.dom Cod\<mu>_Cod\<nu>.leg0"
no_selfloop: shows "\<forall>u w v. (u, w, v) \<in> set L \<longrightarrow> u \<noteq> v"
cnf_of_encode_problem_structure_4: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_all_frame_axioms \<Pi> t) \<subseteq> cnf (\<Phi> \<Pi> t)"
abs_def: fixes R :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and Abs :: "'p set \<Rightarrow> 'q"   and Rep :: "'q \<Rightarrow> 'p set"   and x :: "'p" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
borel_measurable_lfp: fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "sup_continuous F"   and "\<And>f. f \<in> borel_measurable M \<Longrightarrow> F f \<in> borel_measurable M" shows "lfp F \<in> borel_measurable M"
step_none_rejects: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and t :: "(String.literal \<times> value list) list" assumes "step e s r (fst h) (snd h) = None" shows "\<not> recognises_execution e s r (h # t)"
LNil: shows "set xss' \<subseteq> {xs. lnull xs}"
L2_ControlSystem: fixes s :: "nat \<Rightarrow> nat list"   and i :: "nat" assumes "ControlSystemArch s" shows "200 \<le> hd (s i)"
SUPSEQ_def: fixes A :: "'b list set" shows "SUPSEQ A \<equiv> {x. \<exists>y\<in>A. subseq y x}"
proj6_def: shows "\<pi>\<^sub>6 \<equiv> \<lambda>(uu_, uu_, uu_, uu_, nhip, uu_). nhip"
cases: fixes \<I> :: "('a, 'b) \<I>"   and a :: "('a, 'b) resource" assumes "\<I> \<turnstile>res a \<surd>" obtains res :: "('a, 'b) resource" where "a = res" and "\<And>q r res'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, res') \<in> set_spmf (run_resource res q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>"
assms_1: shows "f \<in> carrier \<rightarrow> carrier\<^bsub>T\<^esub>"
incidence_system_axioms: shows "incidence_system (del_point p) (str_del_point_blocks p)"
rel_fun_map_fun2: fixes h :: "'d \<Rightarrow> 'a"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "rel_fun (eq_on (range h)) A f g" shows "rel_fun (BNF_Def.Grp UNIV h)\<inverse>\<inverse> A f (map_fun h id g)"
poly_subst_sum: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "'f \<Rightarrow> ('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and A :: "'f set" shows "poly_subst f (sum p A) = (\<Sum>a\<in>A. poly_subst f (p a))"
assms_3: fixes p :: "'a"   and q :: "'a" assumes "P p q" shows "P q p"
dom_restrict_total: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and f :: "'d \<Rightarrow> 'e"   and X :: "'d set" assumes "sifum_security_init dma \<C>_vars eval" shows "dom (to_partial f |` X) = X"
nonzero_imp_inverse_nonzero: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse a \<noteq> zero"
subring_gen_incl: fixes H :: "'a set"   and K :: "'a set"   and I :: "'a set" assumes "subring H R"   and "subring K R"   and "I \<subseteq> H"   and "I \<subseteq> K" shows "generate_ring (R\<lparr>carrier := K\<rparr>) I \<subseteq> generate_ring (R\<lparr>carrier := H\<rparr>) I"
collision_after_delta: assumes "s\<^sub>o \<le> u_max"   and "u_max < other.s_stop" shows "collision_react {0..} = collision_react {\<delta>..}"
product_nn_integral_insert_rev: fixes M :: "'a \<Rightarrow> 'b measure"   and I :: "'a set"   and i :: "'a"   and f :: "('a \<Rightarrow> 'b) \<Rightarrow> ennreal" assumes "product_sigma_finite M"   and "finite I"   and "i \<notin> I"   and "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)" shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x(i := y)) \<partial>Pi\<^sub>M I M \<partial>M i"
summands_def: shows "summands \<equiv> pi_rec {} (\<lambda>a b P Pa. {a{b}.P}) (\<lambda>P Pa. {\<tau>.P}) (\<lambda>a x P Pa. {a<x>.P}) (\<lambda>a b P Pa. {}) (\<lambda>a b P Pa. {}) (\<lambda>P Q. (\<union>)) (\<lambda>P Q Pa Qa. {}) (\<lambda>x P Pa. if \<exists>a P'. a \<noteq> x \<and> P = a{x}.P' then {<\<nu>x>P} else {}) (\<lambda>P Pa. {})"
simps_2: fixes cx :: "int"   and cy :: "int"   and black :: "bool"   and more :: "'b" shows "bw_point.more \<lparr>cx = cx, cy = cy, black = black, \<dots> = more\<rparr> = more"
lemma_degree_zero: fixes p :: "'a list" shows "(\<forall>c\<in>set p. c = (0::'a)) = (pnormalize p = [])"
wf_2: shows "wf\<^sub>e\<^sub>s\<^sub>t {} \<A>\<^sub>d"
C_eq_RS3: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "not_MT C p" shows "C (list2FWpolicy (rm_MT_rules C p)) = C (list2FWpolicy p)"
from_bool_mask_simp: fixes r :: "bool" shows "from_bool r AND 1 = from_bool r"
divides_mult_lI: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.monoid G"   and "a divides\<^bsub>G\<^esub> b"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "c \<otimes>\<^bsub>G\<^esub> a divides\<^bsub>G\<^esub> c \<otimes>\<^bsub>G\<^esub> b"
SL_def: fixes A :: "'a set" shows "SL A = Pi_pmf A 0 (\<lambda>_. geometric_pmf p)"
sup_V_0_left: fixes a :: "V" shows "[]\<^sub>\<circ> \<union>\<^sub>\<circ> a = a"
ev: fixes nP :: "'a \<Rightarrow> 'b" assumes "wf_list_graph G" shows "sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP"
Id_on_eqI: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" assumes "a = b"   and "a \<in> A" shows "(a, b) \<in> Id_on A"
Box_def: fixes \<alpha> :: "('d, 'e, 'f) hp"   and P :: "('d, 'e, 'f) formula" shows "([[\<alpha>]]P) = ! (\<langle> \<alpha> \<rangle> ! P)"
measure_spmf_bind: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "measure_spmf (p \<bind> f) = measure_spmf p \<bind> measure_spmf \<circ> f"
ring_functions_def: fixes R :: "('c, 'd) ring_scheme" shows "ring_functions R \<equiv> ring R"
fold_A_facet_distance_map: fixes F :: "'a set" assumes "chamber F" shows "face_distance z (fold_A ` F) = face_distance z F"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "g (b1 + b2) = g b1 + g b2"
check_extension_l_no_new_var_err_def: fixes p :: "(char list list \<times> int) list" shows "check_extension_l_no_new_var_err p = SPEC (\<lambda>_. True)"
star_rtc2: fixes y :: "'a"   and x :: "'a" assumes "(1::'a) + y \<cdot> y + x \<le> y" shows "x\<^sup>\<star> \<le> y"
valid_store_sops_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "valid_store_sops ts \<equiv> \<forall>p i is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> (\<forall>sop\<in>store_sops is. valid_sop sop)"
simps_2: fixes \<sigma> :: "nat \<Rightarrow> exp option"   and e :: "exp"   and f :: "nat" shows "substs \<sigma> (FieldProj e f) = FieldProj (substs \<sigma> e) f"
ground_substs_yield_ground_clause: fixes C :: "'b literal set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "ground_on (vars_of_cl C) \<sigma>" shows "ground_clause (subst_cl C \<sigma>)"
False: shows "\<not> dominates g n\<^sub>1 n\<^sub>2"
greatest_dual: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and A :: "'c set" shows "greatest (inv_gorder L) x A = least L x A"
simps: fixes a :: "('test, 'atom) com" shows "discr a = (\<exists>c. a = c \<and> (\<forall>x xa xb. (c, x) \<rightarrow>c (xa, xb) \<longrightarrow> x \<approx> xb \<and> discr xa) \<and> (\<forall>x xa. (c, x) \<rightarrow>t xa \<longrightarrow> x \<approx> xa))"
subgroup_kernel: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h" shows "subgroup (kernel G H h) G"
actions_def: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "actions E = {n. enat n < llength E}"
inverse_clause_renaming: fixes A :: "'at set"   and L :: "'at Literal set" shows "rename_clause A (rename_clause A L) = L"
lifelock_free_def: fixes P :: "'c process" shows "lifelock_free P \<equiv> CHAOS UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
no_better_thansubset_rel: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier"   and "no_better_than y carrier relation \<subseteq> no_better_than x carrier relation" shows "x \<succeq> y"
bounded_iff: fixes a :: "'g multiset"   and b :: "'g multiset"   and c :: "'g multiset" shows "(a \<subseteq># b \<inter># c) = (a \<subseteq># b \<and> a \<subseteq># c)"
adiff_le_adiff: fixes a :: "ant"   and a' :: "ant"   and b :: "ant" assumes "a \<le> a'" shows "a - b \<le> a' - b"
negligible_eq_zero_density_alt: fixes S :: "'a set" shows "negligible S = (\<forall>x\<in>S. \<forall>e>0. \<exists>d U. 0 < d \<and> d \<le> e \<and> S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue U < e * Sigma_Algebra.measure lebesgue (ball x d))"
nths_vars2'_comp: shows "nths dims0 vars2' = dims2"
assms_4: shows "index xs q < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < length xs"
proj_not_active: fixes n :: "nat"   and t :: "cnf llist"   and c :: "'id" assumes "enat n < llength t"   and "\<not> \<parallel>c\<parallel>\<^bsub>lnth t n\<^esub>" shows "\<pi>\<^bsub>c\<^esub>ltake (enat (Suc n)) t = \<pi>\<^bsub>c\<^esub>ltake (enat n) t"
that: fixes a :: "'a"   and v :: "'a llist" assumes "w = <a> $ v"   and "llength v = enat n" shows "thesis"
spec\<^sub>2_def: fixes p :: "prog" shows "spec\<^sub>2 p \<equiv> para p = [''x'', ''y''] \<and> (\<exists>e\<^sub>1 e\<^sub>2. body p = Add e\<^sub>1 e\<^sub>2 \<and> wfe \<Gamma>\<^sub>x\<^sub>y (body p) \<and> (\<forall>x y. eval (\<E>\<^sub>x\<^sub>y x y) (body p) = Some (f x y)) \<and> coste (body p) \<le> 3)"
S_def: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and bT :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" shows "S cf bT \<equiv> \<exists>n. qsend (cf ## bT) = enat n \<and> snd ((cf ## bT) !! n) \<approx> s"
apply_fun_def: fixes A :: "'e \<Rightarrow> 'g \<Rightarrow> 'f"   and B :: "'e \<Rightarrow> 'g" shows "(A .. B) = (\<lambda>x. A x (B x))"
Agent: fixes A :: "agent" shows "eq (cAgent A) (cAgent A)"
restrict_Restrict: fixes i :: "nat"   and \<AA> :: "'i"   and k :: "'k"   and P :: "'v"   and r :: "'k \<Rightarrow> 'v \<Rightarrow> nat \<Rightarrow> bool" assumes "i\<^bsup>\<AA>\<^esup>k = P" shows "restrict k P = satisfies_gen r \<AA> (Restrict k i)"
B: shows "B = four_block_mat B1 B2 B3 B4"
left_lang_half: fixes M :: "'b dfa"   and qs1 :: "hf set"   and qs2 :: "hf set" assumes "dfa M"   and "\<Union> (dfa.left_lang M ` qs1) \<subseteq> \<Union> (dfa.left_lang M ` qs2)"   and "\<And>x. x \<in> qs1 \<Longrightarrow> dfa.left_lang M x \<noteq> {}" shows "qs1 \<subseteq> qs2"
OUT_g_B: fixes x :: "'v" assumes "x \<in> B \<Gamma>" shows "d_OUT (\<lambda>(x, y). g (\<langle>y\<rangle>, \<langle>x\<rangle>)) x = d_IN g \<langle>x\<rangle> - g (SOURCE, \<langle>x\<rangle>)"
in_hom_char: fixes f :: "'c \<times> 'c \<times> 'c"   and a :: "'c \<times> 'c \<times> 'c"   and b :: "'c \<times> 'c \<times> 'c" shows "C.VVV.in_hom f a b = (C.VVV.arr a \<and> C.VVV.arr b \<and> C.VVV.arr f \<and> C.VxVxV.in_hom f a b)"
borel_measurable_cINF: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> bdd_below ((\<lambda>i. F i x) ` I)" shows "(\<lambda>x. INF i\<in>I. F i x) \<in> borel_measurable M"
h1: shows "h \<squnion> f \<circ>\<^sub>K g \<le> g"
merge_guards_block: fixes init :: "'d \<Rightarrow> 'd"   and bdy :: "('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "merge_guards (block init bdy return c) = block init (merge_guards bdy) return (\<lambda>s t. merge_guards (c s t))"
mult_scaleC_right: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and a :: "complex"   and y :: "'a" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times x (scaleC a y) = scaleC a (times x y)"
set_subset_Cons: fixes xs :: "'b list"   and x :: "'b" shows "order.greater_eq (set (x # xs)) (set xs)"
ProofStepInduct_NonTrivial_P_k_pre_exists: shows "\<exists>P_k_pre y P_k_post. ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post"
outstanding_non_volatile_writes_unshared_tl: fixes \<S> :: "nat \<Rightarrow> bool option"   and t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "outstanding_non_volatile_writes_unshared \<S> (t # ts)" shows "outstanding_non_volatile_writes_unshared \<S> ts"
fold_def_alt: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('c, 'd) RBT.rbt" shows "RBT.fold f t = fold (\<lambda>(x, y). f x y) (RBT.entries t)"
simps_3: fixes m :: "'a simple_match"   and rs :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme" shows "simple_fw (SimpleRule m Drop # rs) p = (if simple_matches m p then Decision FinalDeny else simple_fw rs p)"
less_rexp_def: fixes r :: "'a rexp"   and s :: "'a rexp" shows "r < s \<equiv> local.le_rexp r s \<and> r \<noteq> s"
dbl_inc_simps_4: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (uminus one) = uminus one"
ArithmeticException_def: shows "ArithmeticException = STR ''java/lang/ArithmeticException''"
ABinds_env_restr: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ABinds \<Delta>\<cdot>(ae f|` domA \<Delta>) = ABinds \<Delta>\<cdot>ae"
HomFtorDist: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e"   and g :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "(Hom\<^bsub>C\<^esub>[X,f]) ;;\<^bsub>SET\<^esub> (Hom\<^bsub>C\<^esub>[X,g]) = Hom\<^bsub>C\<^esub>[X,f ;;\<^bsub>C\<^esub> g]"
split_n_is: shows "split_n ves (length t1s) = (ves', ves'')"
test_comp_eq_mult: fixes x :: "'a"   and y :: "'a" assumes "is_test x"   and "is_test y" shows "x ; y = x \<cdot> y"
Inv_def: fixes xs :: "'a list"   and ys :: "'a list" shows "Inv xs ys = {(x, y). x < y in xs \<and> y < x in ys}"
bij_betw_ball: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and phi :: "'b \<Rightarrow> bool" assumes "bij_betw f A B" shows "(\<forall>b\<in>B. phi b) = (\<forall>a\<in>A. phi (f a))"
surjective: fixes r :: "('a, 'b, 'c, 'd) object_scheme" shows "r = \<lparr>obj_mark = obj_mark r, obj_fields = obj_fields r, obj_payload = obj_payload r, \<dots> = object.more r\<rparr>"
unless_def: fixes A :: "'b set"   and B :: "'b set" shows "A unless B \<equiv> A - B co A \<union> B"
inj_on_subgroup_iff_trivial_ker: fixes H :: "'a set" assumes "subgroup H (add_monoid R)" shows "inj_on h H = (a_kernel (R\<lparr>carrier := H\<rparr>) S h = {\<zero>})"
nat_of_name_inject: fixes n1 :: "name"   and n2 :: "name" shows "(nat_of_name n1 = nat_of_name n2) = (n1 = n2)"
Diff_single_insert: fixes A :: "'e set"   and x :: "'e"   and B :: "'e set" assumes "A - {x} \<subseteq> B" shows "A \<subseteq> insert x B"
trans_multpw: fixes ns :: "('a \<times> 'a) set" assumes "trans ns" shows "trans (multpw ns)"
index_vec: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "i < n" shows "Matrix.vec n f $v i = f i"
wellorder_Least_lemma_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and k :: "'c" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
inverse_le_iff_le_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero"   and "less b zero" shows "less_eq (inverse a) (inverse b) = less_eq b a"
genideal_self': fixes i :: "'a list" assumes "i \<in> carrier (K [X])" shows "i \<in> Idl\<^bsub>K [X]\<^esub> {i}"
step_op_inl: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and op :: "'op"   and ar :: "nat"   and \<Sigma> :: "'dyn list"   and opinl :: "'opinl"   and x :: "'dyn"   and F' :: "'fenv_inca"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IOp op"   and "\<AA>\<rr>\<ii>\<tt>\<yy> op = ar"   and "ar \<le> length \<Sigma>"   and "\<II>\<nn>\<ll> op (take ar \<Sigma>) = Some opinl"   and "\<II>\<nn>\<ll>\<OO>\<pp> opinl (take ar \<Sigma>) = x"   and "Finca_add F f (rewrite_fundef_body fd pc (Inca.instr.IOpInl opinl)) = F'" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F' H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
set_eq_fold: fixes x :: "'a"   and xs :: "'a list" shows "F (set (x # xs)) = fold f xs x"
rel_distinct_2: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y21 :: "'a"   and y22 :: "('a, 'c) tllist"   and x1 :: "'d" shows "\<not> tllist_all2 R1 R2 (TCons y21 y22) (TNil x1)"
leq: fixes v :: "complex vec" assumes "v \<in> carrier_vec n" shows "0 \<le> inner_prod v ((C - A) *\<^sub>v v)"
segment_eq: shows "closed_segment X (X + H) = line ` {0..1}"
valid_augment_Faults: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set"   and F' :: "'g set" assumes "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"   and "F \<subseteq> F'" shows "\<Gamma>\<Turnstile>\<^bsub>/F'\<^esub> P c Q,A"
in_Imagep: fixes b :: "'c"   and r :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set" shows "(b \<in> r ``` A) = (\<exists>a\<in>A. r a b)"
Receive: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> (S @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
simps_26: shows "cospan_shape.comp cospan_shape.BT cospan_shape.BT = cospan_shape.Null"