hcju/isabelleps huggingface dataset:
Cod_comp: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr"   and g :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) g"   and "concrete_category.Dom g = concrete_category.Cod f" shows "concrete_category.Cod (concrete_category.COMP Obj Hom Comp g f) = concrete_category.Cod g"
entails_def: fixes N1 :: "'f set"   and N2 :: "'f set" shows "(N1 \<Turnstile>\<inter>\<G> N2) = (\<forall>q\<in>Q. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2))"
seq_comp_list_sequential: fixes P :: "'c option process"   and PS :: "'c option process list" assumes "\<And>x. x \<in> set (P # PS) \<Longrightarrow> sequential x" shows "sequential (foldl (;) P PS)"
h_def: shows "h \<equiv> \<lambda>z n. (- 1) ^ n / of_nat (2 * n + 1) * z ^ (2 * n + 1)"
healthy_intros_23: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "healthy (wp a)" shows "healthy (wp (repeat n a))"
disjoint_iff: fixes A :: "'h set"   and B :: "'h set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
obj_at_split_asm: fixes Q :: "bool \<Rightarrow> bool"   and P :: "('e, 'f, 'g) object \<Rightarrow> bool"   and r :: "'g"   and s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state" shows "Q (obj_at P r s) = (\<not> (sys_heap s r = None \<and> \<not> Q False \<or> (\<exists>obj. sys_heap s r = Some obj \<and> \<not> Q (P obj))))"
keys_dom_lookup: fixes m :: "('c, 'd) mapping" shows "Mapping.keys m = dom (Mapping.lookup m)"
mem_case_prodE: fixes z :: "'d"   and c :: "'e \<Rightarrow> 'f \<Rightarrow> 'd set"   and p :: "'e \<times> 'f" assumes "z \<in> (case p of (x, xa) \<Rightarrow> c x xa)" obtains x :: "'e" and y :: "'f" where "p = (x, y)" and "z \<in> c x y"
cmod_divide: fixes a :: "complex"   and b :: "complex" shows "cmod (a / b) = cmod a / cmod b"
interval_neqs_7: fixes r :: "real" shows "{} \<noteq> {r<..}"
intros: fixes x :: "'b"   and ys :: "'b list"   and xs :: "'b list" assumes "x \<in> set ys" shows "double (xs @ ys) (xs @ x # ys)"
inv_solve_left: fixes a :: "cltn2"   and b :: "cltn2"   and c :: "cltn2" assumes "a \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "b \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "c \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "(a = inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> b \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> c) = (c = b \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> a)"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'c"   and x :: "('c, 'd) phi \<Rightarrow> 'c" assumes "eq_onp (\<lambda>xf. True) xa xa"   and "eq_onp (\<lambda>xf. True) x x" shows "Abs_lifted_boolean_algebra xa \<squnion> Abs_lifted_boolean_algebra x = Abs_lifted_boolean_algebra (\<lambda>f. xa f \<squnion> x f)"
maybe_and_one: fixes x :: "trilean" shows "true \<and>? x = x"
supp: fixes x :: "'a"   and y :: "'b" assumes "0 < h x y" shows "(x, y) \<in> R'"
ntrs_called_mon: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrs fg" shows "fst (\<alpha>n fg e) \<subseteq> mon_s fg s'"
tail_suffix_suc: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "s |\<^sub>s Suc n = tail (s |\<^sub>s n)"
simps_3: fixes x :: "'a"   and y :: "'a" shows "(ext0.Val x \<le> ext0.Val y) = (x \<le> y)"
g_set_xx_loc_axioms: shows "g_set_xx_loc ls_ops hs_ops"
False: shows "\<not> (n = 0 \<or> x = 0)"
H: shows "Inum (x # bs) a = Inum (x # bs) (CN 0 n s)"
coeff_X_mult: fixes p :: "'b poly"   and i :: "nat" shows "coeff (X * p) (Suc i) = coeff p i"
disjoint: shows "{0..<Suc m} \<inter> {Suc m} = {}"
finite_span: fixes S :: "'c set" assumes "finite S"   and "S \<subseteq> carrier V" shows "V.module.span S = {V.module.lincomb a S |a. a \<in> S \<rightarrow> carrier K}"
extend_map_lemma_cofinite1: fixes \<F> :: "'a set set"   and U :: "'a set"   and T :: "'b set"   and K :: "'a set"   and h :: "'a \<Rightarrow> 'b" assumes "finite \<F>"   and "\<And>X. X \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x\<in>X \<inter> K. g x = h x)"   and "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"   and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; \<not> X \<subseteq> Y; \<not> Y \<subseteq> X\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K" obtains C :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite C" and "disjnt C U" and "card C \<le> card \<F>" and "continuous_on (\<Union> \<F> - C) g" and "g ` (\<Union> \<F> - C) \<subseteq> T" and "\<And>x. x \<in> (\<Union> \<F> - C) \<inter> K \<Longrightarrow> g x = h x"
dgrad_p_set_leI_insert: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dgrad_p_set_le d F G"   and "dgrad_p_set_le d {f} G" shows "dgrad_p_set_le d (insert f F) G"
differentiable_at_polynomial_function: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and S :: "'c set" assumes "polynomial_function f" shows "f differentiable at a within S"
mat_diag_mult_left: fixes A :: "'b mat"   and n :: "nat"   and nr :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "A \<in> carrier_mat n nr" shows "mat_diag n f * A = mat n nr (\<lambda>(i, j). f i * A $$ (i, j))"
vv'_odd_disjoint: shows "{v, v'} \<inter> odd_nodes_set G = {}"
semiring_def: fixes R :: "('c, 'd) ring_scheme" shows "semiring R \<equiv> abelian_monoid R \<and> Group.monoid R \<and> semiring_axioms R"
lconcat_fixp_parallel_induct: fixes P :: "('c llist llist \<Rightarrow> 'c llist) \<Rightarrow> ('d llist llist \<Rightarrow> 'd llist) \<Rightarrow> bool" assumes "ccpo.admissible (prod_lub llist.lub_fun llist.lub_fun) (rel_prod llist.le_fun llist.le_fun) (\<lambda>x. P (fst x) (snd x))"   and "P (\<lambda>_. lSup {}) (\<lambda>_. lSup {})"   and "\<And>f g. P f g \<Longrightarrow> P (\<lambda>xss. case xss of LNil \<Rightarrow> LNil | LCons xs xss' \<Rightarrow> lappend xs (f xss')) (\<lambda>xss. case xss of LNil \<Rightarrow> LNil | LCons xs xss' \<Rightarrow> lappend xs (g xss'))" shows "P lconcat lconcat"
hom_in_carrier: fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme"   and x :: "'f" assumes "h \<in> hom G H"   and "x \<in> carrier G" shows "h x \<in> carrier H"
make_mat_index: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and k :: "nat"   and B :: "'a mat" assumes "i < n"   and "j < n" shows "make_mat n k B $$ (i, j) = (if i = 0 \<and> j = 0 then B $$ (0, 0) else if i = 0 \<and> j = k then B $$ (0, 1) else if i = k \<and> j = 0 then B $$ (1, 0) else if i = k \<and> j = k then B $$ (1, 1) else if i = j then 1::'a else (0::'a))"
top_def: fixes p :: "alt \<Rightarrow> real"   and b :: "alt" shows "p <\<cdot> b \<equiv> \<forall>a. a \<noteq> b \<longrightarrow> p a < p b"
WellFormedToSetInRangeExplode: fixes T :: "(ZF, ZF) Axioms"   and n :: "nat" assumes "ZFAxioms T" shows "Expr2ZF ` WellFormedToSet aS\<^bsub>T\<^esub> n \<in> range explode"
min_gallery_betw_chamber_distance: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "min_gallery (C # Cs @ [D])" shows "chamber_distance C D = Suc (length Cs)"
a_Unit: fixes \<Gamma> :: "(Syntax.var, ty) fmap" shows "\<Gamma> \<turnstile> Unit, Unit, Unit : One"
opEGood_def: fixes f :: "Val \<Rightarrow> Val \<Rightarrow> Val" shows "opEGood f = (\<forall>\<beta> v v' w w'. (\<beta>, v, v') \<in> twiddleVal \<longrightarrow> (\<beta>, w, w') \<in> twiddleVal \<longrightarrow> (\<beta>, f v w, f v' w') \<in> twiddleVal)"
STAR_sin_pi_divide_HNatInfinite_approx_pi2: fixes n :: "nat star" assumes "n \<in> HNatInfinite" shows "hypreal_of_hypnat n * (*f* sin) (hypreal_of_real pi / hypreal_of_hypnat n) \<approx> hypreal_of_real pi"
iwlsFSwSTR_iterSTR_ipresAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSTR MOD" shows "ipresAbs (iterSTR MOD) (iterAbsSTR MOD) MOD"
id_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun A A id id"
step: fixes r :: "sequent list \<times> sequent"   and R :: "(sequent list \<times> sequent) set"   and m :: "nat" assumes "r \<in> R"   and "fst r \<noteq> []"   and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R" shows "(snd r, m + 1) \<in> derivable R"
evenperm_unique: fixes n :: "nat"   and p :: "'a \<Rightarrow> 'a"   and b :: "bool" assumes "swapidseq n p"   and "even n = b" shows "evenperm p = b"
case_rews_2: fixes f :: "C \<rightarrow> 'a"   and C :: "C" shows "C_case\<cdot>f\<cdot>(C\<cdot>C) = f\<cdot>C"
fBallE: fixes A :: "'c fset"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "fBall A P" obtains "P x"   | "x |\<notin>| A"
subst_domain_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_domain \<sigma> = {x. \<sigma> x \<noteq> Var x}"
group_sort_correct: fixes xs :: "'a list" shows "group_sort xs = local.group (local.sort xs)"
map_2: fixes f :: "'f \<Rightarrow> 'e"   and x2 :: "'f" shows "map_option f \<lfloor>x2\<rfloor> = \<lfloor>f x2\<rfloor>"
AE_point_measure: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" shows "almost_everywhere (point_measure A f) P = (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
arr_assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "ide a"   and "ide b"   and "ide c" shows "arr \<a>[a, b, c]"
line_through_K2_intersect_S_exactly_twice: fixes p :: "proj2"   and l :: "proj2_line" assumes "p \<in> hyp2"   and "proj2_incident p l" shows "\<exists>q r. q \<noteq> r \<and> q \<in> S \<and> r \<in> S \<and> proj2_incident q l \<and> proj2_incident r l \<and> (\<forall>s\<in>S. proj2_incident s l \<longrightarrow> s = q \<or> s = r)"
natural_isomorphism_axioms: shows "natural_isomorphism D C G.F G.F G.F"
shiftpath_shiftpath: fixes g :: "real \<Rightarrow> 'a"   and a :: "real"   and x :: "real" assumes "pathfinish g = pathstart g"   and "a \<in> {0..1}"   and "x \<in> {0..1}" shows "shiftpath (1 - a) (shiftpath a g) x = g x"
bconf_InitBlock: fixes P :: "'a prog"   and sh :: "sheap"   and V :: "'b"   and T :: "ty"   and v :: "val"   and e\<^sub>2 :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b ({V:T; V:=Val v;; e\<^sub>2},b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e\<^sub>2,b) \<surd>)"
intros_10: fixes args :: "'a \<Rightarrow> ('b, 'a) trm"   and p :: "'a" assumes "\<And>i. dsafe (args i)" shows "fsafe ($\<phi> p args)"
simps_1: fixes uu :: "'a ext0" shows "(ext0.Bot \<le> uu) = True"
path_connected_complement_homeomorphic_convex_compact: fixes S :: "'b set"   and T :: "'c set" assumes "S homeomorphic T"   and "convex T"   and "compact T"   and "2 \<le> DIM('b)" shows "path_connected (- S)"
assms_1: shows "qGood X \<or> qGood X' \<or> qGood X''"
that: fixes n' :: "'node SDG_node"   and ns :: "'node SDG_node list" assumes "n' \<in> S"   and "realizable (CFG_node (_Entry_)) ns n'"   and "n \<in> set ns" shows "thesis"
assms_1: shows "(f, g) \<in> A\<^sub>1"
natural_transformation_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" shows "natural_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_transformation_axioms A B F G \<tau>"
neg_equal_iff_equal: fixes minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and uminus :: "'q \<Rightarrow> 'q"   and a :: "'q"   and b :: "'q" assumes "class.group_add minus plus zero uminus" shows "(uminus a = uminus b) = (a = b)"
fls_prpart_coeff: fixes f :: "'a fls"   and n :: "nat" shows "coeff (fls_prpart f) n = (if n = 0 then 0::'a else fls_nth f (- int n))"
a: shows "subset.chain {B. \<not> B \<turnstile> \<phi>} C"
proper_parts_principle: fixes x :: "i"   and y :: "i" assumes "EM"   and "\<exists>z. PP z x"   and "\<forall>z. PP z x \<longrightarrow> P z y" shows "P x y"
cases: fixes a1 :: "boolean_unop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'a OCL_Types.type" assumes "boolean_unop_type a1 a2 a3" obtains \<tau> :: "'a OCL_Types.type" where "a1 = NotOp" and "a2 = \<tau>" and "a3 = \<tau>" and "\<tau> \<le> Boolean[?]"
preciseI: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> assn" assumes "\<And>a a' h p F F'. h \<Turnstile> R a p * F \<and>\<^sub>A R a' p * F' \<Longrightarrow> a = a'" shows "precise R"
case_1: fixes f1 :: "agent \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> 'a"   and f2 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and f3 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and x11 :: "agent"   and x12 :: "agent"   and x13 :: "msg" shows "(case Says x11 x12 x13 of Says x xa xb \<Rightarrow> f1 x xa xb | Gets x xa \<Rightarrow> f2 x xa | Notes x xa \<Rightarrow> f3 x xa) = f1 x11 x12 x13"
evContainsEx0_allRepeats: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and A :: "formula"   and i :: "nat" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f (0, FAll Neg A))" shows "EV (contains f (i, FAll Neg A))"
FrameStatImpClosed: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and G :: "'d frame"   and p :: "(name \<times> name) list" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.FrameStatImp SImp F G" shows "assertionAux.FrameStatImp SImp (p \<bullet> F) (p \<bullet> G)"
simps_1: fixes a :: "nat"   and b :: "nat" shows "arcs a b [] = [(a, b)]"
resCap_alt: fixes p :: "(nat \<times> nat) list" shows "resCap p = Min (cf ` set p)"
USUP_as_Inf_collect: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Squnion> P \<in> A \<bullet> f P) = \<Squnion>(f ` A)"
node_connectTE: fixes ii :: "nat"   and s :: "'c"   and R :: "nat set"   and i :: "nat"   and i' :: "nat"   and s' :: "'c"   and R' :: "nat set"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS ii s R, connect(i, i'), NodeS ii s' R') \<in> node_sos S" obtains "i = ii" and "s' = s" and "R' = insert i' R"   | "i' = ii" and "s' = s" and "R' = insert i R"   | "ii \<noteq> i" and "ii \<noteq> i'" and "s' = s" and "R' = R"
prv_fls: fixes \<phi> :: "fmla" shows "{} \<turnstile> fls IMP \<phi>"
that: fixes a :: "real" assumes "0 < a"   and "{- a..a} \<subseteq> existence_ivl0 x" shows "thesis"
ActiveProcessSilentDecisionValuesIncrease: fixes p :: "'p"   and q :: "'p"   and c :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message"   and c' :: "('p, 'v, 's) configuration" assumes "p = q"   and "c \<turnstile> msg \<mapsto> c'"   and "isReceiverOf p msg"   and "initReachable c" shows "val[q,c] \<subseteq> val[q,c']"
uses_in_\<alpha>n: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> uses g n" shows "n \<in> set (\<alpha>n g)"
prems_2: shows "ys \<noteq> [] \<longrightarrow> last (a # xs) = hd ys"
cod_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.cod (trg\<^sub>B \<mu>) = trg\<^sub>B \<mu>"
nrcons_inv_2: fixes t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_measured_ftree t"   and "is_measured_node nd" shows "is_measured_ftree (nrcons t nd)"
ltake_ltake: fixes n :: "enat"   and m :: "enat"   and xs :: "'c llist" shows "ltake n (ltake m xs) = ltake (min n m) xs"
reduced_run_length: fixes q :: "'state"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action stream"   and l :: "'action list"   and w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "length v\<^sub>1 \<le> length w"
delay_bisimulation_diverge_axioms: shows "delay_bisimulation_diverge trsys1 trsys2 bisim12 tlsim12 \<tau>move1 \<tau>move2"
consistentI1: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and store :: "'c \<Rightarrow> nat" shows "consistent \<sigma> (\<lambda>sv. \<sigma> (fst sv)) store"
encode_f: fixes n :: "nat"   and y :: "lisp"   and x :: "lisp" shows "encode (f n y x) = of_nat n * \<omega>^ (encode y) + encode x"
sharing_consis_nth_update: fixes i :: "nat"   and \<O> :: "nat set"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "sharing_consistent \<S>\<^sub>s\<^sub>b \<O> sb" shows "sharing_consis \<S>\<^sub>s\<^sub>b (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
SeqReds2: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   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 e\<^sub>2 :: "char list exp"   and e\<^sub>2' :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1;; e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2',s\<^sub>2\<rangle>"
aux114: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list" assumes "x \<noteq> DenyAll" shows "removeShadowRules1_alternative_rev (x # xs) = x # removeShadowRules1_alternative_rev xs"
strict_prefix_simps_1: fixes xs :: "'b list" shows "(xs <p \<epsilon>) = False"
winning_path_def: fixes p :: "Player"   and P :: "'a llist" shows "G'.winning_path p P \<equiv> \<not> lfinite P \<and> (\<exists>a\<in>G'.path_inf_priorities P. (\<forall>b\<in>G'.path_inf_priorities P. a \<le> b) \<and> (if p = Even then even else odd) a) \<or> \<not> lnull P \<and> lfinite P \<and> llast P \<in> G'.VV p** \<or> lnull P \<and> p = Even"
load_sub1_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and rd :: "word5"   and s_val :: "word1"   and s :: "'b sparc_state" assumes "s' = snd (fst (load_sub1 instr rd s_val s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
compute_set_of_jordan_blocks: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and ev :: "'a" assumes "jordan_nf A n_as" shows "set (compute_set_of_jordan_blocks A ev) = set n_as \<inter> UNIV \<times> {ev}"
RUN_def: fixes A :: "'a set" shows "RUN A \<equiv> \<mu> x. Mprefix A (\<lambda>xa. x)"
crit_pair_cbelow_on_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "crit_pair_cbelow_on d m F p q = cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) (monomial (1::'c) (term_of_pair (lcs (lp p) (lp q), component_of_term (lt p)))) (\<lambda>a b. red F a b \<or> red F b a) (fst (crit_pair p q)) (snd (crit_pair p q))"
sort_oalist_def: fixes xs :: "('a \<times> 'b) list" shows "sort_oalist_tc xs = foldr update_by_pair_tc xs []"
read_only_read_acquired_unforwarded_acquire_witness: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat"   and X :: "nat set" assumes "non_volatile_owned_or_read_only True \<S> \<O> sb"   and "sharing_consistent \<S> \<O> sb"   and "a \<notin> read_only \<S>"   and "a \<notin> \<O>"   and "a \<in> unforwarded_non_volatile_reads sb X" shows "(\<exists>sop a' v ys zs A L R W. sb = ys @ Write\<^sub>s\<^sub>b True a' sop v A L R W # zs \<and> a \<in> A \<and> a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b ys \<and> a' \<noteq> a) \<or> (\<exists>A L R W ys zs. sb = ys @ Ghost\<^sub>s\<^sub>b A L R W # zs \<and> a \<in> A \<and> a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b ys)"
presFreshAll_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, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "presFreshAll h hA MOD \<equiv> presFresh h MOD \<and> presFreshAbs hA MOD"
filter_distinct_at5: fixes xs :: "'a list"   and as :: "'a list"   and u :: "'a"   and bs :: "'a list"   and us :: "'a list"   and zs :: "'a list" assumes "distinct xs"   and "xs = as @ u # bs"   and "filter (\<lambda>v. v = u \<or> v \<in> set us) xs = u # us"   and "set zs \<inter> set xs \<subseteq> {u} \<union> set as" shows "filter (\<lambda>v. v \<in> set us) (zs @ bs) = us"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_7: fixes l :: "'a strand_label"   and X :: "'c list"   and F :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) list"   and G :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) list"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) # A) = (l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
has_bochner_integral_mult_left: fixes c :: "'b"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b" assumes "c \<noteq> (0::'b) \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x * c) (x * c)"
usolves_odeD_1: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and t0 :: "real"   and T :: "real set"   and X :: "'b set" assumes "(y usolves_ode f from t0) T X" shows "(y solves_ode f) T X"
cyclic_on_f_in: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and A :: "'b set"   and x :: "'b" assumes "f permutes S"   and "cyclic_on f A"   and "f x \<in> A" shows "x \<in> A"
top_greatest: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and a :: "'a" assumes "class.order_top less_eq less top" shows "less_eq a top"
leg_simps_3: shows "C.arr \<eta>.dom.leg1"
wax_def: fixes \<sigma> :: "'b" shows "wax \<sigma> \<equiv> [Pos (Pr (Guard \<sigma>) [Fn (Wit \<sigma>) []])]"
tendsto_arcosh_strong: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "1 \<le> a"   and "\<forall>\<^sub>F x in F. 1 \<le> f x" shows "((\<lambda>x. arcosh (f x)) \<longlongrightarrow> arcosh a) F"
absolutely_integrable_measurable: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "S \<in> sets lebesgue" shows "(f absolutely_integrable_on S) = (f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (norm \<circ> f))"