hcju/isabelleps huggingface dataset:
seeing_class_def2: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D" shows "seeing_class P C M = \<lfloor>D\<rfloor>"
hyps: fixes q :: "'a poly"   and r :: "'a poly"   and dr :: "nat" shows "fst (pseudo_divmod_main lc q r d dr n) = divide_poly_main lc (smult (lc ^ n) q) (smult (lc ^ n) r) d dr n"
measure_distr: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and S :: "'b set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "S \<in> sets N" shows "Sigma_Algebra.measure (distr M N f) S = Sigma_Algebra.measure M (f -` S \<inter> space M)"
GH: fixes z :: "'a \<times> 'b" assumes "dist (x, y) z < r" shows "G (H z) = z"
x_in: shows "row x (Gauss_Jordan A) \<in> {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \<noteq> 0}"
final_def: fixes idx :: "'n" shows "final idx = nullable \<circ> finalize idx"
select_convs_5: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.globs \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = globs"
tllist_of_llist_cong: fixes xs :: "'a llist"   and xs' :: "'a llist"   and b :: "'b"   and b' :: "'b" assumes "xs = xs'"   and "lfinite xs' \<Longrightarrow> b = b'" shows "tllist_of_llist b xs = tllist_of_llist b' xs'"
multh: shows "Var x ^ Suc i = Var x ^ i * Var x"
addfirststut: fixes s :: "nat \<Rightarrow> 'b" assumes "first s = second s" shows "s \<approx> tail s"
simps_2: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and ys :: "'b list" shows "insert_wrt R x (y # ys) = (if (x, y) \<in> R then x # y # ys else y # insert_wrt R x ys)"
nths_reduceIndices: fixes xs :: "'a list"   and iset :: "nat set" shows "nths xs iset = nths xs {i. i < |xs| \<and> i \<in> iset}"
select_fold_exec\<^sub>S\<^sub>e\<^sub>q: fixes \<tau> :: "'d state \<times> 'd state"   and l :: "('d state \<times> 'd state \<Rightarrow> 'e) list" assumes "list_all (\<lambda>f. \<tau> \<Turnstile> \<upsilon> f) l" shows "\<lceil>\<lceil>Rep_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e (foldl UML_Sequence.OclIncluding Sequence{} l \<tau>)\<rceil>\<rceil> = map (\<lambda>f. f \<tau>) l"
asIMOD_igSubstAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and ys :: "'varSort"   and Y :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and y :: "'var"   and A :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs" shows "igSubstAbs (asIMOD SEM) ys Y y A = (\<lambda>val. if sWlsVal SEM val then A (val (y := Y val)_ys) else undefined)"
update_induct: fixes P :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and f :: "'b \<Rightarrow>\<^sub>0 'c" assumes "P 0"   and "\<And>f a b. \<lbrakk>a \<notin> keys f; b \<noteq> (0::'c); P f\<rbrakk> \<Longrightarrow> P (Poly_Mapping.update a b f)" shows "P f"
valapp_def: fixes f :: "('b \<Rightarrow> 'a) \<times> (unit \<Rightarrow> term)"   and x :: "'b \<times> (unit \<Rightarrow> term)" shows "valapp f x = (fst f (fst x), \<lambda>u. App (snd f ()) (snd x ()))"
hom_mult: fixes x :: "'c poly"   and y :: "'c poly"   and a :: "'c" shows "poly (x * y) a = poly x a * poly y a"
norm_bcontfun_def: fixes f :: "'a \<Rightarrow>\<^sub>C 'b" shows "norm f = dist f 0"
bijective_map_axioms: shows "bijective_map \<eta> G G'"
empty_state_list_lemma: fixes s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "[] \<noteq> state_list s as"
simps_9: fixes p :: "fm"   and q :: "fm" shows "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
ih_23: shows "InvariantVarsF (getF state') F0 Vbl"
lvl_insert: fixes x :: "'b"   and t :: "('b \<times> nat) tree" obtains "lvl (insert x t) = lvl t"   | "lvl (insert x t) = lvl t + 1" and "sngl (insert x t)"
l2_11: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet A' B' C'"   and "Cong A B A' B'"   and "Cong B C B' C'" shows "Cong A C A' C'"
load_sub2_privilege: fixes s' :: "'b sparc_state"   and addr :: "word32"   and asi :: "word8"   and r :: "word5"   and win :: "'b word"   and w :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (load_sub2 addr asi r win w s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
assms: shows "f \<in> {..n} \<rightarrow> carrier G"
nth_non_equal_first_eq: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and n :: "nat" assumes "x \<noteq> y" shows "((x # xs) ! n = y) = (xs ! (n - 1) = y \<and> 0 < n)"
assign_upd_blah: fixes x1 :: "char list"   and s :: "char list \<Rightarrow> int"   and x2 :: "aexp" shows "(\<lambda>a. if a = x1 then s x1 else (s(x1 := aval x2 s)) a) = s"
cpu_reg_mod_def: fixes data_w32 :: "word32"   and cpu :: "CPU_register"   and state :: "'b sparc_state" shows "cpu_reg_mod data_w32 cpu state \<equiv> state\<lparr>cpu_reg := (cpu_reg state)(cpu := data_w32)\<rparr>"
ActualDisjE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>(\<phi> \<^bold>\<or> \<psi>) in v]" shows "[\<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<psi> in v]"
prefix_prefix: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "prefix xs ys" shows "prefix xs (ys @ zs)"
img_fst: fixes a :: "'e"   and b :: "'f"   and S :: "('e \<times> 'f) set" assumes "(a, b) \<in> S" shows "a \<in> fst ` S"
eexp_suminf: fixes f :: "nat \<Rightarrow> ereal" assumes "\<not> {- \<infinity>, \<infinity>} \<subseteq> range f"   and "summable f" shows "(\<lambda>n. \<Prod>i<n. eexp (f i)) \<longlonglongrightarrow> eexp (\<Sum>i. f i)"
NatTransExt': fixes \<eta>1' :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and \<eta>2' :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NTDom \<eta>1' = NTDom \<eta>2'"   and "NTCod \<eta>1' = NTCod \<eta>2'"   and "\<And>X. X \<in> obj\<^bsub>NTCatDom \<eta>1'\<^esub> \<Longrightarrow> \<eta>1' $$ X = \<eta>2' $$ X" shows "MakeNT \<eta>1' = MakeNT \<eta>2'"
Cauchy_convergent: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a" assumes "class.complete_space dist uniformity open"   and "uniform_space.Cauchy uniformity X" shows "topological_space.convergent open X"
ndet_idemp: fixes P :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<or> P) = P"
app\<^sub>i_mono: fixes p :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and \<tau> :: "ty list \<times> ty err list"   and \<tau>' :: "ty list \<times> ty err list"   and i :: "instr"   and mxs :: "nat"   and mpc :: "nat"   and rT :: "ty" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'"   and "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')" shows "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>)"
linord_of_list_Field: fixes xs :: "'a list" shows "Field (linord_of_list xs) = set xs"
int_to_real_def: fixes n :: "int" shows "to_real n = real_of_int n"
intro: fixes R :: "('a, 'b) ring_scheme"   and I :: "'a set" assumes "\<exists>i\<in>carrier R. I = Idl\<^bsub>R\<^esub> {i}" shows "principalideal_axioms I R"
antisymmetric_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "antisymmetric x" shows "antisymmetric (x \<sqinter> y)"
hintikka: fixes x :: "\<nu>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<equiv> \<phi> x \<^bold>& (\<^bold>\<forall>z. \<phi> z \<^bold>\<rightarrow> z \<^bold>= x) in dw]"
Sr_mOp_closed: fixes S :: "'a set"   and x :: "'a" assumes "sr R S"   and "x \<in> S" shows "-\<^sub>a x \<in> S"
vpath_self: fixes u :: "'a"   and G :: "('a, 'b) pre_digraph" assumes "u \<in> verts G" shows "vpath [u] G"
antisym: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dpo le"   and "le x y"   and "le y x" shows "x = y"
minus_top_ennreal: fixes x :: "ennreal" shows "x - top = (if x = top then top else 0)"
vsubstAbs_compose2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var"   and y1 :: "'var" assumes "goodAbs A"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "A $[y1 // y]_ys $[z1 // z]_zs = A $[z1 // z]_zs $[y1 @ys[z1 / z]_zs // y]_ys"
zero_power2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero 2 = zero"
tobd_F13_inj: fixes x :: "'d"   and xa :: "('e, 'f, 'd) F1"   and y :: "'d"   and r2 :: "(bd_type_F2 \<times> bd_type_F2) set" assumes "x \<in> F1set3 xa"   and "y \<in> F1set3 xa" shows "(toCard (F1set3 xa) (dir_image (bd_F1 +c r2) Abs_bd_type_F) x = toCard (F1set3 xa) (dir_image (bd_F1 +c r2) Abs_bd_type_F) y) = (x = y)"
map_option_cong: fixes x :: "'b option"   and y :: "'b option"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = y"   and "\<And>a. y = \<lfloor>a\<rfloor> \<Longrightarrow> f a = g a" shows "map_option f x = map_option g y"
a_in_rcs: fixes H :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G" shows "a \<in> H \<bullet> a"
reflP_R2P: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "reflP (S2P A) (R2P r) = refl_on A r"
USUP_upto_expand_first: fixes n :: "nat"   and P :: "nat \<Rightarrow> 'a upred" shows "(\<Squnion> i \<in> {0..<Suc n} \<bullet> P i) = (P 0 \<and> (\<Squnion> i \<in> {1..<Suc n} \<bullet> P i))"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "inj_ring_hom hom"   and "surjective hom" shows "ring_isom hom"
v2_elim: fixes a :: "name"   and T :: "ty"   and \<Gamma> :: "(name \<times> ty) list" assumes "valid ((a, T) # \<Gamma>)" shows "a \<sharp> \<Gamma> \<and> valid \<Gamma>"
shows_attr_append: fixes s :: "char list"   and av :: "char list \<times> char list"   and r :: "char list"   and t :: "char list" shows "(shows_string s \<circ> shows_attr av) (r @ t) = (shows_string s \<circ> shows_attr av) r @ t"
alphabeta: shows "int p = \<alpha>\<^sup>2 + 3 * \<beta>\<^sup>2"
eq_LConsD: fixes xs :: "'a llist"   and y :: "'a"   and ys :: "'a llist" assumes "xs = y #\<^sub>l ys" shows "xs \<noteq> []\<^sub>l \<and> lhd xs = y \<and> ltl xs = ys"
HInfinite_inverse_Infinitesimal: fixes x :: "'b star" assumes "x \<in> HInfinite" shows "inverse x \<in> Infinitesimal"
invariant_1_2I: fixes rai :: "int poly \<times> rat \<times> rat" assumes "invariant_1 rai"   and "1 < degree (poly_real_alg_1 rai)" shows "invariant_1_2 rai"
le_cases: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h"   and y :: "'h" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
cring_simprules_14: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> y = y \<otimes> x"
not_Some_eq: fixes x :: "'m option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
assms_1: shows "run (Array.make n f) \<sigma> \<sigma>' r"
round_robin_reschedule_invar_Some: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and \<sigma>' :: "'t list \<times> nat" assumes "deterministic I"   and "round_robin_reschedule t0 queue n0 s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "round_robin_invar (queue, n0) (dom (thr s))"   and "t0 \<in> set queue"   and "s \<in> I" shows "round_robin_invar \<sigma>' (dom (thr s) \<union> {t. \<exists>x m. NewThread t x m \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>})"
chamberD_simplex: fixes X :: "'c set set"   and x :: "'c set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X x" shows "x \<in> X"
subspace_scale: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b"   and c :: "'a" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S" shows "scale c x \<in> S"
case_1: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f" shows "(case None of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
rfd_rfb_demorgan: fixes R :: "('d \<times> 'c) set" shows "\<partial> \<circ> fd\<^sub>\<R> R = fb\<^sub>\<R> R \<circ> \<partial>"
atLeastAtMost_def: fixes l :: "'b"   and u :: "'b" shows "{l..u} \<equiv> {l..} \<inter> {..u}"
rel_simps_64: shows "- (1::'a) < (0::'a)"
tailrec_body_def: fixes a :: "'d \<Rightarrow> 'b nres"   and b :: "'d \<Rightarrow> 'c nres" shows "tailrec_body a b \<equiv> \<lambda>D s. sup (a s \<bind> D) (b s)"
qVar_alpha_iff: fixes xs :: "'h"   and x :: "'i"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "(qVar xs x #= X') = (X' = qVar xs x)"
ih: fixes ka :: "hmultiset"   and la :: "hmultiset" assumes "{#ka, la#} < {#k, l#}"   and "ka < la" shows "ordinal_of_hmset ka < ordinal_of_hmset la"
a: shows "x \<rightarrow>\<^sup>* z \<and> y \<rightarrow>\<^sup>* z"
axioms_1: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FGModule G smult V" shows "Group G"
lookupDef_induct: fixes n :: "'node"   and g :: "'g"   and v :: "'var"   and P :: "'node \<Rightarrow> bool" assumes "n \<in> set (\<alpha>n g)"   and "\<And>n. \<lbrakk>n \<in> set (\<alpha>n g); v \<in> defs g n; lookupDef g n v = (v, n, SimpleDef)\<rbrakk> \<Longrightarrow> P n"   and "\<And>n. \<lbrakk>n \<in> set (\<alpha>n g); v \<notin> defs g n; length (predecessors g n) \<noteq> 1; lookupDef g n v = (v, n, PhiDef)\<rbrakk> \<Longrightarrow> P n"   and "\<And>n m. \<lbrakk>v \<notin> defs g n; predecessors g n = [m]; m \<in> set (\<alpha>n g); lookupDef g n v = lookupDef g m v; P m\<rbrakk> \<Longrightarrow> P n" shows "P n"
distinct_left: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree"   and y :: "'c" assumes "all_distinct (Node l x False r)"   and "y \<in> set_of l" shows "x \<noteq> y"
abs_trans: fixes ts1 :: "traffic"   and ts2 :: "traffic"   and ts0 :: "traffic" assumes "ts1 \<^bold>\<Rightarrow> ts2"   and "ts0 \<^bold>\<Rightarrow> ts1" shows "ts0 \<^bold>\<Rightarrow> ts2"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and cstate :: "'b cstate" shows "P (case cstate of Void \<Rightarrow> f1 | cstate.Fail \<Rightarrow> f2 | Store x \<Rightarrow> f3 x | cstate.Collect x \<Rightarrow> f4 x) = ((cstate = Void \<longrightarrow> P f1) \<and> (cstate = cstate.Fail \<longrightarrow> P f2) \<and> (\<forall>x3. cstate = Store x3 \<longrightarrow> P (f3 x3)) \<and> (\<forall>x4. cstate = cstate.Collect x4 \<longrightarrow> P (f4 x4)))"
scc_root_unique_scc: fixes v :: "'v"   and scc :: "'v set"   and scc' :: "'v set" assumes "scc_root s v scc"   and "scc_root s v scc'" shows "scc = scc'"
assms: shows "X \<in> sets (M \<Otimes>\<^sub>M tree_sigma M \<Otimes>\<^sub>M tree_sigma M)"
module_hom_on_axioms_def: fixes S1 :: "'c set"   and s1 :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'c \<Rightarrow> 'e" shows "module_hom_on_axioms S1 s1 s2 f \<equiv> (\<forall>b1 b2. b1 \<in> S1 \<longrightarrow> b2 \<in> S1 \<longrightarrow> f (b1 + b2) = f b1 + f b2) \<and> (\<forall>r b. b \<in> S1 \<longrightarrow> f (s1 r b) = s2 r (f b))"
OUTfromVCorrect1_data20: shows "OUTfromVCorrect1 data20"
ins: shows "ins k p r = \<langle>l2, (k2, p2), r2\<rangle>"
cidr_split_prefix: fixes r :: "'b wordinterval" shows "\<Union> (uncurry ipset_from_cidr ` set (cidr_split r)) = wordinterval_to_set r"
add_mono_thms_linordered_semiring_4: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i = j \<and> k = l" shows "i + k = j + l"
safe_distance_5r_def: shows "safe_distance_5r = v\<^sub>e\<^sup>2 / 2 / (a\<^sub>o - a\<^sub>e) + v\<^sub>o\<^sup>2 / 2 / a\<^sub>o + v\<^sub>e * \<delta>"
bijection_def: fixes f :: "'a \<Rightarrow> 'a" shows "bijection f \<equiv> bij f"
X: shows "X \<in> sets (dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) (case_nat undefined \<rho>))"
minus_add_distrib: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "- (x + y) = - x + - y"
app_tl_empty_length: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'b list"   and a :: "'a" assumes "tl (map f [0..<length l] @ [a]) = []" shows "l = []"
of_rat_prod: fixes f :: "'b \<Rightarrow> rat"   and A :: "'b set" shows "of_rat (prod f A) = (\<Prod>a\<in>A. of_rat (f a))"
normal_set_range: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "normal F" shows "normal_set (range F)"
set_sel_1: fixes a :: "'a llist" assumes "\<not> lnull a" shows "lhd a \<in> lset a"
assoc'_naturality: fixes f0 :: "'a"   and f1 :: "'a"   and f2 :: "'a" assumes "arr f0"   and "arr f1"   and "arr f2" shows "((f0 \<otimes> f1) \<otimes> f2) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f0, local.dom f1, local.dom f2] = \<a>\<^sup>-\<^sup>1[cod f0, cod f1, cod f2] \<cdot> (f0 \<otimes> f1 \<otimes> f2)"
gla_var: fixes y :: "'a"   and x :: "'a" assumes "test y" shows "(y \<sqinter> d x \<le> bot) = (y \<sqinter> x \<le> bot)"
allg: fixes upds :: "('a \<times> 'd) set"   and A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'e"   and xs :: "('a \<times> 'f) list"   and l'' :: "'a \<Rightarrow> 'b"   and q :: "'f \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>l s n x. x \<in> fst ` upds \<Longrightarrow> A (l(x := n)) = A l"   and "lesvars xs \<subseteq> fst ` upds" shows "A (ListUpdate l'' xs q) = A l''"
inverse_arrows_can_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u" shows "category.iso V (bicategory.can V H \<a> \<i> src trg u t)"
PREFER_I: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "P x" shows "PREFER P x"
mono_body_fixp: fixes F :: "('f \<Rightarrow> 'g) \<Rightarrow> 'f \<Rightarrow> 'g" assumes "\<And>x. lfp.mono_body (\<lambda>f. F f x)" shows "lfp.fixp_fun F = F (lfp.fixp_fun F)"
hbase_mono: fixes n :: "C"   and b :: "nat" assumes "n \<in> hbase b" shows "n \<in> hbase (Suc b)"
no_pseudominimal_in_pow_is_empty: fixes A :: "'b set" assumes "\<And>x. \<not> order.pseudominimal_in (Pow A) {x}" shows "A = {}"
HFinite_not_Infinitesimal_inverse: fixes x :: "'a star" assumes "x \<in> HFinite - Infinitesimal" shows "inverse x \<in> HFinite - Infinitesimal"
fiter_fstar: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "fiter_id f \<le> fstar f"
le_imp_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.semi_canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" shows "\<exists>c. b = plus a c"
adds_pp_triv: fixes t :: "'a"   and v :: "'t" shows "t adds\<^sub>p t \<oplus> v"
cfg_onI: fixes cfg :: "'s cfg"   and s :: "'s" assumes "state cfg = s"   and "action cfg \<in> K s"   and "\<And>t. t \<in> set_pmf (action cfg) \<Longrightarrow> cont cfg t \<in> cfg_on t" shows "cfg \<in> cfg_on s"
\<alpha>'_simp: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "arr f"   and "arr g"   and "arr h" shows "\<alpha>' (f, g, h) = ((f \<otimes> g) \<otimes> h) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f, local.dom g, local.dom h]"
sint_ge: fixes x :: "'b word" shows "- (2 ^ (LENGTH('b) - 1)) \<le> sint x"
induced_phi_graph_def: fixes g :: "'g"   and P :: "'val set" shows "induced_phi_graph g P \<equiv> Restr {(\<phi>, \<phi>'). phiArg g \<phi> \<phi>'} P"
exec_step_ind_Throw_Null: fixes stk :: "val list"   and P :: "jvm_prog"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "hd stk = Null" shows "exec_step_ind (StepI Throw) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
lincomb_in_finite_cone: fixes x :: "'a vec"   and l :: "'a vec \<Rightarrow> 'a"   and W :: "'a vec set" assumes "x = next_dim.lincomb l W"   and "finite W"   and "\<forall>i\<in>W. (0::'a) \<le> l i"   and "W \<subseteq> carrier_vec (n + 1)" shows "x \<in> next_dim.finite_cone W"
gfp_trans_upperbound: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_utrans t (T t)"   and "\<And>P. unitary P \<Longrightarrow> unitary (t P)" shows "le_utrans t (gfp_trans T)"
l_i_unique: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e"   and b :: "'e" assumes "Group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<cdot>\<^bsub>G\<^esub> a = \<one>\<^bsub>G\<^esub>" shows "\<rho>\<^bsub>G\<^esub> a = b"
inorder2: fixes t :: "'b tree"   and xs :: "'b list" shows "Root_Balanced_Tree.inorder2 t xs = Tree.inorder2 t xs"
SCallInitProcessingReds: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and sfs :: "sfields"   and l\<^sub>2' :: "sfields"   and ef :: "char list exp"   and h\<^sub>3 :: "heap"   and l\<^sub>3 :: "sfields"   and sh\<^sub>3 :: "sheap"   and b\<^sub>3 :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "sh\<^sub>2 D = \<lfloor>(sfs, Processing)\<rfloor>"   and "length vs = length pns"   and "l\<^sub>2' = [pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"   and "final ef" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2, sh\<^sub>3),b\<^sub>3\<rangle>"