hcju/isabelleps huggingface dataset:
uint8_set_bit_def: fixes x :: "uint8"   and n :: "integer"   and b :: "bool" shows "uint8_set_bit x n b = (if n < 0 \<or> 7 < n then undefined set_bit_class.set_bit x n b else set_bit_class.set_bit x (nat_of_integer n) b)"
is_ub_lub: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "omega_complete le lub" shows "le (f n) (lub f)"
Limsup_lessD: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "Limsup F f < y" shows "\<forall>\<^sub>F x in F. f x < y"
assms_1: shows "\<Psi> \<simeq> \<Psi>'"
Anno_change_is_lub: fixes wf_md :: "'b J_prog \<Rightarrow> String.literal \<Rightarrow> (String.literal list \<times> 'b expr) mdecl \<Rightarrow> bool"   and P :: "'b J_prog"   and is_lub1 :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and is_lub2 :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'b expr"   and e' :: "'b expr" assumes "wf_prog wf_md P"   and "\<And>T1 T2 T3. \<lbrakk>is_lub1 T1 T2 T3; is_type P T1; is_type P T2\<rbrakk> \<Longrightarrow> is_lub2 T1 T2 T3"   and "\<And>T1 T2 T3. \<lbrakk>is_lub2 T1 T2 T3; is_type P T1; is_type P T2\<rbrakk> \<Longrightarrow> is_type P T3"   and "is_lub1,P,E \<turnstile> e \<leadsto> e'"   and "ran E \<union> set (block_types e) \<subseteq> types P" shows "is_lub2,P,E \<turnstile> e \<leadsto> e'"
DynPDG_path_not_inner: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>d* n'"   and "\<not> inner_node n'" shows "n = n'"
simps_7: fixes \<C> :: "t_context"   and t :: "t"   and uz :: "relop_i"   and ts :: "checker_type" shows "check_single \<C> (Relop_i t uz) ts = (if is_int_t t then type_update ts [TSome t, TSome t] (Type [T_i32]) else Bot)"
Nil: shows "CoWBasic.prod_list \<epsilon> = \<epsilon>"
delete_expunge_commute: fixes i :: "'c"   and "is" :: "'c set"   and e1 :: "'d"   and e2 :: "'d"   and mo :: "'c" assumes "i \<notin> is" shows "\<langle>Delete is e1\<rangle> \<rhd> \<langle>Expunge e2 mo i\<rangle> = \<langle>Expunge e2 mo i\<rangle> \<rhd> \<langle>Delete is e1\<rangle>"
point_mult2_eq_double: fixes a :: "int"   and p :: "int point" shows "point_mult a 2 p = local.add a p p"
map_gpv'_bind_gpv: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'f"   and gpv :: "('g, 'e, 'f) gpv"   and F :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "map_gpv' f g h (bind_gpv gpv F) = bind_gpv (map_gpv' id g h gpv) (\<lambda>x. map_gpv' f g h (F x))"
full_SetCompr_eq: fixes f :: "'d \<Rightarrow> 'c" shows "{u. \<exists>x. u = f x} = range f"
g_in_hhom: shows "\<guillemotleft>g : trg f \<rightarrow> src f\<guillemotright>"
Derivation_implies_append: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and c :: "symbol list" assumes "Derivation a D b"   and "Derivation b E c" shows "Derivation a (D @ E) c"
equalityE: fixes A :: "'c set"   and B :: "'c set" assumes "A = B" obtains "A \<subseteq> B" and "B \<subseteq> A"
valid_store_sops_nth_update: 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"   and i :: "nat"   and "is" :: "instr list"   and p :: "'b"   and xs :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "valid_store_sops ts"   and "i < length ts"   and "\<forall>sop\<in>store_sops is. valid_sop sop" shows "valid_store_sops (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
HEndPhase2_HInv1: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HInv1 s"   and "HEndPhase2 s s' p" shows "HInv1 s'"
add_cancl: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "D z x"   and "D z y"   and "z \<oplus> x = z \<oplus> y" shows "x = y"
eval_closed_lcl_unchanged: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap" assumes "P \<turnstile> \<langle>e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>"   and "fv e = {}" shows "l' = l"
df_def: shows "df \<equiv> \<lambda>n. (pderiv ^^ n) f"
NC2_not_NonceCCA: fixes C' :: "agent"   and N' :: "nat"   and C :: "agent"   and N :: "nat"   and evs :: "event list"   and KC1 :: "nat"   and B :: "agent"   and p :: "msg" assumes "Hash \<lbrace>Agent C', Nonce N', Agent C, Nonce N\<rbrace> \<in> parts (knows Spy evs)"   and "Nonce N \<notin> analz (knows Spy evs)"   and "evs \<in> set_cr" shows "Crypt KC1 \<lbrace>\<lbrace>Agent B, Nonce N\<rbrace>, Hash p\<rbrace> \<notin> parts (knows Spy evs)"
is_sig_GB_inD: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_GB_in d G u"   and "r \<in> dgrad_sig_set' (length fs) d"   and "lt r = u" shows "sig_red_zero (\<preceq>\<^sub>t) G r"
assms_1: shows "g \<turnstile> nodes_of_phis \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"
maxR_rangeD_2: fixes x :: "'a"   and y :: "'a" assumes "local.MaxR.maxR x y \<noteq> y" shows "local.MaxR.maxR x y = x"
hom_induced_compose_empty: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and Z :: "'d topology"   and g :: "'c \<Rightarrow> 'd"   and p :: "int" assumes "continuous_map X Y f"   and "continuous_map Y Z g" shows "hom_induced p X {} Z {} (g \<circ> f) = hom_induced p Y {} Z {} g \<circ> hom_induced p X {} Y {} f"
oa_contingent_2: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr> in v]"
map_cong0: fixes x :: "('d, 'e, 'f) val"   and f1 :: "'d \<Rightarrow> 'g"   and g1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'h"   and g2 :: "'e \<Rightarrow> 'h"   and f3 :: "'f \<Rightarrow> 'i"   and g3 :: "'f \<Rightarrow> 'i" assumes "\<And>z1. z1 \<in> RID\<^sub>V x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> LID\<^sub>V x \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> set3_val x \<Longrightarrow> f3 z3 = g3 z3" shows "map_val f1 f2 f3 x = map_val g1 g2 g3 x"
pmf_Pi: fixes A :: "'b set"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf"   and f :: "'b \<Rightarrow> 'c" assumes "finite A" shows "pmf (Pi_pmf A dflt p) f = (if \<forall>x. x \<notin> A \<longrightarrow> f x = dflt then \<Prod>x\<in>A. pmf (p x) (f x) else 0)"
gamma1_def: shows "gamma1 = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))"
load_tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>load_tmps is. i < t"
subspace_add: fixes S :: "'d set"   and x :: "'d"   and y :: "'d" assumes "csubspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
r_universal_recfn: fixes n :: "nat" assumes "0 < n" shows "recfn (Suc n) (r_universal n)"
Vinv1_finite_vote_set: fixes s :: "'b v_state_scheme"   and Q :: "process set" assumes "s \<in> Vinv1" shows "finite (vote_set (votes s) Q)"
card_injective_functions_domain_permutation: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_permutation A B) = card B choose card A"
induced_hom_surj: fixes S :: "'a set"   and P :: "('a \<times> bool) list set"   and f :: "'a \<Rightarrow> 'b" assumes "GroupByPresentationInducedFun S P f" shows "GroupByPresentationInducedFun.induced_hom S P f ` Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle> = \<langle>f ` S\<rangle>"
bij_\<phi>: shows "bij_betw \<phi> (elts (tp D)) D"
ord_of_nat_\<omega>1: fixes n :: "nat" shows "ord_of_nat n \<in> elts \<omega>1"
WellDefinedToWellFormedSet: fixes S :: "(ZF, ZF) Signature"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "Sig S \<triangleright> Vx : A \<turnstile> e : B" shows "\<exists>n. e \<in> WellFormedToSet S n"
tan_inj_pi: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) < x"   and "x < pi / 2"   and "- (pi / 2) < y"   and "y < pi / 2"   and "tan x = tan y" shows "x = y"
f1'f1: shows "1 / ((f1' - f1) * c) \<le> l0"
bex_UNIV: fixes P :: "'f \<Rightarrow> bool" shows "Bex UNIV P = Ex P"
in_homE: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "CC.in_hom f a b" obtains "\<guillemotleft>fst f : fst a \<rightarrow> fst b\<guillemotright>" and "\<guillemotleft>snd f : snd a \<rightarrow> snd b\<guillemotright>"
invmode_IntVir_eq: fixes m :: "'a member_scheme"   and e :: "expr" shows "(invmode m e = IntVir) = (\<not> is_static m \<and> e \<noteq> Super)"
degree_add_eq_right: fixes p :: "'c poly"   and q :: "'c poly" assumes "degree p < degree q" shows "degree (p + q) = degree q"
d: fixes f :: "'a ratfps" assumes "f \<in> A" shows "f dvd d"
re_refine: shows "gen_dfs \<le> \<Down> (\<langle>ES\<rangle>restr_simple_state_rel R) a.a.it_dfs"
hcomp_reassoc_2: fixes \<tau> :: "'d"   and \<mu> :: "'d"   and \<nu> :: "'d" assumes "D.arr \<tau>"   and "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<tau> = trg\<^sub>D \<mu>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>" shows "\<tau> \<star>\<^sub>D \<mu> \<star>\<^sub>D \<nu> = \<a>\<^sub>D[D.cod \<tau>, D.cod \<mu>, D.cod \<nu>] \<cdot>\<^sub>D ((\<tau> \<star>\<^sub>D \<mu>) \<star>\<^sub>D \<nu>) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[D.dom \<tau>, D.dom \<mu>, D.dom \<nu>]"
pre_2: fixes \<sigma> :: "'\<sigma>"   and x :: "'x"   and S :: "'x set" assumes "c \<sigma>"   and "x \<in> S0 - S"   and "I S \<sigma>"   and "S \<subseteq> S0"   and "\<forall>y\<in>S0 - S - {x}. R x y"   and "\<forall>y\<in>S. R y x" shows "I (insert x S) (f x \<sigma>)"
subpath_to_frontier_strong: fixes g :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "pathfinish g \<notin> S" obtains u :: "real" where "0 \<le> u" and "u \<le> 1" and "g u \<notin> interior S" and "u = 0 \<or> (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow> subpath 0 u g x \<in> interior S) \<and> g u \<in> closure S"
concat_map_sing_ident: fixes xs :: "'b list" shows "concat (map (\<lambda>x. [x]) xs) = xs"
Fvars: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "Fvars \<phi> \<subseteq> var"
iso4_sharp: fixes x :: "'a" shows "d ((x \<sqinter> nc) \<cdot> nc) \<cdot> nc = (x \<sqinter> nc) \<cdot> nc"
nq: shows "(quat.Re q)\<^sup>2 + (Im1 q)\<^sup>2 + (Im2 q)\<^sup>2 + (Im3 q)\<^sup>2 = 1"
inf_glb: fixes s1 :: "'a"   and s2 :: "'a" shows "is_glb (s1 \<sqinter> s2) s1 s2"
N_2: fixes i :: "'i" shows "{x \<in> space M. \<not> P i x} \<subseteq> N i"
\<phi>_def: shows "\<phi> = lucas_lehmer_hom q"
len\<^sub>C: shows "length cms\<^sub>C = length cms"
empty_delayed_pool_write_privilege: fixes s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "get_delayed_pool s = [] \<and> get_S (cpu_reg_val PSR s) = 0 \<and> s' = delayed_pool_write s" shows "get_S (cpu_reg_val PSR s') = 0"
body_def: shows "body \<equiv> \<mu> body. \<Lambda> r. A"
insert_absorb: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "insert a A = A"
ide_compE: fixes g :: "'c"   and f :: "'c" assumes "C.ide (C g f)" obtains "C.seq g f" and "C.seq f g" and "C g f = C.dom f" and "C g f = C.cod g"
A1: shows "k \<times> x = k' \<otimes> x"
well_order_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "well_order_on A r \<equiv> linear_order_on A r \<and> wf (r - Id)"
axioms_2: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah" assumes "Limit.cone J C D a \<chi>" shows "category J"
infsetsum_nat: fixes f :: "nat \<Rightarrow> 'a"   and A :: "nat set" assumes "f abs_summable_on A" shows "infsetsum f A = (\<Sum>n. if n \<in> A then f n else (0::'a))"
M_in_subprob: fixes s :: "'s" shows "T s \<in> space (subprob_algebra (T s))"
assms_2: shows "\<lbrace>P'\<rbrace> M \<lbrace>\<lambda>x. Q'\<rbrace>"
distinct_cnt: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs" shows "cnt x xs \<le> 1"
dim_col: shows "dim_col U = 2 ^ n * 2 ^ n"
lesseq_b_def: fixes a :: "'a"   and b :: "'a" shows "lesseq_b a b = (a r\<rightarrow> b = (1::'a))"
dsum_sprod_mem: fixes I :: "'h set"   and M :: "'h \<Rightarrow> ('i, 'a, 'j) Module_scheme"   and a :: "'a"   and b :: "'h \<Rightarrow> 'i" assumes "\<forall>i\<in>I. R module M i"   and "a \<in> carrier R"   and "b \<in> carr_dsumag I M" shows "prodM_sprod R I M a b \<in> carr_dsumag I M"
prefix_to_wordset_def: fixes pfx :: "'b prefix_match" shows "prefix_to_wordset pfx = {pfxm_prefix pfx..pfxm_prefix pfx || pfxm_mask pfx}"
poly_roots_bounds: fixes p :: "real poly" assumes "p \<noteq> 0" obtains l :: "real" and u :: "real" where "l \<le> u" and "poly p l \<noteq> 0" and "poly p u \<noteq> 0" and "{x. l < x \<and> x \<le> u \<and> poly p x = 0} = {x. poly p x = 0}" and "\<And>x. x \<le> l \<Longrightarrow> sgn (poly p x) = sgn (poly p l)" and "\<And>x. u \<le> x \<Longrightarrow> sgn (poly p x) = sgn (poly p u)"
Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e_inject: fixes x :: "\<langle>\<langle>'a \<Rightarrow> nat\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "\<langle>\<langle>'a \<Rightarrow> nat\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "x \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> \<lceil>\<lceil>X\<rceil>\<rceil> UML_Types.bot_class.bot = 0}"   and "y \<in> {X. X = UML_Types.bot_class.bot \<or> X = null \<or> \<lceil>\<lceil>X\<rceil>\<rceil> UML_Types.bot_class.bot = 0}" shows "(Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e x = Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e y) = (x = y)"
assms: fixes X :: "nat \<Rightarrow> ereal" assumes "filterlim X (at_right c) sequentially" shows "(\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
map_simp_ide: fixes a :: "'d" assumes "D.ide a" shows "\<eta>.map a = \<eta>o a"
word_not_def: fixes a :: "'b word" shows "NOT a = word_of_int (NOT (uint a))"
lsl_Inf_closed_var: fixes X :: "'b set" shows "\<nu> (\<Sqinter> (\<nu> ` X)) = \<Sqinter> (\<nu> ` X)"
leg_simps_3: shows "C.arr \<pi>\<rho>.leg1"
stimes_case: fixes I :: "('e, 'f, 'd) interp"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" shows "sterm_sem I (Times t1 t2) = (\<lambda>v. sterm_sem I t1 v * sterm_sem I t2 v)"
MPair_synth: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> synth H" obtains "\<lbrace>X, Y\<rbrace> \<in> H"   | "X \<in> synth H" and "Y \<in> synth H"
prod_list_partition: fixes f :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "List.partition f xs = (ys, zs)" shows "prod_list xs = prod_list ys * prod_list zs"
fmlookup'_def: fixes A :: "'b"   and k :: "('b, 'a) fmap" shows "fmlookup' A k = fmlookup k A"
braking_distance_def: fixes ts :: "traffic" shows "braking_distance ts \<equiv> snd (snd (snd (snd (snd (Rep_traffic ts)))))"
filter_mset_image_mset: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c multiset" shows "filter_mset P (image_mset f A) = image_mset f {#x \<in># A. P (f x)#}"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_backto_standard_pos_O m rs (l, r) = (\<exists>ml mr rn. l = Oc \<up> ml \<and> r = Oc \<up> mr @ Bk # Oc \<up> Suc (Suc rs) @ Bk \<up> rn \<and> ml + mr = Suc m \<and> 0 < mr)"
left_eq_app_right: fixes u :: "'a list"   and q :: "hf"   and v :: "'a list" assumes "u \<in> MN.left_lang q"   and "v \<in> MN.left_lang q" shows "(u, v) \<in> eq_app_right MN.language"
rewriteR_comp_comp2: fixes g :: "'e \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'e"   and r1 :: "'f \<Rightarrow> 'd"   and r2 :: "'c \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'g"   and l :: "'f \<Rightarrow> 'g" assumes "g \<circ> h = r1 \<circ> r2"   and "f \<circ> r1 = l" shows "f \<circ> g \<circ> h = l \<circ> r2"
assms: shows "Suc i < length (half_segments_of_aform X)"
sup_least: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> x" shows "sup y z \<le> x"
cf_set_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and e :: "nat \<times> nat"   and cap :: "'capacity" shows "cf_set cf e cap \<equiv> assert (case e of (u, v) \<Rightarrow> u \<in> V \<and> v \<in> V) \<bind> (\<lambda>_. return (cf(e := cap)))"
mtf2_forward_effect2': fixes q :: "'a"   and xs :: "'a list"   and x :: "'a"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q = index xs x" shows "index (mtf2 n q xs) (xs ! index xs x) = index xs q - n \<and> index xs q - n = index (mtf2 n q xs) (xs ! index xs x)"
of_nat_nat: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus"   and "0 \<le> z" shows "semiring_1.of_nat one plus zero (nat z) = ring_1.of_int one plus zero minus z"
dualization_rew_2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<not> \<diamond>F \<equiv> \<box>\<not> F"
X'_def: shows "X' \<equiv> (\<lambda>\<sigma>. <|\<sigma> x, \<sigma> y|>) -` X \<inter> space (state_measure V \<Gamma>)"
subdegree_mod: fixes f :: "'a fps"   and g :: "'a fps" assumes "subdegree f < subdegree g" shows "subdegree (f mod g) = subdegree f"
continuous_closed_vimage: fixes S :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "closed S"   and "\<And>x. isCont f x" shows "closed (f -` S)"
card_of_Fpow_mono: fixes A :: "'a set"   and B :: "'b set" assumes "|A| \<le>o |B|" shows "|Fpow A| \<le>o |Fpow B|"
nth_zip: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'b list" assumes "i < length xs"   and "i < length ys" shows "zip xs ys ! i = (xs ! i, ys ! i)"
A: shows "((u # r, c), ee, s', c') \<in> ntrs fg"
cc_restr_bot: fixes S :: "Vars.var set" shows "\<bottom> G|` S = \<bottom>"
obs_consistent_refl: fixes S :: "('a, 'b) spec" shows "obs_consistent Id id S S"
OclAllInstances_generic_defined: fixes \<tau> :: "'d state \<times> 'd state"   and pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" shows "\<tau> \<Turnstile> \<delta> OclAllInstances_generic pre_post H"
power_sq_le_rd: fixes x :: "'a"   and n :: "nat" shows "proto_dioid.power_sq 1\<^sub>\<sigma> (\<cdot>) sup x n \<le> proto_dioid.power_rd 1\<^sub>\<sigma> (\<cdot>) sup (Sup {}) x (Suc n)"
gbm_empty_def: shows "gbm_empty \<equiv> m1.empty"
bool_assn_proper_5: fixes P :: "heap \<times> nat set \<Rightarrow> bool" assumes "proper P" shows "proper (\<lambda>h. in_range h \<and> \<not> P h)"
lift_simulation_eval: fixes s2 :: "'state2"   and s2' :: "'state2"   and i1 :: "'index"   and s1 :: "'state1" assumes "L2.eval s2 s2'"   and "match i1 s1 s2" shows "\<exists>i2 s1'. L1.eval s1 s1' \<and> match i2 s1' s2'"
assms_3: fixes s :: "'c" assumes "I g s" shows "P s"
PB2: shows "P2 * B2 \<in> carrier_mat m m"
proper_subgroup_imp_bound_on_card: fixes H :: "nat set" assumes "H \<subset> carrier G"   and "subgroup H G"   and "finite (carrier G)" shows "card H \<le> order G div 2"
map_filter_finite: fixes m :: "'c \<Rightarrow> 'd option"   and P :: "'c \<Rightarrow> bool" assumes "finite (dom m)" shows "finite (dom (map_filter P m))"
size_bst_insert: fixes x :: "'a"   and t :: "'a tree" assumes "x \<notin> set_tree t" shows "size (Tree_Set.insert x t) = Suc (size t)"
disc_inter_ocircline_set: fixes H :: "ocircline" shows "disc H \<inter> ocircline_set H = {}"
cp_array_def: fixes xs :: "'b list" shows "cp_array xs \<equiv> let ys = op_array_replicate (length xs) (0::'b) in nfoldli [0..<length xs] (\<lambda>_. True) (\<lambda>i ys. ASSERT (i < length xs \<and> i < length ys) \<bind> (\<lambda>_. RETURN (ys[i := xs ! i]))) ys \<bind> RETURN"
assms_2: shows "pirreducible\<^bsub>L\<^esub> (carrier L) q"
binomial_gbinomial: fixes n :: "nat"   and k :: "nat" shows "of_nat (n choose k) = of_nat n gchoose k"
leftlang: shows "\<L> = {v |v. is_word v \<and> is_leftderivation v}"
distinct_disc_4: fixes pr_operation' :: "pr_operation'" assumes "is_SAT_PUSH' pr_operation'" shows "\<not> is_RELABEL' pr_operation'"
ideD_3: fixes a :: "'s" assumes "ide a" shows "cod a = a"
all_dupE: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<forall>x. P x" obtains "P x" and "\<forall>x. P x"
IH: fixes Y :: "'a multiset" assumes "M \<in> multisets A"   and "Y \<in> multisets A"   and "\<forall>y. y \<in># Y \<longrightarrow> (\<exists>x. x \<in># M \<and> P y x)" shows "mulex_on P A Y M"
select_convs_2: fixes ob :: "'c set"   and ar :: "'b set"   and dom :: "'b \<Rightarrow> 'c"   and cod :: "'b \<Rightarrow> 'c"   and id :: "'c \<Rightarrow> 'b"   and comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'d" shows "Ar\<^bsub>\<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr>\<^esub> = ar"
member_def: fixes xs :: "'d list"   and x :: "'d" shows "List.member xs x = (x \<in> set xs)"
simp: shows "mk_eq (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) (Abs T s)) (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) (Abs T t)) = mk_eq (Abs T (subst_bvs1' s 1 (map (\<lambda>(x, y). Fv x y) vs))) (Abs T (subst_bvs1' t 1 (map (\<lambda>(x, y). Fv x y) vs)))"
splitAt_dist_ram_all: fixes vs :: "'a list"   and a :: "'a list"   and ram1 :: "'a"   and b :: "'a list"   and ram2 :: "'a"   and c :: "'a list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(a, b) = splitAt ram1 (fst (splitAt ram2 vs)) \<and> (c, []) = splitAt ram1 (snd (splitAt ram2 vs)) \<and> (a, []) = splitAt ram2 (fst (splitAt ram1 vs)) \<and> (b, c) = splitAt ram2 (snd (splitAt ram1 vs)) \<and> c = snd (splitAt ram2 vs) \<and> a = fst (splitAt ram1 vs)"
keysFor_def: fixes H :: "msg set" shows "keysFor H \<equiv> invKey ` {K. \<exists>X. Crypt K X \<in> H}"
simps: fixes a1 :: "nat"   and a2 :: "'a set"   and a3 :: "'a set" shows "dimension a1 a2 a3 = ((\<exists>K. a1 = 0 \<and> a2 = K \<and> a3 = {\<zero>}) \<or> (\<exists>v E n K. a1 = Suc n \<and> a2 = K \<and> a3 = line_extension K v E \<and> v \<in> carrier R \<and> v \<notin> E \<and> dimension n K E))"
consequence_relation_family_def: fixes Bot :: "'a set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" shows "consequence_relation_family Bot Q entails_q \<equiv> Q \<noteq> {} \<and> (\<forall>q\<in>Q. consequence_relation Bot (entails_q q))"
restrict_relp_DomainpD: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and x :: "'d" assumes "Domainp (R \<upharpoonleft> P \<otimes> Q) x" shows "Domainp R x \<and> P x"
f: shows "(f \<longlongrightarrow> (0::'a)) F"
assms: fixes i :: "nat" shows "(0::'a) \<le> f i"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (sub_instr instr s1)) \<and> t2 = snd (fst (sub_instr instr s2))"
lnth_inf_llist: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "lnth (inf_llist f) n = f n"
hp2: shows "\<turnstile>\<^sub>p {adjoint (U2 * U1) * P * (U2 * U1)} Utrans U1 {adjoint U2 * P * U2}"
algebraic_uminus_iff: fixes x :: "'b" shows "algebraic (- x) = algebraic x"
delta_nonneg: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and deltaG :: "'a itself \<Rightarrow> real" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "0 \<le> deltaG TYPE('a)"
cod_simps_3: shows "dom_\<pi>.cod.leg1 = doms.\<pi>.leg1"