hcju/isabelleps huggingface dataset:
monad_reader_state_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'d \<Rightarrow> 'b \<Rightarrow> 'b" shows "monad_reader_state return bind ask get put \<equiv> monad_reader return bind ask \<and> monad_state return bind get put \<and> monad_reader_state_axioms ask get put"
SourcesLevelLimit: fixes i :: "AbstrLevelsID"   and chSet :: "chanID set" shows "\<Union> (Sources i ` outSetOfComponents i chSet) \<subseteq> AbstrLevel i"
prob_space_point_measure: fixes S :: "'a set"   and p :: "'a \<Rightarrow> ennreal" assumes "finite S"   and "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> p s"   and "sum p S = 1" shows "prob_space (point_measure S p)"
set_iterator_image_filter_foldli_conv: fixes g :: "'f \<Rightarrow> 'e option"   and xs :: "'f list" shows "set_iterator_image_filter g (foldli xs) = foldli (List.map_filter g xs)"
binrelchain_Cons_reduce: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "binrelchain P (x # xs)" shows "binrelchain P xs"
tensor_from_lookup_def: fixes ds :: "nat list"   and e :: "nat list \<Rightarrow> 'a" shows "tensor_from_lookup ds e = tensor_from_vec ds (tensor_vec_from_lookup ds e)"
HEndPhase2_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4d s q" shows "HInv4d s' q"
sup_least: fixes y :: "'b multiset"   and x :: "'b multiset"   and z :: "'b multiset" assumes "y \<subseteq># x"   and "z \<subseteq># x" shows "y \<union># z \<subseteq># x"
1: shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u \<in> VHom (Dom u) (Cod t)"
pdivmod_monic_def: fixes f :: "'b poly"   and g :: "'b poly" shows "pdivmod_monic f g \<equiv> let cg = coeffs g; cf = coeffs f; (q, r) = divmod_poly_one_main_list [] (rev cf) (rev cg) (1 + length cf - length cg) in (poly_of_list q, poly_of_list (rev r))"
less_imp_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
qp_insert: fixes i :: "name"   and i' :: "name" assumes "atom i \<sharp> p"   and "atom i' \<sharp> (i, p)" shows "quote_perm ((atom i \<rightleftharpoons> atom i') + p) (insert i V)"
2: assumes "\<lbrakk>M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>; \<A>\<rbrakk>\<^sub>c \<I>" shows "\<lbrakk>M; \<A>\<rbrakk>\<^sub>c' \<I>"
zip_map_fst_snd: fixes zs :: "('e \<times> 'f) list" shows "map fst zs || map snd zs = zs"
mono: fixes c :: "'c \<times> 'd"   and a :: "'c \<times> 'd"   and d :: "'c \<times> 'd"   and b :: "'c \<times> 'd" assumes "less_eq_prod c a"   and "less_eq_prod d b" shows "less_eq_prod (sup_prod c d) (sup_prod a b)"
diff: fixes u :: "'a set" assumes "u \<in> U" shows "diff k (src.charts_submanifold u) charts2 f"
strict_coboundedI1: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a < c" shows "min a b < c"
dm_downset_var: fixes x :: "'b" shows "dm (\<eta> x) = \<down> x"
ip_cidr_set_intersect_subset_helper: fixes i1 :: "'b word"   and r1 :: "nat"   and i2 :: "'b word"   and r2 :: "nat" assumes "ip_cidr_set i1 r1 \<inter> ip_cidr_set i2 r2 \<noteq> {}"   and "r1 \<le> r2" shows "ip_cidr_set i2 r2 \<subseteq> ip_cidr_set i1 r1"
nble: shows "nat (bitlen \<bar>mantissa x\<bar>) \<le> Suc LENGTH('f)"
span_insert_idI: fixes r :: "'b"   and B :: "'b set" assumes "r \<in> ideal B" shows "ideal (insert r B) = ideal B"
starP2_eq_iff: shows "*p2* (=) = (=)"
nn_integral_cong_simp: fixes M :: "'c measure"   and u :: "'c \<Rightarrow> ennreal"   and v :: "'c \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M =simp=> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
doesnt_have_mode\<^sub>C: fixes x :: "'Var\<^sub>A"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and m :: "Mode" assumes "x \<notin> mds\<^sub>A_of mds\<^sub>C m" shows "var\<^sub>C_of x \<notin> mds\<^sub>C m"
iprefixE: fixes xs :: "'d list"   and f :: "nat \<Rightarrow> 'd" assumes "xs \<sqsubseteq> f" obtains g :: "nat \<Rightarrow> 'd" where "f = xs \<frown> g"
uint_sub_if': fixes a :: "'a word"   and b :: "'a word" shows "uint (a - b) = (if uint b \<le> uint a then uint a - uint b else uint a - uint b + 2 ^ LENGTH('a))"
abc_only_cba_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "\<not> [[a c b]]"
antichain_inverse: fixes y :: "'b set" assumes "y \<in> {A. finite A \<and> incomparable A}" shows "set_antichain (antichain y) = y"
map_sel: fixes a :: "'d option"   and f :: "'d \<Rightarrow> 'e" assumes "a \<noteq> None" shows "the (map_option f a) = f (the a)"
pre: shows "pre_splitFace g ram1 ram2 oldF newVs"
simps_19: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "char list"   and ve :: "aexp"   and vf :: "'a" shows "({v} WHILE va DO vb {vc} \<le> vd ::= ve {vf}) = False"
simps: fixes V :: "nat set"   and s :: "nat set"   and t :: "'a option list set" shows "projectTable V (s, t) = (s \<inter> V, restrict V ` t)"
simps_69: fixes x6 :: "nat"   and x4 :: "nat" shows "Passwd x6 \<noteq> Key x4"
trivial_group_subgroup_generated_eq: fixes G :: "('a, 'b) monoid_scheme"   and s :: "'a set" assumes "Group.group G" shows "trivial_group (subgroup_generated G s) = (carrier G \<inter> s \<subseteq> {\<one>\<^bsub>G\<^esub>})"
bind_pmf_cong: fixes p :: "'b pmf"   and q :: "'b pmf"   and f :: "'b \<Rightarrow> 'c pmf"   and g :: "'b \<Rightarrow> 'c pmf" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
assms_1: shows "rel_stacktraces (Fubx_get F) st1 st2 (Some f)"
4: shows "\<Union> (A ` {0..<n}) \<in> lambda_system \<Omega> M f"
iteration_unfoldr_distl: fixes y :: "'a"   and x :: "'a" shows "y + y \<cdot> x\<^sup>\<infinity> \<cdot> x = y \<cdot> x\<^sup>\<infinity>"
lowest_tops_cases: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex"   and var :: "'a" assumes "lowest_tops [i, t, e] = Some var" shows "ifex_top_var i = Some var \<or> ifex_top_var t = Some var \<or> ifex_top_var e = Some var"
components_iff: fixes S :: "'b set"   and U :: "'b set" shows "(S \<in> components U) = (\<exists>x. x \<in> U \<and> S = connected_component_set U x)"
simps_4: fixes f :: "atom \<Rightarrow> nat \<Rightarrow> atom"   and \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and n :: "nat" shows "map_fm_binders f (And \<phi> \<psi>) n = And (map_fm_binders f \<phi> n) (map_fm_binders f \<psi> n)"
from_nat_CARD: shows "mod_type_class.from_nat CARD('d) = (0::'d)"
fpairwiseI: fixes M :: "'a fset"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>m n. \<lbrakk>m |\<in>| M; n |\<in>| M\<rbrakk> \<Longrightarrow> P m n" shows "fpairwise P M"
insert_remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set (A - {x}))"
shrK_not_AKcryptSK: fixes evs :: "event list"   and K :: "nat"   and A :: "agent" assumes "evs \<in> kerbIV" shows "\<not> AKcryptSK K (shrK A) evs"
is_borel_def: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" shows "Borel_Space.is_borel f M = (f \<in> borel_measurable M)"
case_1: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and f3 :: "'d \<Rightarrow> 'a"   and x1 :: "'b" shows "(case In1 x1 of In1 x \<Rightarrow> f1 x | In2 x \<Rightarrow> f2 x | In3 x \<Rightarrow> f3 x) = f1 x1"
Rep_preal_inverse: fixes x :: "preal" shows "Abs_preal (Rep_preal x) = x"
prod_list_zero_iff: fixes xs :: "'b list" shows "(prod_list xs = (0::'b)) = ((0::'b) \<in> set xs)"
AB_4: fixes i :: "'a"   and n :: "nat" assumes "i \<in> Basis" shows "2 * (B (Suc n) \<bullet> i - A (Suc n) \<bullet> i) \<le> B n \<bullet> i - A n \<bullet> i"
bij_betw_imp_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "inj_on f A"
rem_UnPath_parity_v: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "finite E"   and "is_trail v ps v'" shows "(v \<noteq> v') = (odd (degree v (rem_unPath ps G)) = even (degree v G))"
exhaust_sel: fixes rest_scheme :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme" obtains "rest_scheme = Rest (rinit rest_scheme) (rfunc_adv rest_scheme) (rfunc_usr rest_scheme)"
inverse_aform_def: fixes p :: "nat"   and a :: "real \<times> real pdevs" shows "inverse_aform p a = (let l = Inf_aform' p a; u = Sup_aform' p a in if l \<le> 0 \<and> 0 \<le> u then None else if l \<le> 0 then Some (apfst uminus_aform (inverse_aform' p (uminus_aform a))) else Some (inverse_aform' p a))"
PTrans: shows "P \<longmapsto> a<x> \<prec> P'"
leg_in_hom_1: shows "\<guillemotleft>\<mu>\<nu>\<pi>.cod.leg0 : \<mu>\<nu>\<pi>.cod.apex \<rightarrow>\<^sub>C \<mu>\<nu>\<pi>.cod.src\<guillemotright>"
hv_eq_zero: fixes v :: "(real, 3) vec" shows "(Hv v = 0) = (v = 0)"
g_source_in: fixes x :: "'v" shows "g (x, source \<Delta>) = 0"
elims_3: fixes x :: "('c, 'd, 'e, 'f) edge_kind"   and xa :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "\<not> local.CFG.pred x xa" obtains Q :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = (Q__)\<^sub>\<surd>" and "xa = cf__ # cfs__" and "\<not> Q__ (fst cf__)"   | Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool" and r :: "'e" and p :: "'f" and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = Qa__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfa__ # cfsa__" and "\<not> Qa__ (fst cfa__, r)"   | Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool" and p :: "'f" and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "\<not> (Q cf \<and> cfs \<noteq> [])"   | et :: "('c, 'd, 'e, 'f) edge_kind" where "x = et" and "xa = []"
unlabel_mem_has_label: fixes x :: "'e"   and A :: "('f \<times> 'e) list" assumes "x \<in> set (unlabel A)" shows "\<exists>l. (l, x) \<in> set A"
Par1B: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "x \<sharp> P"   and "x \<sharp> Q"   and "x \<sharp> a" shows "P \<parallel> Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<parallel> Q"
assms_5: shows "is_Groebner_basis (fst ` set gs)"
\<rho>_def: shows "\<rho> \<equiv> run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w"
inv_partition_preserv: fixes E :: "'b set set"   and S :: "'b set"   and X :: "'b set"   and P :: "'b set set"   and x :: "'b" assumes "Graph_E E"   and "Graph_E.inv_partition E S X P"   and "x \<in> Graph_E.vertices E - X" shows "Graph_E.inv_partition E (insert x S) (X \<union> Graph_E.neighbors E x \<union> {x}) (insert ({x} \<union> Graph_E.neighbors E x) P)"
hyps: fixes qs :: "'b list"   and tab :: "'a \<Rightarrow> 'b option" assumes "length ps = length qs" shows "tab(ps [\<mapsto>] qs, xs [\<mapsto>] ys) = tab(ps @ xs [\<mapsto>] qs @ ys)"
lexext_compat_list: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and xs :: "'b list"   and xs' :: "'b list" assumes "gt y x" shows "lexext gt (xs @ y # xs') (xs @ x # xs')"
strict_is_neg_transitive: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier \<and> y \<in> carrier \<and> z \<in> carrier"   and "x \<succeq> y \<and> (y, x) \<notin> relation" shows "x \<succeq> z \<and> (z, x) \<notin> relation \<or> z \<succeq> y \<and> (y, z) \<notin> relation"
BuildSeq2_induct: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and a :: "hf"   and a' :: "hf"   and P :: "hf \<Rightarrow> hf \<Rightarrow> bool" assumes "BuildSeq2 B C s k a a'"   and "\<And>x x'. B x x' \<Longrightarrow> P x x'"   and "\<And>x x' y y' z z'. \<lbrakk>C x x' y y' z z'; P y y'; P z z'\<rbrakk> \<Longrightarrow> P x x'" shows "P a a'"
swapEnv_substEnv: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good Y"   and "goodEnv rho" shows "rho &[Y / y]_ys &[z1 \<and> z2]_zs = rho &[z1 \<and> z2]_zs &[Y #[z1 \<and> z2]_zs / y @ys[z1 \<and> z2]_zs]_ys"
wlp_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "local.wlp c Q s \<equiv> \<forall>t. \<pi>: (c, s) \<Rightarrow> t \<longrightarrow> Q t"
bv_Cons: fixes x :: "bool"   and xs :: "bool list"   and y :: "bool"   and ys :: "bool list" shows "x # xs \<preceq>\<^sub>b y # ys = ((x \<longrightarrow> y) \<and> xs \<preceq>\<^sub>b ys)"
cong_dvd_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "[b = c] (mod a)" shows "(a dvd b) = (a dvd c)"
simps: fixes a1 :: "char list \<Rightarrow> ty option"   and a2 :: "expr list"   and a3 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a4 :: "expr list"   and a5 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,a1 \<turnstile> \<langle>a2,a3\<rangle> [\<rightarrow>] \<langle>a4,a5\<rangle> = ((\<exists>E e s e' s' es. a1 = E \<and> a2 = e # es \<and> a3 = s \<and> a4 = e' # es \<and> a5 = s' \<and> P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>) \<or> (\<exists>E es s es' s' v. a1 = E \<and> a2 = Val v # es \<and> a3 = s \<and> a4 = Val v # es' \<and> a5 = s' \<and> P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>))"
case_prod_comp: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'f"   and x :: "'d \<times> 'e" shows "(case x of (a, b) \<Rightarrow> (f \<circ> g) a b) = f (g (fst x)) (snd x)"
d_a_galois2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<cdot> d y \<le> d z) = (d x \<le> d z + ad y)"
eval_imp: fixes tt :: "'b set"   and p :: "'b pl"   and q :: "'b pl" shows "tt[[p -> q]] = (tt[[p]] \<longrightarrow> tt[[q]])"
Equivalence_def: shows "{(x, y). (x, y) \<in> (R \<times> S) \<times> R \<times> S \<and> x \<sim> y} = {(a, b). \<exists>A\<in>rel.Partition. a \<in> A \<and> b \<in> A}"
in_theFTOne_theNFTBoth_2: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and n1 :: "nat"   and cl :: "('c, 'd, 'e) cmd list"   and dl :: "('f, 'g, 'h) cmd list"   and n2 :: "nat" assumes "PL_Indis cval indis"   and "n1 \<in> theFTOne cl dl"   and "n2 \<in> theNFTBoth cl dl" shows "n2 \<noteq> n1"
bij_betw_the_inv_into_onto: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "the_inv_into A f ` B = A"
unit_eq_div2: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "is_unit b" shows "(a = c div b) = (a * b = c)"
f_def: shows "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
simps_18: fixes x21 :: "('b, 'a) tm"   and x22 :: "('b, 'a) tm" shows "vars (App x21 x22) = vars x21 \<union> vars x22"
continuous_on_vector_derivative: fixes S :: "real set"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)" shows "continuous_on S f"
removeNones_def: fixes vOptionList :: "'a option list" shows "removeNones vOptionList \<equiv> concat (map (\<lambda>x. if x \<noteq> None then [the x] else []) vOptionList)"
red_lcl_add: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and l\<^sub>0 :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>" shows "P,E \<turnstile> \<langle>e,(h, l\<^sub>0 ++ l)\<rangle> \<rightarrow> \<langle>e',(h', l\<^sub>0 ++ l')\<rangle>"
induct: fixes P :: "pol \<Rightarrow> (mon \<times> pol) list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "(mon \<times> pol) list"   and a2 :: "nat" assumes "\<And>P1 n. P P1 [] n"   and "\<And>P1 M P2 mps n. P (pnsubst1 P1 M P2 n) mps n \<Longrightarrow> P P1 ((M, P2) # mps) n" shows "P a0 a1 a2"
rel_simps_27: fixes n :: "num" shows "(0::'b) \<le> numeral n"
ortho_antimono: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "- y \<le> - x"
abss_S: fixes l :: "'s"   and u :: "'c \<Rightarrow> real" assumes "(l, u) \<in> S" shows "abss (l, u) = (l, [u]\<^sub>\<R>)"
Diff_insert: fixes A :: "'k set"   and a :: "'k"   and B :: "'k set" shows "A - insert a B = A - B - {a}"
S5Basic_6: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<box>\<phi> in v]"
set_union_code: fixes xs :: "'d list"   and ys :: "'d list" shows "set xs \<union> set ys = set (xs @ ys)"
prems: shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t (s # S)"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "C.ide (trg\<^sub>C a)"
q_Ex_def: fixes x :: "hf" shows "q_Ex x \<equiv> \<langle>htuple 5, x\<rangle>"
list_split: fixes n :: "nat"   and bss :: "'a list" assumes "n \<le> length bss" shows "\<exists>b bs. bss = b @ bs \<and> length b = n"
intros_1: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "chain R (LCons x LNil)"
degree_mult_eq: fixes p :: "'d poly"   and q :: "'d poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "degree (p * q) = degree p + degree q"
assms_5: shows "rk {A', B', P, Q} = 4"
sym: fixes \<tau> :: "'\<AA> state \<times> '\<AA> state"   and x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" assumes "\<tau> \<Turnstile> f x y" shows "\<tau> \<Turnstile> f y x"
lt__nle: 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 D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Lt Bet Cong A B C D" shows "\<not> Tarski_neutral_dimensionless.Le Bet Cong C D A B"
bij_betw_extend: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set"   and V :: "'a set" assumes "bij_betw f A B"   and "A \<subseteq> V"   and "B \<subseteq> V"   and "finite A" shows "\<exists>g. finite {x. g x \<noteq> x} \<and> (\<forall>x\<in>UNIV - (A \<union> B). g x = x) \<and> (\<forall>x\<in>A. g x = f x) \<and> bij_betw g V V"
sup_absorb1: fixes y :: "'b multiset"   and x :: "'b multiset" assumes "y \<subseteq># x" shows "x \<union># y = x"
negligible_on_intervals: fixes s :: "'a set" shows "negligible s = (\<forall>a b. negligible (s \<inter> cbox a b))"
bin_ops_same_1: fixes x :: "int" shows "x AND x = x"
Plus_eq_empty_conv: fixes A :: "'d set"   and B :: "'e set" shows "(A <+> B = {}) = (A = {} \<and> B = {})"
senior_states_finite: fixes q :: "'b"   and n :: "nat" shows "finite (senior_states q n)"
trees_root: fixes t :: "'a" assumes "t \<in> trees" shows "root t \<in> F"
sublens_refl: fixes X :: "'e \<Longrightarrow> 'f" shows "X \<subseteq>\<^sub>L X"
prod_emb_PiE: fixes J :: "'c set"   and I :: "'c set"   and E :: "'c \<Rightarrow> 'd set"   and M :: "'c \<Rightarrow> 'd measure" assumes "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" shows "prod_emb I M J (Pi\<^sub>E J E) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))"
prj_in_hom_3: shows "\<guillemotleft>\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0 : HH\<mu>H\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<rho>.apex\<guillemotright>"
lex_pp_lin: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "lex_pp s t \<or> lex_pp t s"
octo_of_real_sel_3: fixes x :: "real" shows "Im2 (octo_of_real x) = 0"
merge_finite: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j"   and "finite (local.merge j)" shows "finite (local.merge i)"
has_derivative_bounded_linear: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and F :: "'c filter" assumes "(f has_derivative f') F" shows "bounded_linear f'"
float_val_le_largest: fixes a :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "1 < LENGTH('a)" shows "valof a \<le> largest TYPE(('a, 'b) IEEE.float)"
nth_tl: fixes ys :: "'b list"   and a :: "'b" assumes "ys ! 0 = a"   and "ys \<noteq> []" shows "ys = a # tl ys"
HaddP_Mem_I: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and z :: "tm" assumes "H \<turnstile> HaddP x y z"   and "H \<turnstile> OrdP x" shows "H \<turnstile> x IN SUCC z"
simps_1: fixes l :: "'b set list" shows "FWNormalisationCore.sort [] l = []"
lQuots_append: fixes v :: "'c list"   and w :: "'c list"   and A :: "'c list set" shows "(v \<in> lQuots w A) = (w @ v \<in> A)"
Setcompr_eq_image: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "{uu_. \<exists>x. uu_ = f x \<and> x \<in> A} = f ` A"
GF_advice_prop_entailment_2: fixes \<A> :: "'b ltln set"   and X :: "'b ltln set"   and \<phi> :: "'b ltln" assumes "false\<^sub>n \<notin> \<A>"   and "{\<psi>. \<psi>[X]\<^sub>\<nu> \<in> \<A>} \<Turnstile>\<^sub>P \<phi>" shows "\<A> \<Turnstile>\<^sub>P \<phi>[X]\<^sub>\<nu>"
equiv_Sup_Liminf: fixes C :: "'a"   and Ns :: "'a set llist" assumes "C \<in> Sup_llist Ns"   and "C \<notin> Liminf_llist Ns" shows "\<exists>i\<in>{i. enat (Suc i) < llength Ns}. C \<in> lnth Ns i - lnth Ns (Suc i)"
relative_polar_form_eq: fixes Q :: "(real, 2) vec"   and P :: "(real, 2) vec" shows "Q + dist P Q *\<^sub>R of_radiant (radiant_of (P - Q)) = P"
pcr_fmap_def: fixes T :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Ta :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "pcr_fmap T Ta \<equiv> (T ===> rel_option Ta) OO cr_fmap"
sqrt_float_interval: fixes X :: "float interval"   and prec :: "nat" shows "sqrt ` set_of (real_interval X) \<subseteq> set_of (real_interval (sqrt_float_interval prec X))"
singleton_sup_aux: fixes default :: "unit \<Rightarrow> 'a"   and A :: "'a pred"   and B :: "'a pred" shows "singleton default (A \<squnion> B) = (if A = \<bottom> then singleton default B else if B = \<bottom> then singleton default A else singleton default (single (singleton default A) \<squnion> single (singleton default B)))"
assms: shows "(f, g) \<in> Id \<rightarrow> Id \<rightarrow> \<langle>Id\<rangle>nres_rel"
finite_nlists: fixes A :: "'b set"   and n :: "nat" assumes "finite A" shows "finite (nlists A n)"
simps_2: fixes v :: "String.literal"   and va :: "typ" shows "is_variable (Ct v va) = False"
orthogonal_basis_subspace: fixes S :: "'a set" assumes "subspace S" obtains B :: "'a set" where "(0::'a) \<notin> B" and "B \<subseteq> S" and "pairwise orthogonal B" and "independent B" and "card B = dim S" and "span B = S"