hcju/isabelleps huggingface dataset:
Inf_pres_radj: fixes g :: "'c \<Rightarrow> 'd" assumes "Inf_pres g" shows "\<exists>f. f \<stileturn> g"
A: fixes \<Gamma> :: "'b formula list" shows "\<bottom> # \<Gamma> \<Rightarrow> []"
snd_set_zip: fixes xs :: "'a list"   and ys :: "'b list" assumes "length xs = length ys" shows "snd ` set (zip xs ys) = set ys"
connect_cong_trace: fixes A :: "'y set"   and res1 :: "('y, 'z) resource"   and res2 :: "('y, 'z) resource"   and \<I> :: "('y, 'z) \<I>"   and d :: "('y, 'z) distinguisher" assumes "A \<turnstile>\<^sub>R res1 \<approx> res2"   and "\<I> \<turnstile>g d \<surd>"   and "outs_gpv \<I> d \<subseteq> A"   and "\<I> \<turnstile>res res1 \<surd>"   and "\<I> \<turnstile>res res2 \<surd>" shows "connect d res1 = connect d res2"
real_arch_pow_inv: fixes y :: "real"   and x :: "real" assumes "0 < y"   and "x < 1" shows "\<exists>n. x ^ n < y"
Says_clientK_unique: fixes A' :: "agent"   and B' :: "agent"   and Na :: "nat"   and Nb :: "nat"   and PMS :: "nat"   and NA :: "nat"   and NB :: "nat"   and Y :: "msg"   and evs :: "event list"   and A :: "agent"   and B :: "agent" assumes "Says A' B' (Crypt (clientK (Na, Nb, PRF (PMS, NA, NB))) Y) \<in> set evs"   and "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "evs \<in> tls"   and "A' \<noteq> Spy" shows "A = A'"
pullback_commutes': fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.cospan f g" shows "f \<odot> Maps.PRJ\<^sub>1 f g = g \<odot> Maps.PRJ\<^sub>0 f g"
init: shows "\<turnstile> initPsi \<longrightarrow> I"
sup_ge2: fixes y :: "'g multiset"   and x :: "'g multiset" shows "y \<subseteq># x \<union># y"
psi_bounds_ln_fact_1: fixes n :: "nat" shows "ln (fact n) - 2 * ln (fact (n div 2)) \<le> psi n"
simps_2: fixes h :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "blind_blindable h (Unblinded x) = Blinded (Content (h x))"
prems_1: shows "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"
normalizeQ_def: fixes p :: "((int \<times> int) set set, int) Combinators" shows "normalizeQ p = removeAllDuplicates (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) (Nets_List p))))"
slist_eq_FF_snil_1: fixes xs :: "[:'a:]" shows "(eq\<cdot>xs\<cdot>[::] = FF) = (\<exists>y ys. y \<noteq> \<bottom> \<and> ys \<noteq> \<bottom> \<and> xs = y :# ys)"
assms: fixes m :: "nat" shows "g m = (\<Sum>n. f (m, n))"
lstrict_prefix_llength_less: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lstrict_prefix xs ys" shows "llength xs < llength ys"
in_arcs_def: fixes G :: "('c, 'b) pre_digraph"   and v :: "'c" shows "in_arcs G v \<equiv> {e \<in> arcs G. head G e = v}"
transE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "Preliminaries.var" shows "transE \<xi> x = proj (tpOfV x) (\<xi> x)"
absorb1: fixes b :: "'b word"   and a :: "'b word" assumes "b \<le>s a" shows "signed.max a b = a"
wellformed: fixes b :: "'a set" assumes "b \<in># \<B> + mset_set \<G>" shows "b \<subseteq> \<V>"
Alist_impl_of: fixes xs :: "('a, 'b) alist" shows "Alist (impl_of xs) = xs"
sheaf_spec_is_sheaf: shows "sheaf_of_rings cring0.spectrum cring0.is_zariski_open cring0.sheaf_spec cring0.sheaf_spec_morphisms cring0.\<O>b cring0.add_sheaf_spec cring0.mult_sheaf_spec cring0.zero_sheaf_spec cring0.one_sheaf_spec"
to_from_Gromov_completion: fixes x :: "'a Gromov_completion" assumes "x \<notin> Gromov_boundary" shows "to_Gromov_completion (from_Gromov_completion x) = x"
distinct_3: fixes x1 :: "'b" shows "Fin x1 \<noteq> -\<infinity>"
cases: fixes a :: "fmla" assumes "a \<in> boolean_axioms" obtains A :: "fmla" where "a = Ab__ IMP Ab__"   | A :: "fmla" and B :: "fmla" where "a = Ac__ IMP Ac__ OR Ba__"   | A :: "fmla" where "a = Ad__ OR Ad__ IMP Ad__"   | A :: "fmla" and B :: "fmla" and C :: "fmla" where "a = Ae__ OR Bb__ OR C__ IMP (Ae__ OR Bb__) OR C__"   | C :: "fmla" and A :: "fmla" and B :: "fmla" where "a = C OR A IMP (C IMP B) IMP A OR B"
simps_1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "binrelchain P [] = True"
word_add_le_dest: fixes i :: "'b word"   and k :: "'b word"   and j :: "'b word" assumes "i + k \<le> j + k"   and "unat i + unat k < 2 ^ LENGTH('b)"   and "unat j + unat k < 2 ^ LENGTH('b)" shows "i \<le> j"
FinalAllow_approximating_in_doubt_allow: fixes \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and rs :: "'a rule list"   and \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and p :: "'b" assumes "matcher_agree_on_exact_matches (\<gamma>::'a \<Rightarrow> 'b \<Rightarrow> bool) (\<beta>::'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue)"   and "good_ruleset (rs::'a rule list)"   and "(\<Gamma>::char list \<Rightarrow> 'a rule list option),(\<gamma>::'a \<Rightarrow> 'b \<Rightarrow> bool),(p::'b)\<turnstile> \<langle>(rs::'a rule list), Undecided\<rangle> \<Rightarrow> Decision FinalAllow" shows "(\<beta>::'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue, in_doubt_allow),(p::'b)\<turnstile> \<langle>(rs::'a rule list), Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow"
lass_val_of_nsub_RI: fixes e :: "'a exp"   and a :: "'a \<times> val" assumes "lass_val_of e = \<lfloor>a\<rfloor>" shows "\<not> sub_RI e"
FEval_finst: fixes mo :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and A :: "form"   and u :: "nat" shows "FEval mo e (finst A u) = FEval mo (case_nat (e u) e) A"
PiE_restrict: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "f \<in> Pi\<^sub>E A B" shows "restrict f A = f"
less_iff_sym_greater: fixes cmp :: "'b comparator"   and b :: "'b"   and a :: "'b" shows "(compare cmp b a = Less) = (compare cmp a b = Greater)"
sep_add_disjD: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj (plus x y) z"   and "sep_disj x y" shows "sep_disj x z \<and> sep_disj y z"
list_all2_map_map: fixes \<phi> :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'e"   and Tl :: "'g list"   and g :: "'g \<Rightarrow> 'f" shows "list_all2 \<phi> (map f Tl) (map g Tl) = list_all (\<lambda>T. \<phi> (f T) (g T)) Tl"
hyp: shows "lincomb f (set ws) = lincomb_list (\<lambda>i. if \<exists>j<i. ws ! i = ws ! j then 0::'a else f (ws ! i)) ws"
set_sset_stake: fixes n :: "nat"   and xs :: "'c stream" shows "set (stake n xs) \<subseteq> sset xs"
p_l_r_def_1: shows "p \<equiv> poly_real_alg_1 y"
COMP_def: 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 g :: "('a, 'b) concrete_category.arr"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp" shows "concrete_category.COMP Obj Hom Comp g f \<equiv> if (f \<noteq> concrete_category.Null \<and> concrete_category.Dom f \<in> Obj \<and> concrete_category.Cod f \<in> Obj \<and> concrete_category.Map f \<in> Hom (concrete_category.Dom f) (concrete_category.Cod f)) \<and> (g \<noteq> concrete_category.Null \<and> concrete_category.Dom g \<in> Obj \<and> concrete_category.Cod g \<in> Obj \<and> concrete_category.Map g \<in> Hom (concrete_category.Dom g) (concrete_category.Cod g)) \<and> concrete_category.Dom g = concrete_category.Cod f then concrete_category.MkArr (concrete_category.Dom f) (concrete_category.Cod g) (Comp (concrete_category.Cod g) (concrete_category.Dom g) (concrete_category.Dom f) (concrete_category.Map g) (concrete_category.Map f)) else concrete_category.Null"
snoc_3: shows "snd (cs c cid) \<noteq> NotStarted"
reachable_Pr: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and x :: "nat"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list"   and y :: "nat" assumes "valid ((Pr n f g, x # xs, []) # rest)"   and "\<And>xs rest. valid ((f, xs, []) # rest) \<Longrightarrow> correct ((f, xs, []) # rest, None)"   and "\<And>xs rest. valid ((g, xs, []) # rest) \<Longrightarrow> correct ((g, xs, []) # rest, None)"   and "y \<le> x"   and "eval (Pr n f g) (y # xs) \<down>" shows "reachable ((Pr n f g, x # xs, []) # rest, None) ((Pr n f g, x # xs, trace n f g xs y) # rest, None)"
fls_div_zero: fixes g :: "'a fls" shows "0 / g = 0"
strong_inf_map: fixes \<iota> :: "'f inference" assumes "\<iota> \<in> Inf_F"   and "\<G>_I \<iota> \<noteq> None" shows "concl_of ` the (\<G>_I \<iota>) \<subseteq> \<G>_F (concl_of \<iota>)"
rsp_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "rsp_on A f = {(x, y). \<exists>B\<subseteq>A. x \<in> Rf f B \<and> y \<in> f B}"
sets_PiM_finite: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {Pi\<^sub>E I X |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
execute: fixes p :: "'c"   and A :: "('d, 'c) nba"   and a :: "'d \<times> 'c" assumes "p \<in> nodes A"   and "fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p" shows "snd a \<in> nodes A"
step_preserves_defass: fixes 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 e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "\<D> e \<lfloor>dom (lcl s)\<rfloor>" shows "\<D> e' \<lfloor>dom (lcl s')\<rfloor>"
p: assumes "n \<noteq> 0" shows "eval p [f \<triangleright> n] \<down>= pdec1 (the (parallel (the (f 0)) (the (f 1)) (the (inconsist (amalg01 f) (f \<triangleright> n)))))"
subst_psubst_compose: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "goodEnv rho" shows "X #[rho] #[Y / y]_ys = X #[rho &[Y / y]_ys]"
analz_Un_implAuthSet_aux_3: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implAuthSet_aux G \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
simps_1: fixes A :: "'c mat" shows "A ^\<^sub>m 0 = 1\<^sub>m (dim_row A)"
fg_next_config_def: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and p :: "'f"   and HO :: "'f set"   and SHO :: "'f set"   and crd :: "'f"   and config :: "('g, 'f, 'h, 'j) config_scheme"   and config' :: "('g, 'f, 'h, 'j) config_scheme" shows "fg_next_config A p HO SHO crd config config' \<equiv> (\<exists>q. fg_send_msg A p q config config') \<or> (\<exists>q. fg_rcv_msg p q HO SHO config config') \<or> fg_local A p HO crd config config'"
list_set_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>list_set_rel \<equiv> \<langle>R\<rangle>list_rel O br set distinct"
simps_2: fixes t :: "(val \<times> val) list" shows "T (VFun t) = TFun (Tf t)"
assms: shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A @ B) Sec"
RInit: fixes P :: "(char list list \<times> char list exp) prog"   and e' :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh'' :: "sheap"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e',s\<rangle> \<Rightarrow> \<langle>Val v,(h', l', sh')\<rangle>"   and "sh' C = \<lfloor>(sfs, i)\<rfloor>"   and "sh'' = sh'(C \<mapsto> (sfs, Done))"   and "C' = last (C # Cs)"   and "P \<turnstile> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h', l', sh'')\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>RI (C,e') ; Cs \<leftarrow> e,s\<rangle> \<Rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>"
scnj_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and F :: "'c set" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls" shows "Syntax_with_Connectives_False.scnj cnj imp fls F = Syntax_with_Connectives_False.lcnj cnj imp fls (asList F)"
Q'Chain: shows "\<Psi> \<otimes> \<Psi>' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q''"
comp_left_increasing_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<le> (x \<squnion> z) * y"
subset_snd_imageI: fixes A :: "'b set"   and B :: "'c set"   and S :: "('b \<times> 'c) set"   and x :: "'b" assumes "A \<times> B \<subseteq> S"   and "x \<in> A" shows "B \<subseteq> snd ` S"
eval_monom: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and r :: "'c"   and s :: "'e"   and n :: "nat" assumes "UP_pre_univ_prop R S h"   and "r \<in> carrier R"   and "s \<in> carrier S" shows "eval R S h s (monom (UP R) r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"
fmpredI: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x y" shows "fmpred P m"
principal_prod_principal: fixes A :: "'a set"   and B :: "'b set" shows "principal A \<times>\<^sub>F principal B = principal (A \<times> B)"
dom_def: fixes m :: "'a \<Rightarrow> 'b option" shows "dom m = {a. m a \<noteq> None}"
output_def: fixes resumption :: "('g, 'f, 'h) Resumption.resumption" shows "Resumption.resumption.output resumption = (case resumption of Resumption.resumption.Pause x21 x22 \<Rightarrow> x21)"
uminus_add_minus_vec: fixes l :: "'b vec"   and n :: "nat"   and r :: "'b vec" assumes "l \<in> carrier_vec n"   and "r \<in> carrier_vec n" shows "- (l + r) = - l - r"
case_cong_weak: fixes option :: "'a option"   and option' :: "'a option"   and f1 :: "'b"   and f2 :: "'a \<Rightarrow> 'b" assumes "option = option'" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option' of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x)"
counterex_reducing_inference_system_axioms: shows "Standard_Redundancy_Criterion.counterex_reducing_inference_system {{#}} (\<TTurnstile>e) Inf I_of"
hunion_iff: fixes x :: "hf"   and a :: "hf"   and b :: "hf" shows "(x \<^bold>\<in> a \<squnion> b) = (x \<^bold>\<in> a \<or> x \<^bold>\<in> b)"
eval_simps'_2: fixes f :: "'a E.term" assumes "E.Arr f" shows "src\<^sub>B (EVAL f) = EVAL (E.Src f)"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_rec: fixes n :: "nat" shows "FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n = (\<lambda>fs. Mprefix (Tr\<^sub>F n fs) (\<lambda>e. FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (Up\<^sub>F fs e)))"
finite_sub_card_eq_mult_imp_comp: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "finite H"   and "finite J"   and "card (H <#> J) = card J * card H" shows "complementary H J"
exhaust: fixes y :: "'a blue_witness" obtains "y = NO_CYC"   | x21 :: "'a" and x22 :: "'a list" where "y = REACH x21 x22"   | x31 :: "'a list" and x32 :: "'a list" where "y = CIRC x31 x32"
add_less_cancel_right: fixes a :: "'d multiset"   and c :: "'d multiset"   and b :: "'d multiset" shows "(a + c \<subset># b + c) = (a \<subset># b)"
trapped_backward_def: fixes x :: "'a"   and K :: "'a set" shows "trapped_backward x K = (flow0 x ` (existence_ivl0 x \<inter> {..0}) \<subseteq> K)"
subset_imageE: fixes B :: "'g set"   and f :: "'h \<Rightarrow> 'g"   and A :: "'h set" assumes "B \<subseteq> f ` A" obtains C :: "'h set" where "C \<subseteq> A" and "B = f ` C"
pp_gets_def: fixes v :: "char list"   and e :: "(char list \<Rightarrow> 'b) \<Rightarrow> 'b" shows "v ::= e = {ppath.Cons s (Node (s(v := e s))) |s. True}"
weak_consis': shows "weak_sharing_consistent (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
simply_connected_eq_contractible_loop_any: fixes S :: "'a set" shows "simply_connected S = (\<forall>p a. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<and> a \<in> S \<longrightarrow> homotopic_loops S p (linepath a a))"
vec_of_def: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "vec_of sp = (if spp_inv sp then Eps (spp_rel sp) else 0)"
WCFG_Entry: fixes prog :: "cmd" shows "prog \<turnstile> (_Entry_) -(\<lambda>s. True)\<^sub>\<surd>\<rightarrow> (_ 0 _)"
negated_euclidean_axiom: shows "\<exists>a b c d t. a \<in> unit_disc \<and> b \<in> unit_disc \<and> c \<in> unit_disc \<and> d \<in> unit_disc \<and> t \<in> unit_disc \<and> poincare_between a d t \<and> poincare_between b d c \<and> a \<noteq> d \<and> (\<forall>x y. x \<in> unit_disc \<and> y \<in> unit_disc \<and> poincare_between a b x \<and> poincare_between x t y \<longrightarrow> \<not> poincare_between a c y)"
len64: fixes x :: "64 itself" shows "len_of x = 64"
\<Z>_Vset_\<omega>2_vsubset_Vset: shows "Vset (\<omega> + \<omega>) \<subseteq>\<^sub>\<circ> Vset \<alpha>"
crsp_step_inc: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "abc_fetch as ap = Some (Inc n)" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some (Inc n))) (steps (s, l, r) (ci ly (start_of ly as) (Inc n), start_of ly as - Suc 0) stp) ires"
finprod_empty: fixes f :: "'d \<Rightarrow> 'a" shows "finprod G f {} = \<one>"
hyps: assumes "distinct (map fst xs)" shows "set (sort_oalist xs) = {kv \<in> set xs. snd kv \<noteq> (0::'b)}"
prod_cases: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and p :: "'b \<times> 'c" assumes "\<And>a b. P (a, b)" shows "P p"
bounded_norm_comp: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "bounded ((\<lambda>x. \<parallel>f x\<parallel>) ` S) = bounded (f ` S)"
monotone_convergence_increasing: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and S :: "'c set"   and g :: "'c \<Rightarrow> real" assumes "\<And>k. f k integrable_on S"   and "\<And>k x. x \<in> S \<Longrightarrow> f k x \<le> f (Suc k) x"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"   and "bounded (range (\<lambda>k. integral S (f k)))" shows "g integrable_on S \<and> (\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
eucl_induct: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "\<And>b. P b (0::'a)"   and "\<And>a b. \<lbrakk>b \<noteq> (0::'a); P b (a mod b)\<rbrakk> \<Longrightarrow> P a b" shows "P a b"
lt_set_card_lt: fixes A :: "nat set"   and x :: "nat" assumes "finite A"   and "x \<in> A" shows "card {y \<in> A. y < x} < card A"
next_action_after_cswitch: shows "\<forall>s n d aseqs. fst (control (cswitch n s) d aseqs) = fst (control s d aseqs)"
infs_snth: fixes A :: "'b set"   and w :: "'b stream" shows "infs A w = (\<forall>n. \<exists>k\<ge>n. w !! k \<in> A)"
rb_spp_aux_1: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and rword_strict :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and dgrad :: "'a \<Rightarrow> nat"   and z :: "nat" assumes "is_strict_rewrite_ord fs rword_strict"   and "dickson_grading dgrad"   and "distinct fs"   and "0 \<notin> set fs"   and "hom_grading dgrad" shows "punit.is_Groebner_basis (set (map snd (fst (fst (rb_spp_aux fs rword_strict (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z))))))"
inv_end_nonempty_7: fixes x :: "nat"   and list :: "cell list" shows "inv_end5 x ([], Oc # list) = False"
Some_in_opt: fixes x :: "'b"   and A :: "'b set" shows "(\<lfloor>x\<rfloor> \<in> opt A) = (x \<in> A)"
translation_apply: fixes a :: "'a"   and b :: "'a" assumes "a \<in> G"   and "b \<in> G" shows "(a)\<^sub>R b = b \<cdot> a"
closure_closed_segment: fixes a :: "'b"   and b :: "'b" shows "closure (closed_segment a b) = closed_segment a b"
lConsts_def: shows "lConsts = {(f, 0) |f. least f}"
assms_4: shows "(\<xi>, p) \<in> reachable A I"
DDH_advantage_bound: fixes n :: "nat"   and \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> bool spmf" shows "local.ddh.DDH_advantage n \<A> \<le> local.ddh.advantage n \<A> + local.ddh.advantage n (local.ddh.DDH_\<A>' n \<A>)"
invrules_4: shows "B\<^sub>2 \<notin> P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2"
le_cases: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" obtains "le_of_key_order ko x y"   | "le_of_key_order ko y x"
row_length_mat_to_cols_list: fixes A :: "complex Matrix.mat" assumes "0 < dim_col A" shows "mult.row_length (mat_to_cols_list A) = dim_row A"
Sum_any_left_distrib: fixes g :: "'c \<Rightarrow> 'd"   and r :: "'d" assumes "finite {a. g a \<noteq> (0::'d)}" shows "Sum_any g * r = (\<Sum>n. g n * r)"
elims_3: fixes x :: "('b \<times> nat) tree" assumes "\<not> sngl x" obtains "x = \<langle>\<rangle>"   | uw_ :: "('b \<times> nat) tree" and ux_ :: "'b" and lva :: "nat" and uy_ :: "('b \<times> nat) tree" and uz_ :: "'b" and lvb :: "nat" and va_ :: "('b \<times> nat) tree" where "x = \<langle>uw_, (ux_, lva), \<langle>uy_, (uz_, lvb), va_\<rangle>\<rangle>" and "\<not> lvb < lva"
MapsToMorDomCod_2: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "dom\<^bsub>C\<^esub> (f ;;\<^bsub>C\<^esub> g) = dom\<^bsub>C\<^esub> f"
gen_subgroup_is_subgroup: fixes gens :: "'a set" assumes "gens \<subseteq> carrier G" shows "\<langle>gens\<rangle> \<le> G"
x_in_dom': shows "x \<in> dom (g (to_partial mem |` X))"
utp_test_def: fixes b :: "'a hrel" shows "utp_test b = (\<I>\<I> \<sqsubseteq> b)"
union_set_fold: fixes xs :: "'d list"   and A :: "'d set" shows "set xs \<union> A = fold insert xs A"
f': fixes x :: "'a \<times> 'b" assumes "x \<in> S" shows "(f has_derivative blinfun_apply (f' x)) (at x)"
fps_deriv_lr_inverse_comm_2: fixes x :: "'a"   and f :: "'a fps" assumes "x * fps_nth f 0 = (1::'a)" shows "fps_deriv (fps_right_inverse f x) = - fps_deriv f * (fps_right_inverse f x)\<^sup>2"
simps: fixes n :: "nat"   and l :: "nat" shows "ins (n, l) = (n + 1, if n < l then l else if l = 0 then 1 else nat \<lceil>c * real l\<rceil>)"
mult_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero b"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"