hcju/isabelleps huggingface dataset:
simps_1: fixes x :: "'b"   and y :: "'b"   and p :: "'c"   and z :: "('b, 'c) Combinators list" shows "removeShadowRules2 (AllowPortFromTo x y p # z) = (if DenyAllFromTo x y \<in> set z then removeShadowRules2 z else AllowPortFromTo x y p # removeShadowRules2 z)"
assms_3: fixes a :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and t :: "'a" assumes "a \<in> pmdl B"   and "P a"   and "p \<in> B"   and "c \<noteq> (0::'b)" shows "P (a + monom_mult c t p)"
f_expand_0: fixes xs :: "'d message_af list" shows "xs \<odot> 0 = []"
adh_lemma: fixes x :: "'a llist"   and P :: "'a llist set" assumes "x \<in> A\<^sup>\<infinity>"   and "P \<subseteq> A\<^sup>\<infinity>" shows "(x adh P) = (\<forall>r\<in>finpref A x. \<exists>s\<in>A\<^sup>\<infinity>. r @@ s \<in> P)"
FunctorId2: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme"   and X :: "'g" assumes "PreFunctor F"   and "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "F @@ X \<in> obj\<^bsub>CatCod F\<^esub> \<and> F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
assms_6: shows "(aaa, aaaa, baa) \<in> set eq"
NOTAF_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "??.Almost_Full.NOTAF A P = (SOME x. x \<in> A \<and> \<not> af A (\<lambda>y z. P y z \<or> P x y))"
axioms_1: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_bij A n f" shows "v11 f"
norm_triangle_le: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y \<le> e" shows "norm (plus x y) \<le> e"
fmrel_on_fsetI: fixes S :: "'e fset"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "'e \<rightharpoonup>\<^sub>f 'f"   and n :: "'e \<rightharpoonup>\<^sub>f 'g" assumes "\<And>x. x |\<in>| S \<Longrightarrow> rel_option P (fmlookup m x) (fmlookup n x)" shows "fmrel_on_fset S P m n"
nqfree_qfree: fixes \<phi> :: "'a fm" assumes "nqfree \<phi>" shows "qfree \<phi>"
split_asm: 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) = (\<not> (cstate = Void \<and> \<not> P f1 \<or> cstate = cstate.Fail \<and> \<not> P f2 \<or> (\<exists>x3. cstate = Store x3 \<and> \<not> P (f3 x3)) \<or> (\<exists>x4. cstate = cstate.Collect x4 \<and> \<not> P (f4 x4))))"
nat_pow_0: fixes x :: "'a" shows "x [^] 0 = \<one>"
word_sless_msb_less: fixes x :: "'b word"   and y :: "'b word" shows "(x <s y) = ((msb y \<longrightarrow> msb x) \<and> (msb x \<and> \<not> msb y \<or> x < y))"
less_V_def: fixes x :: "V"   and y :: "V" shows "x \<subset>\<^sub>\<circ> y \<equiv> x \<subseteq>\<^sub>\<circ> y \<and> x \<noteq> y"
a_5: shows "list_all stateful_strand_step.is_Send (unlabel (transaction_send T))"
neg_less_iff_less: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus b) (uminus a) = less a b"
Collect_case_prod_Sigma: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
assms_4: shows "index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2"
usubstappf_pred_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and p :: "char"   and \<theta> :: "trm" assumes "usubstappf \<sigma> U (Pred p \<theta>) \<noteq> undeff" shows "usubstappt \<sigma> U \<theta> \<noteq> undeft \<and> (SPreds \<sigma> p = undeff \<or> (\<exists>r. SPreds \<sigma> p = Afml r \<and> FVF r \<inter> U = {}))"
all_nstri_imp_lex_nstri: fixes ys :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and xs :: "'b list"   and m :: "nat" assumes "\<forall>i<length ys. snd (f (xs ! i) (ys ! i))"   and "length ys \<le> length xs"   and "length xs = length ys \<or> length ys \<le> m" shows "snd (lex_ext f m xs ys)"
lfilter_conv_lnths: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lfilter P xs = lselect {n. enat n < llength xs \<and> P (xs ?! n)} xs"
Res: fixes P :: "pi"   and Q :: "pi"   and a :: "name" assumes "(P, Q) \<in> Rel''" shows "(<\<nu>a>P, <\<nu>a>Q) \<in> Rel''"
FGModuleHomSetD_Im: fixes T :: "'a \<Rightarrow> 'b"   and G :: "'c set"   and fgsmult :: "('d, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and V :: "'a set"   and fgsmult' :: "('d, 'c) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and W :: "'b set" assumes "T \<in> FGModuleHomSet G fgsmult V fgsmult' W" shows "T ` V \<subseteq> W"
simps_5: fixes i :: "char list \<Rightarrow> tv"   and p :: "fm"   and q :: "fm" shows "eval i (Eql p q) = (if eval i p = eval i q then Det True else Det False)"
goedel_first_theHardHalf: assumes "\<omega>consistent" shows "\<not> prv (neg \<phi>G)"
ccSup_empty: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
s_a_p_1: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "int"   and n :: "int" assumes "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "m < 0"   and "0 < n" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<^sub>f+ m\<Odot>a\<^bsub>f,i,z\<^esub> = (n + m)\<Odot>a\<^bsub>f,i,z\<^esub>"
success_run: fixes f :: "'d Heap"   and h :: "heap" assumes "success f h" obtains h' :: "heap" and r :: "'d" where "run f (Some h) (Some h') r"
assms_1: shows "l \<equiv> limsup (\<lambda>n. ereal (root n (norm (f n))))"
is_nextElem_last_ne: fixes vs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct vs"   and "is_nextElem vs x y"   and "x = last vs" shows "vs \<noteq> []"
periodic_mult: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and a :: "nat"   and m :: "nat" assumes "dcharacter n \<chi>" shows "\<chi> (a + m * n) = \<chi> a"
treesize_Branching: fixes T :: "tree"   and n :: "nat" assumes "treesize T = Suc n" shows "\<exists>l r. T = Branching l r"
semiconfluentp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "semiconfluentp r = (r\<inverse>\<inverse> OO r\<^sup>*\<^sup>* \<le> r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>*)"
mHom_func: fixes f :: "'i \<Rightarrow> 'j"   and M :: "('i, 'a, 'k) Module_scheme"   and N :: "('j, 'a, 'l) Module_scheme" assumes "f \<in> mHom R M N" shows "f \<in> carrier M \<rightarrow> carrier N"
b_1: shows "b \<in> local.degenerate_lotteries"
measurable_restrict_space1: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and \<Omega> :: "'d set" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N"
b: shows "homotopic_with_canon (\<lambda>x. True) T T id (\<lambda>x. b)"
p_pow_res: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "([p] \<cdot> \<one> [^] n) m = 0"
The_on_def: fixes U :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "The_on U P = (if \<exists>!x. x \<in> U \<and> P x then Some (THE x. x \<in> U \<and> P x) else None)"
L_ss: fixes q :: "'v"   and l :: "'L" assumes "G'.L q l" shows "q \<in> G'.V"
oVeblen_fixed: fixes y :: "ordinal" shows "\<forall>x<y. \<forall>a. oVeblen F x (oVeblen F y a) = oVeblen F y a"
iD: shows "isnFDERIV DIM('a) fas [0..<DIM('a)] [DIM('a)..<2 * DIM('a)] (list_of_eucl x @ list_of_eucl d) n"
mult_right_mono_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times a c) (times b c)"
assms: shows "\<forall>p. P p \<and> l < f p \<longrightarrow> (\<exists>p'. P p' \<and> f p' < f p)"
analz_insert_eq_I: fixes X :: "msg"   and H :: "msg set" assumes "\<And>x. x \<in> analz (insert X H) \<Longrightarrow> x \<in> insert X (analz H)" shows "analz (insert X H) = insert X (analz H)"
None_in_opt: fixes A :: "'b set" shows "None \<in> opt A"
list_greater_non_memb: fixes xs :: "'b list"   and oid :: "'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x < oid"   and "oid \<in> set xs" shows "False"
assms_1: shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>"
assms_4: shows "\<forall>q. List.member qs q \<longrightarrow> coprime p q"
ebind_refine': fixes mi :: "('c + 'd) nres"   and E :: "('c \<times> 'e) set"   and R' :: "('d \<times> 'f) set"   and m :: "('e + 'f) nres"   and fi :: "'d \<Rightarrow> ('c + 'g) nres"   and R :: "('g \<times> 'h) set"   and f :: "'f \<Rightarrow> ('e + 'h) nres" assumes "mi \<le> \<Down>\<^sub>E E R' m"   and "\<And>xi x. \<lbrakk>(xi, x) \<in> R'; inres mi (Inr xi); inres m (Inr x); nofail mi; nofail m\<rbrakk> \<Longrightarrow> fi xi \<le> \<Down>\<^sub>E E R (f x)" shows "ebind mi fi \<le> \<Down>\<^sub>E E R (ebind m f)"
simps: fixes cs :: "(String.literal \<times> String.literal) list"   and arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "translate_osig (cs, arss) = (if exe_osig_conds (cs, arss) then (set cs, translate_ars arss) else ({}, [STR ''A'' \<mapsto> [STR ''A'' \<mapsto> [insert STR ''A'' full_sort]]]))"
types_preserved_br_table: fixes c :: "i32"   and "is" :: "nat list"   and i :: "nat"   and i' :: "nat"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[$C ConstInt32 c, $Br_table is i]\<rparr> \<leadsto> \<lparr>[$Br i']\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c, $Br_table is i] : ts _> ts'"   and "i' = is ! nat_of_int c \<and> nat_of_int c < length is \<or> i' = i" shows "\<S>\<bullet>\<C> \<turnstile> [$Br i'] : ts _> ts'"
hd: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) d = D"
induced_arrowI_1: fixes \<chi>' :: "'j \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "\<guillemotleft>\<chi>.induced_arrow a' \<chi>' : a' \<rightarrow> a\<guillemotright>"
sel_14: fixes x4 :: "real" shows "extract_real (RealVal x4) = x4"
big_iff_small: fixes P :: "(char list list \<times> char list exp) prog"   and s :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "wwf_J_prog P"   and "iconf (shp s) e"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> = (P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',False\<rangle> \<and> final e')"
variants_iff_subsumes: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "variants C D = (subsumes C D \<and> subsumes D C)"
prod_coprime_right: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'b set"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>i. i \<in> A \<Longrightarrow> algebraic_semidom.coprime times one a (f i)" shows "algebraic_semidom.coprime times one a (comm_monoid_mult.prod times one f A)"
invert_opposite_sides_of_square: fixes f :: "'d"   and g :: "'d"   and h :: "'d"   and k :: "'d" assumes "D.seq f g"   and "f \<cdot>\<^sub>D g = h \<cdot>\<^sub>D k"   and "D.iso f"   and "D.iso k" shows "D.seq g (D.inv k) \<and> D.seq (D.inv f) h \<and> g \<cdot>\<^sub>D D.inv k = D.inv f \<cdot>\<^sub>D h"
not_outside_2: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "Order_Predicates.preorder_on carrier le"   and "le x y" shows "y \<in> carrier"
coboundedI1: fixes a :: "'k multiset"   and c :: "'k multiset"   and b :: "'k multiset" assumes "a \<subseteq># c" shows "a \<inter># b \<subseteq># c"
dvd_div_eq_2: fixes c :: "nat"   and a :: "nat"   and b :: "nat" assumes "0 < c"   and "a dvd c"   and "b dvd c"   and "c div a = c div b" shows "a = b"
frequently_mp: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x"   and "frequently P F" shows "frequently Q F"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "P One"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Fun x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Syntax.Sum x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Syntax.Prod x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (Mu x1 x2)"   and "\<And>x. P (Alpha x)"   and "\<And>x. P x \<Longrightarrow> P (AuthT x)" shows "P ty"
prim_invar_impl_init: shows "prim_invar_impl (Q_update r 0 Q_empty) M_empty"
xfer_waits: fixes s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and cs :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)"   and t :: "nat" assumes "sim_rel s cs" shows "is_WAIT (tts cs t) = is_WAIT (tts s t)"
stdin_of_println: fixes str :: "String.literal"   and world :: "\<^url>" shows "stdin_of (exec (println str) world) = stdin_of world"
path_endpoints_1: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path g u p v"   and "p \<noteq> []" shows "u \<in> nodes g"
g_meas: shows "g \<in> borel_measurable Invariants"
Atomic_def: fixes y :: "hf" shows "Atomic y \<equiv> \<exists>t u. Term t \<and> Term u \<and> (y = q_Eq t u \<or> y = q_Mem t u)"
prems: shows "poss (add_mset A AA) \<subseteq># map_clause f C"
map_of_SomeD: fixes xs :: "('b \<times> 'a) list"   and k :: "'b"   and y :: "'a" assumes "table_of xs k = Some y" shows "(k, y) \<in> set xs"
conv: fixes P :: "'a measure" shows "weak_conv_m (\<lambda>n. distr P borel (\<lambda>x. birkhoff_sum f n x / B n)) Z"
arr_iff_in_hom: fixes f :: "'a \<times> 'a \<times> 'a" shows "VVV.arr f = VVV.in_hom f (VVV.dom f) (VVV.cod f)"
assms_2: shows "m \<notin> keys (mapping_of p1)"
tendsto_rabs_zero_cancel: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
is_monomial_set_def: fixes A :: "('d \<Rightarrow>\<^sub>0 'e) set" shows "is_monomial_set A = (\<forall>p\<in>A. is_monomial p)"
pred_inject_2: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'a option" shows "pred_phantom2 P1 P2 (PH22 a) = pred_option P1 a"
H2N_def: fixes b :: "nat"   and n :: "C" shows "H2N b n = evalC b n"
dep_tc_def: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set" shows "dep_tc PROB = TC (dep PROB)"
gen_gba_impl_rel_sv: fixes Rl :: "('a \<times> ('b \<Rightarrow> 'c \<Rightarrow> bool)) set"   and Rm :: "('d \<times> 'e) set" assumes "single_valued Rl"   and "single_valued Rm" shows "single_valued (\<langle>Rm, Rl\<rangle>gen_gba_impl_rel_eext)"
a: shows "\<not> mod_type_class.from_nat i \<le> a"
cert_ok_def: fixes cert :: "'a list"   and n :: "nat"   and T :: "'a"   and B :: "'a"   and A :: "'a set" shows "cert_ok cert n T B A = ((\<forall>i<n. cert ! i \<in> A \<and> cert ! i \<noteq> T) \<and> cert ! n = B)"
seqSubstEqvt: fixes p :: "(name \<times> name) list"   and P :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "p \<bullet> P[<\<sigma>>] = (p \<bullet> P)[<(p \<bullet> \<sigma>)>]"
sorted_on_single: fixes x :: "'a" shows "sorted_on r [x] = (x \<in> Field r)"
sets_bind: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e measure"   and N :: "'e measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N"   and "space M \<noteq> {}" shows "sets (M \<bind> f) = sets N"
asIMOD_igFreshAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and ys :: "'varSort"   and y :: "'var"   and A :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs" shows "igFreshAbs (asIMOD SEM) ys y A = (\<forall>val val'. sWlsVal SEM val \<and> sWlsVal SEM val' \<and> eqBut val val' ys y \<longrightarrow> A val = A val')"
Diff_Int2: fixes A :: "'e set"   and C :: "'e set"   and B :: "'e set" shows "A \<inter> C - B \<inter> C = A \<inter> C - B"
simps: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
simps_1: fixes verts :: "'c set"   and arcs :: "'d set"   and tail :: "'d \<Rightarrow> 'c"   and head :: "'d \<Rightarrow> 'c"   and more :: "'e" shows "verts \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = verts"
nonpos_IntsI: fixes x :: "'a" assumes "x \<in> \<int>"   and "x \<le> (0::'a)" shows "x \<in> \<int>\<^sub>\<le>\<^sub>0"
assms_3: fixes n :: "nat" shows "dist (u n) (v n) \<le> C"
graph_nodes_it_axioms_def: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) graph"   and invar :: "'a \<Rightarrow> bool"   and nodes_list_it :: "'a \<Rightarrow> ('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list" shows "graph_nodes_it_axioms \<alpha> invar nodes_list_it \<equiv> \<forall>g. invar g \<longrightarrow> set_iterator (nodes_list_it g) (nodes (\<alpha> g))"
divide_le_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a b) zero = (less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b)"
array_length_list: fixes a :: "'c array" shows "array_length a = length (list_of_array a)"
simps_5: fixes v :: "((int \<times> int) set set, int) Combinators"   and va :: "((int \<times> int) set set, int) Combinators"   and xs :: "((int \<times> int) set set, int) Combinators list" shows "wellformed_policy2 (v \<oplus> va # xs) = ((\<forall>c a b. c = DenyAllFromTo a b \<and> c \<in> set xs \<longrightarrow> dom (C (v \<oplus> va)) \<inter> dom (C c) = {}) \<and> wellformed_policy2 xs)"
sorted_mid_iff2: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "sorted (x # xs @ y # ys) = (sorted (x # xs) \<and> x < y \<and> sorted (xs @ [y]) \<and> sorted (y # ys))"
wls_freshAbs_swapAbs_compose_aux: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and xs :: "'varSort" assumes "wlsAbs (us, s) A"   and "wlsPar P" shows "\<forall>x y z. {x, y, z} \<subseteq> varsOfS P xs \<and> freshAbs xs y A \<and> freshAbs xs z A \<longrightarrow> A $[y \<and> x]_xs $[z \<and> y]_xs = A $[z \<and> x]_xs"
a: fixes x :: "'a" assumes "x \<in> S" shows "closed_segment a x \<subseteq> S"
domintros_1: fixes n :: "nat"   and l :: "nat list" assumes "ackloop_dom (Suc n # l)" shows "ackloop_dom (n # 0 # l)"
is_SNF_intro: fixes P :: "'a mat"   and A :: "'a mat"   and Q :: "'a mat"   and S :: "'a mat" assumes "P \<in> carrier_mat (dim_row A) (dim_row A)"   and "Q \<in> carrier_mat (dim_col A) (dim_col A)"   and "invertible_mat P"   and "invertible_mat Q"   and "Smith_normal_form_mat S"   and "S = P * A * Q" shows "is_SNF A (P, S, Q)"
fstzero_add_zerosnd: shows "no_real_inner.fstzero + no_real_inner.zerosnd = id_blinfun"
algebraic_int_of_nat: fixes k :: "nat" shows "algebraic_int (of_nat k)"
hcomp_simps_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.hseq \<nu> \<mu>" shows "src\<^sub>C (\<nu> \<star>\<^sub>C \<mu>) = src\<^sub>C \<mu>"
p'_def: fixes t :: "real" shows "other.p' t = v\<^sub>o + a\<^sub>o * t"
antimono_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd" assumes "class.order less_eq less" shows "order.antimono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f y \<le> f x)"
r_zero: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_monoid G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<zero>\<^bsub>G\<^esub> = x"
nth_butlast: fixes n :: "nat"   and xs :: "'b list" assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
All_D: fixes H :: "fm set"   and i :: "name"   and A :: "fm"   and x :: "tm" assumes "H \<turnstile> SyntaxN.All i A" shows "H \<turnstile> A(i::=x)"
inverse_less_imp_less: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less (inverse a) (inverse b)"   and "less zero a" shows "less b a"
f: shows "f \<in> borel_measurable (stream_space (count_space UNIV))"
me: shows "match_iface (oiface r) (p_oiface p)"
hyps_2: fixes c\<^sub>2 :: "com" assumes "\<langle>c\<^sub>1,s\<rangle> \<rightarrow> \<langle>c\<^sub>2,s\<^sub>2\<rangle>" shows "c\<^sub>1' = c\<^sub>2 \<and> s' = s\<^sub>2"
heap_def2: fixes s :: "st"   and a :: "loc" shows "heap s a = globs s (Inl a)"
timpl_closurep_timpl_closure_eq: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "timpl_closurep t (\<lambda>x xa. (x, xa) \<in> TI) = (\<lambda>x. x \<in> timpl_closure t TI)"