hcju/isabelleps huggingface dataset:
intro: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_element known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr" shows "l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs create_element known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
right_translations_of_group_def: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "right_translations_of_group G composition unit \<equiv> group G composition unit"
map_sel_2: fixes a :: "('f, 'g, 'h) Resumption.resumption"   and f1 :: "'f \<Rightarrow> 'j"   and f2 :: "'g \<Rightarrow> 'i" assumes "\<not> Resumption.resumption.is_Done a" shows "Resumption.resumption.output (Resumption.resumption.map_resumption f1 f2 a) = f2 (Resumption.resumption.output a)"
rexp_of''_def: fixes n :: "nat"   and \<phi> :: "'a formula" shows "rexp_of'' n \<phi> = rexp.Inter (rexp_of_alt' n \<phi>) (ENC n (FOV \<phi>))"
nn_intergal_power_times_exp_Ici: fixes k :: "nat" shows "\<integral>\<^sup>+x\<in>{0..}. ennreal (x ^ k * exp (- x))\<partial>lborel = ennreal (real (fact k))"
simps_4: fixes vc :: "'b"   and va :: "action"   and rs :: "'b rule list" shows "rmMatchFalse (Rule (MatchNot (Match vc)) va # rs) = Rule (MatchNot (Match vc)) va # rmMatchFalse rs"
shows_prec_char_def: fixes p :: "nat"   and c :: "char" shows "shows_prec p c = (#) c"
nn_integral_convolution: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "finite_measure M"   and "finite_measure N"   and "sets N = sets borel"   and "sets M = sets borel"   and "f \<in> borel_measurable borel" shows "integral\<^sup>N (M \<star> N) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x + y) \<partial>N \<partial>M"
irc_on_discard: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and C :: "'a set"   and B :: "'a set" assumes "irc_on A f"   and "finite C"   and "B \<union> C \<subseteq> A"   and "f (B \<union> C) \<inter> C = {}" shows "f (B \<union> C) = f B"
fin: shows "finite {uu_. \<exists>x y. uu_ = x \<and> x = Abs_fin_string y \<and> y \<in> {s. length s \<le> MAX_STR}}"
app_const_distinct: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a"   and name :: "name" shows "app u\<^sub>1 u\<^sub>2 \<noteq> const name"
less_eq_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "(x \<le> y) = (fst x < fst y \<or> fst x \<le> fst y \<and> snd x \<le> snd y)"
C_eq_Lemmas_sep_14: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "p \<noteq> []" shows "qsort p l \<noteq> []"
esqrt_simps_1: shows "esqrt 0 = 0"
simps_58: fixes x81 :: "nat"   and x82 :: "msg"   and x5 :: "nat" shows "Crypt x81 x82 \<noteq> Key x5"
Var: 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 x :: "'a" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "x \<in> var" shows "Var x \<in> trm"
mask_overlap_zero: fixes a :: "nat"   and b :: "nat"   and p :: "'b word" assumes "a \<le> b" shows "(p AND mask a) AND NOT (mask b) = 0"
simple_valid_cfg: fixes ct :: "'s \<Rightarrow> 's pmf"   and s :: "'s" assumes "ct \<in> Pi S K"   and "s \<in> S" shows "simple ct s \<in> valid_cfg"
get_tag_name_locs_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs" shows "get_tag_name_locs = (\<lambda>element_ptr. {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name)})"
acquired_reads_mono': fixes x :: "nat"   and b :: "bool"   and xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "x \<in> acquired_reads b xs A"   and "acquired_reads b xs B = {}"   and "A \<subseteq> B" shows "False"
elims: fixes x :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and y :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "osig x = y" obtains uu_ :: "String.literal \<Rightarrow> typ option" and uv_ :: "String.literal \<Rightarrow> nat option" and oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" where "x = (uu_, uv_, oss)" and "y = oss"
d_def: shows "d = Min (set (def # map (\<lambda>(x, y). x - y) xys))"
ctr_transfer: fixes R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "(R2 ===> rel_nondetT' R1 R2) NondetT NondetT"
cod_eqI': fixes a :: "'b"   and f :: "'b" assumes "a \<in> C'.codomains f" shows "a = C'.cod f"
union_impr: fixes a :: "'a"   and b :: "'a" shows "(a r\<rightarrow> b) \<squnion>1 (b r\<rightarrow> a) = (1::'a)"
adjoint_pair_def: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory Span.vcomp Span.hcomp Span.assoc Span.unit Span.src Span.trg f g \<eta> \<epsilon>"
h1: shows "Controller_L y (fin_inf_append [Zero] l) l z"
bottom_below_in: fixes P :: "'a set"   and x :: "'a" assumes "has_bottom P"   and "x \<in> P" shows "bottom (P .\<^bold>\<le> x) = bottom P"
idem1: fixes s :: "'a"   and r :: "'b \<times> 'c" assumes "contains s r" shows "s \<bullet> r = s"
map_fst_append1: fixes xs :: "('a \<times> 'b) list"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "\<forall>i\<in>set (map fst xs). P i"   and "P x" shows "\<forall>i\<in>set (map fst (xs @ [(x, y)])). P i"
ffold_finsert_idem: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and x :: "'d"   and A :: "'d fset" assumes "comp_fun_idem f" shows "ffold f z (finsert x A) = f x (ffold f z A)"
get_attribute_is_l_get_attribute: shows "l_get_attribute ShadowRootClass.type_wf get_attribute get_attribute_locs"
word_and_not: fixes x :: "'b word" shows "x && ~~ x = 0"
pop_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier A"   and "b \<in> carrier A" shows "a \<plusminus> b = b \<plusminus> a"
is_root_merge: fixes h1 :: "'b tree"   and h2 :: "'b tree" assumes "is_root h1"   and "is_root h2" shows "is_root (merge h1 h2)"
chain_shift: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'b"   and j :: "nat" assumes "class.po below"   and "po.chain below Y" shows "po.chain below (\<lambda>i. Y (i + j))"
case_option_collapse: fixes x :: "'b" shows "case_option x (\<lambda>_. x) = (\<lambda>_. x)"
distinct_3: fixes x1 :: "nat"   and x3 :: "dB" shows "Var x1 \<noteq> Abs x3"
mult_cancellation_less: fixes a :: "V"   and x :: "V"   and r :: "V"   and y :: "V"   and s :: "V" assumes "a * x + r < a * y + s"   and "r \<sqsubset> a"   and "s \<sqsubset> a" obtains "x < y"   | "x = y" and "r < s"
Ke0_imp_inj: assumes "carrier (V.vs kerT) = {\<zero>\<^bsub>V\<^esub>}" shows "inj_on T (carrier V)"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Cod \<mu>)"
cong: fixes guard :: "'b \<Rightarrow> bool"   and body :: "'b \<Rightarrow> 'b spmf" shows "loop_spmf.iter guard body \<equiv> loop_spmf.iter guard body"
intro: fixes f :: "'c \<Rightarrow> 'd"   and s1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (s1 r b) = s2 r (f b)" shows "module_hom_axioms s1 s2 f"
PiE_ext: fixes x :: "'g \<Rightarrow> 'h"   and k :: "'g set"   and s :: "'g \<Rightarrow> 'h set"   and y :: "'g \<Rightarrow> 'h" assumes "x \<in> Pi\<^sub>E k s"   and "y \<in> Pi\<^sub>E k s"   and "\<And>i. i \<in> k \<Longrightarrow> x i = y i" shows "x = y"
chine_simps_2: shows "C.dom f.chine = f.dom.apex"
pbeta_Upd: fixes t :: "sterm"   and t' :: "sterm"   and L :: "char list set"   and u :: "sterm"   and u' :: "sterm"   and l :: "Label" assumes "t => t'"   and "lc t"   and "finite L"   and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. u\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> u' = \<sigma>[s,p] t'')"   and "body u" shows "Upd t l u => Upd t' l u'"
vec_to_iarray_morph: fixes A :: "('c, 'd) vec"   and B :: "('c, 'd) vec" shows "(A = B) = (vec_to_iarray A = vec_to_iarray B)"
strict_trans2: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "strict_prefix b a"   and "prefix c b" shows "strict_prefix c a"
f: shows "take p (take p w \<cdot> w) = take p w"
normalize_def: fixes p :: "(real, 2) vec" shows "normalize p = (if p = 0 then axis 1 1 else (1 / norm p) *\<^sub>R p)"
prec_spec_def: fixes h :: "heap \<times> nat set"   and \<Gamma> :: "assn"   and \<Gamma>' :: "assn" shows "prec_spec h \<Gamma> \<Gamma>' \<equiv> h \<Turnstile> \<Gamma> * true \<and>\<^sub>A \<Gamma>' * true"
Arg_times_of_real2: fixes r :: "real"   and z :: "complex" assumes "0 < r" shows "Arg (z * complex_of_real r) = Arg z"
Let_def: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "Let s f \<equiv> f s"
Interleaves_equal_fst: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and ys' :: "'a list" assumes "xs \<cong> {ys, zs, P}"   and "xs \<cong> {ys', zs, P}" shows "ys = ys'"
ins_tree_rank_bound: fixes t' :: "'b tree"   and t :: "'b tree"   and ts :: "'b tree list"   and t\<^sub>0 :: "'c tree" assumes "t' \<in> set (ins_tree t ts)"   and "\<forall>t'\<in>set ts. rank t\<^sub>0 < rank t'"   and "rank t\<^sub>0 < rank t" shows "rank t\<^sub>0 < rank t'"
S18: shows "pr_gr upb = g_comp (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair n x)"
Upper_cong: fixes A :: "'a set"   and A' :: "'a set" assumes "A \<subseteq> carrier L"   and "A' \<subseteq> carrier L"   and "A {.=} A'" shows "Upper L A = Upper L A'"
prefix_tl_subset: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs ys" shows "set (tl xs) \<subseteq> set (tl ys)"
basis_inD_1: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> X" shows "\<E>.indep_in \<E> X"
map_def: fixes f :: "'a" shows "local.map f = (if C.arr f then f else C.null)"
can_assoc: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "ide a"   and "ide b"   and "ide c" shows "can \<a>[a, b, c]"
intros_4: fixes r :: "nat"   and m :: "nat" shows "i_set_indp [ r, mod m ]"
finite_ReZ_segments_poly_of_real: fixes p :: "complex poly"   and z :: "complex" shows "finite_ReZ_segments (poly p \<circ> complex_of_real) z"
w_def: shows "w = w1 @ w2 @ w3 @ w4 @ w5"
foldli_transform: fixes iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R" shows "\<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
polys_to_mat_index: fixes i :: "nat"   and ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and j :: "nat"   and ts :: "'c list" assumes "i < length ps"   and "j < length ts" shows "polys_to_mat ts ps $$ (i, j) = lookup (ps ! i) (ts ! j)"
ideD_2: fixes a :: "'j" assumes "J.ide a" shows "J.dom a = a"
coboundedI1: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<le> a" shows "c \<le> sup a b"
C: shows "sinks I D u (x # xs @ [x']) = (if (u, D x') \<in> I \<or> (\<exists>v\<in>sinks I D u (x # xs). (v, D x') \<in> I) then insert (D x') (sinks I D u (x # xs)) else sinks I D u (x # xs))"
unit: fixes H :: "'a set" shows "\<zero> \<in> additive.generate H"
cons1: fixes P :: "'a state \<Rightarrow> exp0 \<Rightarrow> 'a state \<times> (v, v) result \<Rightarrow> bool"   and s1 :: "'a state"   and e :: "exp0"   and s2 :: "'a state"   and v :: "v"   and es :: "exp0 list"   and s3 :: "'a state"   and vs :: "v list" assumes "P s1 e (s2, Rval v)"   and "Big_Step_Unclocked_Single.evaluate_list P s2 es (s3, Rval vs)" shows "Big_Step_Unclocked_Single.evaluate_list P s1 (e # es) (s3, Rval (v # vs))"
hyps_1: shows "p' \<in> grid b ({d} \<union> ds)"
real_of_1_0_1: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "invariant_1 (p, l, r)" shows "(real_of_1 (p, l, r) = 0) = (r = 0)"
chine_eq_apex: shows "HH\<nu>\<pi>\<rho>.chine = HH\<nu>\<pi>\<rho>.apex"
pHom_dec_deg: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and f :: "'a \<Rightarrow> 'e"   and p :: "'a" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "p \<in> carrier R" shows "deg A B Y (f p) \<le> deg R S X p"
sublist_butlast: fixes xs :: "'a list" shows "sublist (butlast xs) xs"
take_eq_Nil: fixes n :: "nat"   and xs :: "'e list" shows "(take n xs = []) = (n = 0 \<or> xs = [])"
norm_sgn: fixes x :: "'b" shows "norm (sgn x) = (if x = (0::'b) then 0 else 1)"
collapse_1: fixes tm :: "('a, 'b) tm" assumes "is_Hd tm" shows "Hd (head tm) = tm"
lookup_fresh': fixes z :: "name"   and \<theta> :: "(name \<times> trm) list" assumes "z \<sharp> \<theta>" shows "lookup \<theta> z = Var z"
take_0_2: fixes xs :: "['b]" shows "take\<cdot>(MkI\<cdot>0)\<cdot>xs = []"
order_refl: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<preceq> x"
that_1: fixes i :: "nat" assumes "a = aa"   and "b = bb @ [i]"   and "bc = in_port_at (aa, bb) i"   and "hyps (nodeOf (a, b)) ba = None" shows "thesis"
mod_less_eq_dividend: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a" shows "a mod b \<le> a"
hom_minus: fixes x :: "'a poly"   and y :: "'a poly" shows "map_poly hom (x - y) = map_poly hom x - map_poly hom y"
cyc_free_diag_dest': fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd"   and i :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len m i i xs"   and "i \<le> n" shows "\<one> \<le> m i i"
tfr: shows "\<forall>f U. Fun f U \<in> subterms s \<union> subterms t \<longrightarrow> U = [] \<or> (\<exists>u\<in>set U. u \<notin> Var ` X)"
updVal_simp: fixes val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and x :: "'var"   and sX :: "'b"   and xs :: "'varSort"   and ys :: "'varSort"   and y :: "'var" shows "(val (x := sX)_xs) ys y = (if ys = xs \<and> y = x then sX else val ys y)"
Key_analz_image_Key: fixes evs :: "event list"   and SK :: "nat"   and KK :: "nat set" assumes "evs \<in> kerbIV"   and "SK \<in> symKeys \<and> KK \<subseteq> - range shrK"   and "\<forall>K\<in>KK. \<not> AKcryptSK K SK evs" shows "(Key SK \<in> analz (Key ` KK \<union> knows Spy evs)) = (SK \<in> KK \<or> Key SK \<in> analz (knows Spy evs))"
order_lesseq_imp: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less" shows "(\<forall>z. less_eq x z \<longrightarrow> less_eq y z) = less_eq y x"
ray_cases: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> ray a b" shows "[[a x b]] \<or> [[a b x]] \<or> x = b"
unitary_simps_1: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "unitary A" shows "adjoint A * A = 1\<^sub>m n"
LIM_offset_zero: fixes f :: "'c \<Rightarrow> 'd"   and L :: "'d"   and a :: "'c" assumes "f \<midarrow>a\<rightarrow> L" shows "(\<lambda>h. f (a + h)) \<midarrow>0::'c\<rightarrow> L"
dom_comp: fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" assumes "J.seq g f" shows "J.dom (g \<cdot>\<^sub>J f) = J.dom f"
intro: fixes A :: "V"   and f :: "V" assumes "nop A (1\<^sub>\<nat>) f" shows "unop A f"
inject_3: fixes x31 :: "'b match_expr"   and x32 :: "'b match_expr"   and y31 :: "'b match_expr"   and y32 :: "'b match_expr" shows "(MatchAnd x31 x32 = MatchAnd y31 y32) = (x31 = y31 \<and> x32 = y32)"
mult1_singleton: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" shows "(({#x#}, {#y#}) \<in> mult1 R) = ((x, y) \<in> R)"
groupsort_def: fixes f :: "'d \<Rightarrow> 'e"   and merge :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d list \<Rightarrow> 'f" shows "groupsort f merge g \<equiv> ((\<forall>x y. f x = f y \<longrightarrow> f (merge x y) = f x) \<and> (\<forall>xs ys. mset xs = mset ys \<longrightarrow> g xs = g ys)) \<and> (\<forall>x y. f x = f y \<longrightarrow> g [x, y] = g [merge x y]) \<and> (\<forall>xs1 xs2 ys1 ys2. g xs1 = g xs2 \<longrightarrow> g ys1 = g ys2 \<longrightarrow> g (xs1 @ ys1) = g (xs2 @ ys2))"
unit_disc_fix_f_circline_direct: fixes M :: "moebius"   and f :: "complex_homo \<Rightarrow> complex_homo"   and H :: "circline" assumes "unit_disc_fix M"   and "f = moebius_pt M" shows "unit_disc_fix_f_circline f H = moebius_circline M H"
x_z: shows "(f ^^ funpow_dist1 f x z) x = z"
elemTIselemHT: fixes t :: "'a event list"   and P :: "'a process"   and A :: "'a set" assumes "t \<in> T P" shows "trace_hide t (ev ` A) \<in> T (P \ A)"
simps_1: fixes x :: "int" shows "fromA (A x) = Some x"
min_dist_split_1: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and d1 :: "nat"   and w :: "nat"   and d2 :: "nat"   and v :: "nat" assumes "Graph.dist c u d1 w"   and "Graph.dist c w d2 v"   and "Graph.min_dist c u v = d1 + d2" shows "Graph.min_dist c u w = d1"
induct: fixes P :: "((variable \<times> typ) \<times> Core.term) list \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "((variable \<times> typ) \<times> Core.term) list"   and a1 :: "Core.term" assumes "\<And>insts c T. P insts (Ct c T)"   and "\<And>insts idn T. P insts (Fv idn T)"   and "\<And>uu_ n. P uu_ (Bv n)"   and "\<And>insts T t. P insts t \<Longrightarrow> P insts (Abs T t)"   and "\<And>insts t u. \<lbrakk>P insts t; P insts u\<rbrakk> \<Longrightarrow> P insts (t $ u)" shows "P a0 a1"
textbuffer_rel_def: fixes A :: "('a \<times> 'b) set" shows "\<langle>A\<rangle>textbuffer_rel \<equiv> {(BUF a b, BUF a' b') |a b a' b'. (a, a') \<in> \<langle>A\<rangle>list_rel \<and> (b, b') \<in> \<langle>A\<rangle>list_rel}"
msum_aforms'_refine: shows "(msum_aforms', \<lambda>xs ys. {map2 (+) x y |x y. x \<in> xs \<and> y \<in> ys}) \<in> aforms_rel \<rightarrow> aforms_rel \<rightarrow> aforms_rel"
wf_graph_def: fixes G :: "'c graph" shows "wf_graph G \<equiv> (fst ` edges G \<subseteq> nodes G \<and> snd ` edges G \<subseteq> nodes G) \<and> finite (edges G) \<and> finite (nodes G)"
elims_2: fixes x :: "'a list" assumes "local.sorted x" obtains "x = []"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "Ball (set ys) (le xa) \<and> local.sorted ys"
simps: fixes a :: "nat"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" shows "c1.incs a x (xs, s) = (let t = s + a * x in if cond1 b ys (x # xs) t then (x # xs, t) # c1.incs a (Suc x) (xs, s) else [])"
pth_4_1: fixes x :: "'a" shows "0 *\<^sub>R x \<equiv> 0::'a"
tm_floatarith: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and xs :: "float list"   and f :: "floatarith"   and t :: "taylor_model"   and x :: "real"   and e :: "nat \<Rightarrow> real" assumes "tm_floatarith prec ord I xs f = Some t"   and "map real_of_float xs all_in I"   and "x \<in>\<^sub>r I ! 0"   and "xs \<noteq> []"   and "\<And>x. x \<in>\<^sub>r I ! 0 \<Longrightarrow> isDERIV 0 f ((map real_of_float xs)[0 := x])"   and "\<And>i. \<lbrakk>0 < i; i < length xs\<rbrakk> \<Longrightarrow> e i = real_of_float (xs ! i)"   and "x - real_of_float (xs ! 0) = e 0" shows "interpret_floatarith f ((map real_of_float xs)[0 := x]) \<in>\<^sub>i range_tm e t"
subgame_maximal_path: fixes V' :: "'a set"   and P :: "'a llist" assumes "V' \<subseteq> V\<^bsub>G'\<^esub>"   and "G'.maximal_path P"   and "lset P \<subseteq> V'" shows "Digraph.maximal_path (G'.subgame V') P"
varDQ: fixes b :: "BExpr"   and q :: "L"   and G :: "Var \<Rightarrow> L"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "(b, Q q G, s, t) \<in> varD" shows "Q q G s t"
sqnf_kno: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:14} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> sqnf (rt \<xi>) (dip \<xi>) = kno)"
fix_ind2: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a \<rightarrow> 'a" assumes "adm P"   and "P \<bottom>"   and "P (F\<cdot>\<bottom>)"   and "\<And>x. \<lbrakk>P x; P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (F\<cdot>(F\<cdot>x))" shows "P (fix\<cdot>F)"
in_front': shows "s\<^sub>e < s\<^sub>o"
mk_vec_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "Matrix.mk_vec n f \<equiv> \<lambda>i. if i < n then f i else undef_vec (i - n)"
set_pmf_iff: fixes x :: "'c"   and M :: "'c pmf" shows "(x \<in> set_pmf M) = (pmf M x \<noteq> 0)"
fresh_swap: fixes a :: "atom"   and b :: "atom"   and c :: "atom" shows "a \<sharp> (b \<rightleftharpoons> c) = (sort_of b \<noteq> sort_of c \<or> b = c \<or> a \<sharp> b \<and> a \<sharp> c)"
dist_riemann_sphere_r3_inner: fixes M1 :: "real \<times> real \<times> real"   and M2 :: "real \<times> real \<times> real" assumes "M1 \<in> unit_sphere"   and "M2 \<in> unit_sphere" shows "(dist_riemann_sphere_r3 M1 M2)\<^sup>2 = 2 - 2 * inner M1 M2"
assms: shows "orientable_1 (x \<squnion> (1::'a))"
len: shows "length Strat2 = length (Lxy qs {x, y})"
cart_basis'_def: shows "cart_basis' = (\<chi>i. axis i (1::'c))"
suba2o: shows "{a<--<flow0 x t2} \<subseteq> {a--b}"
list_3_cases: fixes xs :: "'a list" obtains "xs = []"   | x :: "'a" where "xs = [xb__]"   | x :: "'a" and y :: "'a" and ys :: "'a list" where "xs = x # y # ys"
order_bot_ow_axioms: shows "order_bot_ow U\<^sub>1 bot\<^sub>1 (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>1) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>1)"
Bit_def: fixes k :: "int"   and b :: "bool" shows "k BIT b = (if b then 1 else 0) + k + k"