hcju/isabelleps huggingface dataset:
get_shadow_root_ptr_simp3: fixes element_ptr :: "(_) element_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and f :: "('m, 'n, 'o, 'w option) RShadowRoot_scheme"   and h :: "(_) heap" shows "get element_ptr (put shadow_root_ptr f h) = get element_ptr h"
VectorSpaceEnd_def: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" shows "VectorSpaceEnd smult V T \<equiv> VectorSpaceHom smult V smult T \<and> VectorSpaceEnd_axioms V T"
euclidean_circle_unit_circle: shows "euclidean_circle unit_circle = (0, 1)"
fps_nth_inverse: fixes x :: "'b fps" shows "Abs_fps (($) x) = x"
between_imp_mod_between: fixes b :: "nat"   and m :: "nat"   and k :: "nat"   and a :: "nat"   and n :: "nat" assumes "b < m"   and "m * k + a \<le> n"   and "n \<le> m * k + b" shows "a \<le> n mod m \<and> n mod m \<le> b"
supp_transform: fixes a :: "Arity"   and e :: "exp" shows "supp (ccTransform a e) \<subseteq> supp e"
card_arity_transform_safe: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and r :: "Vars.var list" assumes "c \<Rightarrow>\<^sup>* c'"   and "\<not> boring_step c'"   and "heap_upds_ok_conf c"   and "consistent (ae, ce, a, as, r) c" shows "\<exists>ae' ce' a' as' r'. consistent (ae', ce', a', as', r') c' \<and> conf_transform (ae, ce, a, as, r) c \<Rightarrow>\<^sub>G\<^sup>* conf_transform (ae', ce', a', as', r') c'"
that: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "x = \<langle>a, f a\<rangle>" shows "thesis"
ts_tuple_rel_mono: fixes ys :: "(nat \<times> 'a option list set) set"   and zs :: "(nat \<times> 'a option list set) set" assumes "ys \<subseteq> zs" shows "ts_tuple_rel ys \<subseteq> ts_tuple_rel zs"
simps_4: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "valid (a{b}.P) = valid P"
cfcomp_LAM: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'c \<rightarrow> 'd" assumes "cont g" shows "f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))"
assoc_naturality: fixes f0 :: "'a"   and f1 :: "'a"   and f2 :: "'a" assumes "arr f0"   and "arr f1"   and "arr f2" shows "\<a>[cod f0, cod f1, cod f2] \<cdot> ((f0 \<otimes> f1) \<otimes> f2) = (f0 \<otimes> f1 \<otimes> f2) \<cdot> \<a>[local.dom f0, local.dom f1, local.dom f2]"
S_ENTRY_PAT: fixes P :: "'p multiset" assumes "{#[entry fg p]. p \<in># P#} \<subseteq># ch" shows "(v, mon_w fg w, P) \<in> S_cs fg (size P)"
lookupd_fresh: fixes a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and y :: "name" assumes "a \<sharp> \<theta>_c" shows "lookupd y a \<theta>_c = Ax y a"
assms_3: shows "op \<in> set (\<Pi>\<^sub>\<O>)"
valid_implies_valid_prog: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "valid ts" shows "valid_prog p"
snd_set_prevs: fixes base :: "'b"   and tail :: "'a list"   and n :: "nat" shows "snd ` set_prevs base tail n = (\<lambda>as. foldr f as base) ` {as. subseq_of_length n tail as}"
inf_cases: fixes c :: "com"   and cs :: "com list"   and s :: "state" assumes "PHoareTotal.inf (c # cs) s" shows "PHoareTotal.inf [c] s \<or> (\<exists>t. s -c\<rightarrow> t \<and> PHoareTotal.inf cs t)"
norm_cblinfun_def: shows "norm \<equiv> map_fun cblinfun_apply id onorm"
rightreg_scalar_multD2: fixes R :: "'a set"   and x :: "'a"   and r :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "Ring1 R"   and "x \<notin> R" shows "Ring1.rightreg_scalar_mult R r f x = (0::'b)"
subst_bv2_preserves_beta2: fixes r :: "Core.term"   and s :: "Core.term"   and t :: "Core.term"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "subst_bv2 t i r \<rightarrow>\<^sub>\<beta>\<^sup>* subst_bv2 t i s"
exhaust: fixes y :: "'a option" obtains "y = None"   | x2 :: "'a" where "y = Some x2"
snocl: fixes xs :: "'a list"   and x :: "'a" shows "P (xs @ [x]) []"
Suc_times_binomial: fixes k :: "nat"   and n :: "nat" shows "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)"
hd_rand_det: fixes e :: "expr"   and V' :: "nat set"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and dst :: "pdf_dist" assumes "randomfree e"   and "free_vars e \<subseteq> V'" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Random dst e \<Rightarrow> (\<lambda>\<rho> x. branch_prob (V, V', \<Gamma>, \<delta>) \<rho> * dist_dens dst (expr_sem_rf \<rho> e) x)"
encode2_lt: fixes i :: "nat" assumes "i < st.d" shows "st.encode2 i < st.d2"
curve_elt_opp: fixes p\<^sub>1 :: "'b point"   and x\<^sub>1 :: "'b"   and y\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b point"   and x\<^sub>2 :: "'b"   and y\<^sub>2 :: "'b"   and a :: "'b"   and b :: "'b" assumes "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"   and "p\<^sub>2 = Point x\<^sub>2 y\<^sub>2"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "x\<^sub>1 = x\<^sub>2" shows "p\<^sub>1 = p\<^sub>2 \<or> p\<^sub>1 = opp p\<^sub>2"
list_wset_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>list_wset_rel = br set top O \<langle>R\<rangle>set_rel"
right_neutral: fixes a :: "('d, 'e) ctxt" shows "a \<circ>\<^sub>c \<box> = a"
H_def: shows "H = (THE H. H \<in> subgroups_of_size q)"
sub_sup_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<le> - y" shows "- z \<squnion> - x \<le> - z \<squnion> - y"
insert_sort_sorted: fixes xs :: "'c list" shows "sorted (insert_sort xs)"
vintersection_complement: fixes I :: "V"   and J :: "V" shows "(I -\<^sub>\<circ> J) \<inter>\<^sub>\<circ> J = []\<^sub>\<circ>"
abs_eq: shows "Q' = Abs_poly (\<lambda>k. if n < k then 0 else (- 1) ^ (n - k) * e' (n - k))"
list_succ_permutes: fixes xs :: "'b list" assumes "distinct xs" shows "list_succ xs permutes set xs"
DESTRimpl_rule: fixes s :: "bdd"   and ni :: "nat"   and n :: "nat ifex" assumes "bdd_sane s"   and "(ni, n) \<in> Rmi s" shows "ospec (destrmi' ni s) (\<lambda>r. case n of Trueif \<Rightarrow> r = TD | Falseif \<Rightarrow> r = FD | IF v t e \<Rightarrow> \<exists>tn en. r = IFD v tn en \<and> (tn, t) \<in> Rmi s \<and> (en, e) \<in> Rmi s)"
fps_inverse_unique: fixes f :: "'a fps"   and g :: "'a fps" assumes "f * g = 1" shows "inverse f = g"
isosceles_triangle_orthogonal_on_midpoint: fixes C :: "'b"   and A :: "'b"   and B :: "'b" assumes "dist C A = dist C B" shows "orthogonal (C - midpoint A B) (A - midpoint A B)"
insert: fixes a :: "'a"   and A :: "'a set" shows "Gcd\<^sub>f\<^sub>i\<^sub>n (insert a A) = gcd a (Gcd\<^sub>f\<^sub>i\<^sub>n A)"
size_4: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "size (Fun x21 x22) = size_list size x22 + Suc 0"
A1: fixes n :: "nat" shows "candidate (@) (map wrap [0..<n + 1]) = ups n"
Constrains_INT: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i Co A' i" shows "F \<in> \<Inter> (A ` I) Co \<Inter> (A' ` I)"
reflclp_idemp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "P\<^sup>=\<^sup>=\<^sup>=\<^sup>= = P\<^sup>=\<^sup>="
assms_1: shows "s \<rightarrow>\<^sub>\<beta>\<^sup>* t"
him_eq: fixes p :: "quat"   and q :: "quat" shows "(HIm p = HIm q) = (Im1 p = Im1 q \<and> Im2 p = Im2 q \<and> Im3 p = Im3 q)"
pp_dist_comp: fixes x :: "'a"   and y :: "'a" shows "- - (x \<sqinter> y) = - - x \<sqinter> - - y"
dist_not_less_zero: fixes x :: "'a"   and y :: "'a" shows "\<not> dist x y < 0"
all1_vec_def: fixes n :: "nat" shows "all1_vec n = vec n (\<lambda>i. 1)"
Pbij_inverse_DomRng: fixes \<gamma> :: "(Location \<times> Location) set"   and \<beta> :: "(Location \<times> Location) set" assumes "\<gamma> = Pbij_inverse \<beta>" shows "Pbij_Dom \<beta> = Pbij_Rng \<gamma> \<and> Pbij_Dom \<gamma> = Pbij_Rng \<beta>"
assms: shows "filter (\<lambda>x. x \<noteq> (0::'a)) xs = []"
trm\<^sub>s\<^sub>t\<^sub>p_subst: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and a :: "('d, 'c) strand_step" assumes "subst_domain \<theta> \<inter> set (bvars\<^sub>s\<^sub>t\<^sub>p a) = {}" shows "trms\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta>) = trms\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
circ_decompose_10: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> * x\<^sup>\<star> * y\<^sup>\<star>"
optimize_matches_preserves: fixes rs :: "'b rule list"   and P :: "'b match_expr \<Rightarrow> bool"   and f :: "'b match_expr \<Rightarrow> 'b match_expr" assumes "\<And>r. r \<in> set rs \<Longrightarrow> P (f (get_match r))" shows "\<forall>r\<in>set (optimize_matches f rs). P (get_match r)"
leg0_commutes: shows "\<eta>.cod.leg0 \<cdot> \<eta>.chine = \<eta>.dom.leg0"
rep_eq: shows "list_of_dlist DList_Set.empty = []"
payload_analz: fixes X :: "msg"   and S :: "msg set" assumes "X \<in> analz S"   and "S \<subseteq> payload" shows "X \<in> payload"
hpair_neq_succ: fixes x :: "hf"   and y :: "hf"   and k :: "hf" shows "\<langle>x, y\<rangle> \<noteq> succ k"
simps: fixes a1 :: "complex"   and b1 :: "complex"   and c1 :: "complex"   and d1 :: "complex"   and a2 :: "complex"   and b2 :: "complex"   and c2 :: "complex"   and d2 :: "complex" shows "(a1, b1, c1, d1) *\<^sub>m\<^sub>m (a2, b2, c2, d2) = (a1 * a2 + b1 * c2, a1 * b2 + b1 * d2, c1 * a2 + d1 * c2, c1 * b2 + d1 * d2)"
simps_11: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "rec_latin_type f1 f2 f3 Col = f2"
termFSwSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme" shows "termFSwSbImorph h hA MOD \<equiv> termFSwImorph h hA MOD \<and> ipresSubstAll h hA MOD"
red_def: shows "red \<equiv> \<lambda>P. {((xa, x), xc, xb). redp P xa x xc xb}"
reduced_labelling_zero: fixes j :: "nat"   and n :: "nat"   and x :: "nat \<Rightarrow> nat" assumes "j < n"   and "x j = 0" shows "reduced n x \<noteq> j"
continuous_map_pairedI: fixes Z :: "'b topology"   and X :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'d topology"   and g :: "'b \<Rightarrow> 'd" assumes "continuous_map Z X f"   and "continuous_map Z Y g" shows "continuous_map Z (prod_topology X Y) (\<lambda>x. (f x, g x))"
HAInitState_HARoot: fixes A :: "('a, 'b, 'c) hierauto" shows "HAInitState A \<in> States (HARoot A)"
consistent_def: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set" shows "consistent A S \<equiv> \<nexists>S'. set S' \<subseteq> S \<and> A \<turnstile> S'"
assert_def: fixes P :: "bool" shows "assert P \<equiv> if P then return () else fail ()"
rel_fundefs_def: fixes f :: "'e \<Rightarrow> ('dyn, 'f, 'g, 'h, 'opinl) Inca.instr fundef option"   and g :: "'e \<Rightarrow> ('dyn, 'f, 'g, 'h, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef option" shows "rel_fundefs f g = (\<forall>x. rel_option (rel_fundef (\<lambda>y z. y = norm_instr z)) (f x) (g x))"
detB: shows "Determinant.det (submatrix B (g ` {0..<n}) UNIV) = (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i)))"
find_eigenvector_def: fixes A :: "'b mat"   and e :: "'b" shows "find_eigenvector A e = find_base_vector (fst (gauss_jordan (char_matrix A e) (0\<^sub>m (dim_row A) 0)))"
exe_wf_osig_def: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "exe_wf_osig oss \<equiv> exe_wf_subclass (fst oss) \<and> exe_wf_tcsigs (fst oss) (snd oss) \<and> exe_osig_conds oss"
lift_def: fixes h :: "'c \<Rightarrow> 'b"   and inp :: "'a \<Rightarrow> 'c option" shows "lift h inp \<equiv> \<lambda>i. case inp i of None \<Rightarrow> None | Some v \<Rightarrow> Some (h v)"
lcm_gcd: fixes a :: "'b"   and b :: "'b" shows "lcm a b = normalize (a * b div gcd a b)"
set_pmf_subset_singleton: shows "(set_pmf \<mu> \<subseteq> {x}) = (\<mu> = return_pmf x)"
VarP_fresh_iff: fixes a :: "atom"   and x :: "tm" shows "a \<sharp> VarP x = a \<sharp> x"
simps_1: shows "extN.Bot + extN.Bot = extN.N 0"
GSMP_mono: fixes N :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "N \<subseteq> M" shows "GSMP N \<subseteq> GSMP M"
cInf_lower: fixes x :: "'c multiset"   and X :: "'c multiset set" assumes "x \<in> X"   and "subset_mset.bdd_below X" shows "Inf X \<subseteq># x"
exhaust_sel: fixes list :: "'b list" obtains "list = []"   | "list = hd list # tl list"
subprob_density_distr_real_eq: fixes M :: "real measure"   and f :: "real \<Rightarrow> ennreal"   and h :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> ennreal" assumes "has_subprob_density M lborel f"   and "h \<in> borel_measurable borel"   and "g \<in> borel_measurable borel"   and "\<And>a b. a \<le> b \<Longrightarrow> emeasure (distr (density lborel f) lborel h) {a..b} = emeasure (density lborel g) {a..b}" shows "has_subprob_density (distr M borel h) lborel g"
typed: shows "\<Gamma> \<turnstile>\<^sub>a e \<in> Low"
exhaust_sel: fixes list :: "'g list" obtains "list = []"   | "list = hd list # tl list"
ideal_not_empty: fixes I :: "'a set" assumes "ideal I" shows "I \<noteq> {}"
bhv2cnf_greater_lActive: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1 < n" shows "dynamic_component.lActive active c t < dynamic_component.bhv2cnf tCMP active c t n"
classes_above_sees_method2: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P' \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C'" shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C'"
ce_impl_def: fixes s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "ce_impl s \<equiv> let (a, S, B, I, P) = s in ASSERT (B \<noteq> []) \<bind> (\<lambda>_. let bls = Collect (on_stack_less I (last B)); ls = Collect (on_stack_ge I (last B)) in RETURN (Some (bls, ls), a, S, B, I, P))"
3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
top_empty_eq2: shows "top = (\<lambda>x y. (x, y) \<in> UNIV)"
simps_5: fixes a :: "Cooper.num" shows "bound0 (Gt a) = numbound0 a"
dis_rews_4: fixes x :: "CValue" shows "(is_CB\<cdot>x = \<bottom>) = (x = \<bottom>)"
in_read_only_augment_conv: fixes a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option"   and W :: "nat set"   and R :: "nat set" shows "(a \<in> read_only (\<S> \<oplus>\<^bsub>W\<^esub> R)) = (if a \<in> R then a \<notin> W else a \<in> read_only \<S>)"
find_index_def: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "find_index P xs = (case find_indices P xs of [] \<Rightarrow> None | i # x \<Rightarrow> Some i)"
zip_Nil: fixes ys :: "'b list" shows "zip [] ys = []"
assms_4: fixes ys :: "'varSort"   and y :: "'var" shows "freshAbs ys y A \<or> rho ys y = rho' ys y"
inf0: fixes a :: "'c" shows "a \<equiv> a \<sqinter> \<top>"
affine_dependent_translation_eq: fixes S :: "'a set"   and a :: "'a" shows "affine_dependent S = affine_dependent ((+) a ` S)"
assms_2: shows "\<forall>b\<in>basis. continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"
induced_automorphism_unique_automorphism': fixes s :: "'a \<Rightarrow> 'a" assumes "ChamberComplexAutomorphism X s"   and "s \<noteq> id"   and "fixespointwise s (C0 \<inter> D0)" shows "fun_eq_on s \
 C0"
cnj_mix_minus: fixes z1 :: "complex"   and z2 :: "complex" shows "cnj z1 * z2 - z1 * cnj z2 = \<i> * cnj_mix (rot90 z1) z2"
that: fixes m :: "'c multiset" assumes "count m i = length Qs + count n i"   and "indexed_eval (P # Qs) i m \<noteq> \<zero>" shows "thesis"
eWlsAbs_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and us :: "'varSort"   and s :: "'sort"   and A :: "'d" shows "eWlsAbs MOD (us, s) (OK A) = igWlsAbs MOD (us, s) A"
pointermapieq_exhaust: fixes a :: "'b pointermap"   and b :: "'b pointermap" assumes "entries a = entries b"   and "getentry a = getentry b" shows "a = b"
rcoset_rel_conv_lcoset_rel: shows "rcoset_rel G = map_prod uminus uminus ` lcoset_rel G"
H_def: shows "(H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. (d::'a::refinement_lattice) \<sqinter> (c::'a::refinement_lattice) ; x)"
rec: fixes f :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> 'a"   and x1 :: "bool"   and x2 :: "bool"   and x3 :: "bool"   and x4 :: "bool"   and x5 :: "bool"   and x6 :: "bool"   and x7 :: "bool"   and x8 :: "bool" shows "rec_char f (Char x1 x2 x3 x4 x5 x6 x7 x8) = f x1 x2 x3 x4 x5 x6 x7 x8"
relpow_Suc_D2: fixes x :: "'b"   and z :: "'b"   and n :: "nat"   and R :: "('b \<times> 'b) set" assumes "(x, z) \<in> R ^^ Suc n" shows "\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n"
choose_strategy: fixes v :: "'a" assumes "v \<in> S" shows "strategy p (choose v)"
conv_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.aggregation_algebra one times uminus inf less_eq less sup bot top plus conv" shows "conv x = x"
rcliff_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and xs :: "(accent \<times> 'b) list" shows "local.rcliff r a b xs = ((\<exists>as b' cs a' bs. snd ` set as \<subseteq> under r a \<inter> under r b \<and> b' \<in> {[(Grave, b)], []} \<and> snd ` set cs \<subseteq> under r b \<and> a' = [(Macron, a)] \<and> snd ` set bs \<subseteq> under r b \<and> xs = as @ b' @ cs @ a' @ bs) \<or> (\<exists>as b' cs. snd ` set as \<subseteq> under r a \<and> b' \<in> {[(Grave, b)], []} \<and> snd ` set cs \<subseteq> under r a \<union> under r b \<and> xs = as @ b' @ cs))"
cases: fixes a :: "trm \<Rightarrow> (char list \<times> trm) list" assumes "a \<in> rctxt" obtains E :: "trm \<Rightarrow> trm" and l :: "char list" and fs :: "(char list \<times> trm) list" where "a = (\<lambda>t. (l__, E__ t) \<Colon> fs)" and "E__ \<in> ctxt"   | v :: "trm" and E :: "trm \<Rightarrow> (char list \<times> trm) list" and l :: "char list" where "a = (\<lambda>t. (l, v) \<Colon> E t)" and "v \<in> value" and "E \<in> rctxt"
Ord_cases_lemma: fixes k :: "V" assumes "Ord k" shows "k = 0 \<or> (\<exists>j. k = ZFC_in_HOL.succ j) \<or> Limit k"
map_2: fixes f :: "'d \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d list" shows "map f (x21 # x22) = f x21 # map f x22"
strong_induct: fixes x1 :: "binding list"   and x2 :: "ty"   and x3 :: "ty"   and P :: "'a \<Rightarrow> binding list \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1\<turnstile>x2<:x3"   and "\<And>\<Gamma> S y. \<lbrakk>\<turnstile> \<Gamma> ok; S closed_in \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> S Top"   and "\<And>\<Gamma> X y. \<lbrakk>\<turnstile> \<Gamma> ok; X \<in> ty_dom \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Tvar X) (Tvar X)"   and "\<And>X S \<Gamma> T y. \<lbrakk>TVarB X S \<in> set \<Gamma>; \<Gamma>\<turnstile>S<:T; \<And>z. P z \<Gamma> S T\<rbrakk> \<Longrightarrow> P y \<Gamma> (Tvar X) T"   and "\<And>\<Gamma> T\<^sub>1 S\<^sub>1 S\<^sub>2 T\<^sub>2 y. \<lbrakk>\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1; \<And>z. P z \<Gamma> T\<^sub>1 S\<^sub>1; \<Gamma>\<turnstile>S\<^sub>2<:T\<^sub>2; \<And>z. P z \<Gamma> S\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (S\<^sub>1 \<rightarrow> S\<^sub>2) (T\<^sub>1 \<rightarrow> T\<^sub>2)"   and "\<And>\<Gamma> T\<^sub>1 S\<^sub>1 X S\<^sub>2 T\<^sub>2 y. \<lbrakk>X \<sharp> y; \<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1; \<And>z. P z \<Gamma> T\<^sub>1 S\<^sub>1; (TVarB X T\<^sub>1 # \<Gamma>)\<turnstile>S\<^sub>2<:T\<^sub>2; \<And>z. P z (TVarB X T\<^sub>1 # \<Gamma>) S\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P y \<Gamma> (\<forall>X<:S\<^sub>1. S\<^sub>2) (\<forall>X<:T\<^sub>1. T\<^sub>2)" shows "P z x1 x2 x3"
f_expand_gr0_f_expand_Suc: fixes k :: "nat"   and xs :: "'b message_af list" assumes "0 < k" shows "xs \<odot> k = xs \<odot>\<^bsub>fSuc\<^esub> (k - Suc 0)"
obj_sumE: fixes s :: "'b + 'c"   and P :: "bool" assumes "\<forall>x. s = Inl x \<longrightarrow> P"   and "\<forall>x. s = Inr x \<longrightarrow> P" shows "P"
1129: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- (x \<squnion> y) \<squnion> z) \<sqinter> x = z \<sqinter> x"
label_const: fixes vs :: "e list"   and n :: "nat"   and es :: "e list" assumes "const_list vs" shows "\<lparr>[Label n es vs]\<rparr> \<leadsto> \<lparr>vs\<rparr>"
rOpen: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and x :: "name"   and C :: "'d"   and y :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P (M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle>) P' A\<^sub>P \<Psi>\<^sub>P"   and "x \<in> supp N"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> A\<^sub>P"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* yvec"   and "xvec \<sharp>* yvec"   and "distinct xvec"   and "distinct yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M"   and "A\<^sub>P \<sharp>* C"   and "x \<sharp> C"   and "xvec \<sharp>* C"   and "yvec \<sharp>* C"   and "y \<noteq> x"   and "y \<sharp> \<Psi>"   and "y \<sharp> P"   and "y \<sharp> M"   and "y \<sharp> xvec"   and "y \<sharp> yvec"   and "y \<sharp> N"   and "y \<sharp> P'"   and "y \<sharp> A\<^sub>P"   and "y \<sharp> \<Psi>\<^sub>P"   and "y \<sharp> C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) (M\<lparr>\<nu>*(xvec @ y # yvec)\<rparr>\<langle>([(x, y)] \<bullet> N)\<rangle>) ([(x, y)] \<bullet> P') (x # A\<^sub>P) \<Psi>\<^sub>P"
hyps_2: fixes T\<^sub>1 :: "POPLmark.type"   and T\<^sub>2 :: "POPLmark.type" assumes "S = (\<forall><:T\<^sub>1. T\<^sub>2)"   and "t \<in> value" shows "\<exists>ta S. t = (\<lambda><:S. ta)"