hcju/isabelleps huggingface dataset:
a_shadow_root_valid_def: fixes get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_shadow_root_valid get_shadow_root get_tag_name h = (\<forall>shadow_root_ptr\<in>fset (shadow_root_ptr_kinds h). \<exists>host\<in>fset (element_ptr_kinds h). |h \<turnstile> get_tag_name host|\<^sub>r \<in> safe_shadow_root_element_types \<and> |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root_ptr)"
tsubstT_simulates_subst_typ: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" shows "subst_typ insts T = tsubstT T (\<lambda>idn S. the_default (Tv idn S) (lookup (\<lambda>x. x = (idn, S)) insts))"
class_of_to_rel: fixes rel :: "('c \<times> 'd, 'e) eq_object_scheme"   and r :: "'c"   and s :: "'d" shows "class_of\<^bsub>rel\<^esub> (r, s) = (r |\<^bsub>rel\<^esub> s)"
assms: shows "f \<in> [UU]\<^bsup>2\<^esup> \<rightarrow> {..<Suc (Suc 0)}"
that: fixes x :: "'a"   and y :: "'a" assumes "sphere (0::'a) r = {x, y} \<and> dist x y = 2 * r" shows "thesis"
id_eq_prop_prop_6_a: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and G :: "\<Pi>\<^sub>2"   and H :: "\<Pi>\<^sub>2" shows "[F \<^bold>= G \<^bold>& G \<^bold>= H \<^bold>\<rightarrow> F \<^bold>= H in v]"
simps_2: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and p :: "real mpoly" shows "linear_substitution var a b (Less p) = (let d = MPoly_Type.degree p var; P = \<Sum>i = 0..<d + 1. isolate_variable_sparse p var i * a ^ i * b ^ (d - i) in Less (P * b ^ (d mod 2)))"
HInit_Inv2a: fixes s :: "state" shows "HInit s \<longrightarrow> Inv2a s"
A: shows "\<forall>\<psi>\<in>\<^bold>G (af\<^sub>G \<phi> [x]). w' \<Turnstile> \<psi> = (xs \<frown> w') \<Turnstile> \<psi>"
assms: shows "\<lbrace>P\<rbrace> c \<lbrace>Q\<rbrace>"
cancels_to_preserves_generators: fixes l :: "(bool \<times> 'c) list"   and l' :: "(bool \<times> 'c) list"   and gens :: "'c set" assumes "cancels_to l l'"   and "l \<in> lists (UNIV \<times> gens)" shows "l' \<in> lists (UNIV \<times> gens)"
is_processT3_SR: fixes s :: "'c event list"   and t :: "'c event list"   and X :: "'c event set"   and P :: "'c process" assumes "(s @ t, X) \<in> F P" shows "(s, {}) \<in> F P"
in_idem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "x \<^bold>* set.F A = set.F A"
iIN_iff: fixes x :: "nat"   and n :: "nat"   and d :: "nat" shows "(x \<in> [n\<dots>,d]) = (n \<le> x \<and> x \<le> n + d)"
leg_simps_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "partial_magma.arr C (Leg0 S)"
primitive_part_const_poly: fixes x :: "'b" shows "primitive_part [:x:] = [:unit_factor x:]"
proj3_addpre: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and pre :: "nat set" shows "\<pi>\<^sub>3 (addpre v pre) = \<pi>\<^sub>3 v"
ext_inject: fixes static :: "bool"   and more :: "'b"   and static' :: "bool"   and more' :: "'b" shows "(\<lparr>static = static, \<dots> = more\<rparr> = \<lparr>static = static', \<dots> = more'\<rparr>) = (static = static' \<and> more = more')"
reachable_induce_subgraphD: fixes S :: "'a set"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"   and "S \<subseteq> verts G" shows "u \<rightarrow>\<^sup>* v"
cblinfun_comp_butterfly: fixes a :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and \<psi> :: "'c"   and \<phi> :: "'a" shows "a o\<^sub>C\<^sub>L butterfly \<psi> \<phi> = butterfly (a *\<^sub>V \<psi>) \<phi>"
type: shows "type_definition Rep_bit0 Abs_bit0 {0..<int CARD('b bit0)}"
cindex_poly_mod: fixes a :: "real"   and b :: "real"   and q :: "real poly"   and p :: "real poly" shows "cindex_poly a b q p = cindex_poly a b (q mod p) p"
relevant_atoms_same_semantics: fixes F :: "'a formula"   and \<A>\<^sub>1 :: "'a \<Rightarrow> bool"   and \<A>\<^sub>2 :: "'a \<Rightarrow> bool" assumes "\<forall>k\<in>atoms F. \<A>\<^sub>1 k = \<A>\<^sub>2 k" shows "\<A>\<^sub>1 \<Turnstile> F = \<A>\<^sub>2 \<Turnstile> F"
MaxR_opt_is_greatest: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "local.MaxR.MaxR_opt A = Some x"   and "y \<in> A \<inter> Field r" shows "(y, x) \<in> r"
fps_lr_inverse_gp_ring1'_2: shows "fps_right_inverse (Abs_fps (\<lambda>n. 1::'a)) (1::'a) = 1 - fps_X"
config_rand_distinct: fixes A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and init :: "'b list"   and qs :: "'b list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). distinct (fst x) = distinct init"
Gamma_residues: fixes n :: "nat" shows "(\<lambda>z. Gamma z * (z + of_nat n)) \<midarrow>- of_nat n\<rightarrow> (- (1::'b)) ^ n / fact n"
compl_connI: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "\<And>d m gs bs ps sps p q data. \<lbrakk>dickson_grading d; fst ` set gs \<subseteq> dgrad_p_set d m; is_Groebner_basis (fst ` set gs); fst ` set bs \<subseteq> dgrad_p_set d m; set ps \<subseteq> set bs \<times> (set gs \<union> set bs); sps \<noteq> []; set sps \<subseteq> set ps; unique_idx (gs @ bs) data; (p, q) \<in> set sps; fst p \<noteq> 0; fst q \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs) \<union> fst ` set (fst (compl gs bs (ps -- sps) sps data))) (fst p) (fst q)" shows "compl_conn compl"
coset_join2: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier R"   and "subgroup H (add_monoid R)"   and "x \<in> H" shows "H #>\<^bsub>add_monoid R\<^esub> x = H"
U_arOf_def: shows "U_arOf \<equiv> length \<circ> TE_arOf"
2: shows "\<forall>y<LEAST x. x < upper \<and> P x. \<not> P y"
update_since_def: fixes args :: "Monitor.args"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and nt :: "nat"   and aux :: "'msaux" shows "update_since args rel1 rel2 nt aux = (let aux0 = join_msaux args rel1 (add_new_ts_msaux args nt aux); aux' = add_new_table_msaux args rel2 aux0 in (result_msaux args aux', aux'))"
f1: shows "\<forall>a\<in>A. \<exists>b\<in>B. map f a \<cong> b"
\<Phi>_FUN_simp: fixes a' :: "'c"   and x :: "'s" assumes "C.ide a'"   and "x \<in> Hom.set (a', a)" shows "\<Phi>.FUN a' x = Cones.FUN (\<psi> (a', a) x) (\<iota> \<chi>)"
inject: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset"   and y1 :: "'b literal multiset multiset"   and y2 :: "'b literal multiset"   and y3 :: "'b literal multiset" shows "(Infer x1 x2 x3 = Infer y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
join_shunt: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y = \<top>) = (- x \<le> y)"
empty: fixes c :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" shows "(c, [], c) \<in> trcl t"
autoref_opt_1: fixes R :: "('a \<times> 'b) set" shows "(None, None) \<in> \<langle>R\<rangle>option_rel"
lit_fun_simps_3: fixes g :: "'e \<Rightarrow> 'f \<Rightarrow> 'c"   and x :: "'e"   and y :: "'f" shows "\<guillemotleft>g x y\<guillemotright> = bop g \<guillemotleft>x\<guillemotright> \<guillemotleft>y\<guillemotright>"
bit_1_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit one n = (one \<noteq> zero \<and> n = 0)"
name_fresh: fixes a :: "name"   and b :: "name" shows "a \<sharp> b = (a \<noteq> b)"
retract_of_UNIV: fixes S :: "'c set" shows "(S retract_of UNIV) = (AR S \<and> closed S)"
ord_1: fixes i :: "nat" shows "g i \<le> f i"
pt_fresh_left_ineq: fixes a :: "'c"   and pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "cp TYPE('a) TYPE('b) TYPE('c)" shows "a \<sharp> pi \<bullet> x = rev pi \<bullet> a \<sharp> x"
left_hcomp_closed: fixes \<mu> :: "'a"   and x :: "'a"   and y :: "'a"   and \<nu> :: "'a"   and c :: "'a"   and d :: "'a" assumes "\<guillemotleft>\<mu> : x \<Rightarrow>\<^sub>S y\<guillemotright>"   and "\<guillemotleft>\<nu> : c \<Rightarrow> d\<guillemotright>"   and "\<nu> \<star> \<mu> \<noteq> null" shows "\<guillemotleft>\<nu> \<star> \<mu> : c \<star> x \<Rightarrow>\<^sub>S d \<star> y\<guillemotright>"
in_var_indep: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b" assumes "x \<bowtie> y" shows "$x \<bowtie> $y"
sup_left_commute: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
simps_4: fixes m :: "'a"   and x :: "nat" shows "parent_node (Actual_in (m, x)) = m"
elims_1: fixes x :: "'a"   and xa :: "'a \<Rightarrow> 'b"   and xb :: "('a \<Rightarrow> 'b option) \<times> ('a \<Rightarrow> 'b option)"   and y :: "bool" assumes "in_bounds x xa xb = y" obtains xc :: "'a" and v :: "'a \<Rightarrow> 'b" and lb :: "'a \<Rightarrow> 'b option" and ub :: "'a \<Rightarrow> 'b option" where "x = xc" and "xa = v" and "xb = (lb, ub)" and "y = (v xc \<ge>\<^sub>l\<^sub>b lb xc \<and> v xc \<le>\<^sub>u\<^sub>b ub xc)"
simps_10: fixes \<phi> :: "'b formula"   and I :: "\<I>"   and \<psi> :: "'b formula" shows "safe_formula (Since \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> (safe_formula \<phi> \<or> (case \<phi> of Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False)) \<and> safe_formula \<psi>)"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "alternate (x # xs) = x # alternate (tl xs)"
eqA: shows "A = adjoint P * (P * A * adjoint P) * P"
step_invariant_arbitraryI: fixes A :: "('e \<times> 'f, 'g) automaton"   and I :: "'g \<Rightarrow> bool"   and P :: "('e \<times> 'f) \<times> 'g \<times> 'e \<times> 'f \<Rightarrow> bool" assumes "\<And>\<xi> p a \<xi>' p'. \<lbrakk>(\<xi>, p) \<in> reachable A I; ((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P ((\<xi>, p), a, \<xi>', p')" shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"
extf_compat_append_left: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and ys :: "('s, 'v) tm list"   and xs :: "('s, 'v) tm list"   and zs :: "('s, 'v) tm list" assumes "extf f gt ys xs" shows "extf f gt (zs @ ys) (zs @ xs)"
serialize_actions_def: fixes pids :: "char list \<Rightarrow> char list"   and a :: "of_action list" shows "serialize_actions pids a \<equiv> if length a = 0 then ''drop'' else (intersperse CHR '','' \<circ> map (serialize_action pids)) a"
phis_in_\<alpha>n: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list" assumes "phis g (n, v) = Some vs" shows "n \<in> set (\<alpha>n g)"
sWlsNE_imp_sWlsValNE: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and x :: "'var" assumes "sWlsNE SEM" shows "sWlsValNE SEM x"
dual_equivalence: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "equivalence_in_bicategory V H \<a> \<i> src trg g f (category.inv V \<epsilon>) (category.inv V \<eta>)"
linorder_on_cases: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "linorder_on A R"   and "x \<in> A"   and "y \<in> A" shows "x = y \<or> (x, y) \<in> R \<and> (y, x) \<notin> R \<or> (y, x) \<in> R \<and> (x, y) \<notin> R"
sup_inf_absorb: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) x (y::'b::type)) = x"
g1_g2_continuous_1: shows "continuous_on (cbox a b) g1"
e1_def: shows "e1 = prj1 (ECC.prod f1 (local.dom f1) \<cdot> ECC.dup (local.dom f1)) (ECC.prod f0 (local.dom f0) \<cdot> ECC.dup (local.dom f0))"
length_vec_to_iarray: fixes xa :: "('d, 'e) vec" shows "IArray.length (vec_to_iarray xa) = CARD('e)"
unit_circle_cmat_def: shows "unit_circle_cmat = (1, 0, 0, - 1)"
prm_set_distributes_difference: fixes \<pi> :: "'b prm"   and S :: "'b set"   and T :: "'b set" shows "\<pi> {$} (S - T) = \<pi> {$} S - \<pi> {$} T"
f_def: shows "f \<equiv> \<lambda>x. if x \<notin> A' then 0::'a else f' x"
Domain_Collect_case_prod: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Domain {(x, y). P x y} = {x. \<exists>y. P x y}"
equivalentFormulaeSymmetry: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list" shows "equivalentFormulae formula1 formula2 = equivalentFormulae formula2 formula1"
W0_eq: shows "W0 = psi -` {..<e} \<inter> U \<times> C"
not_arr_null: shows "\<not> B.VVV.arr B.VVV.null"
g: fixes y :: "'a" shows "mcont luba orda lub_spmf (ord_spmf (=)) (g y)"
tinres_step1: fixes l :: "cell list"   and l' :: "cell list"   and ss :: "nat"   and r :: "cell list"   and t :: "(action \<times> nat) list"   and sa :: "nat"   and la :: "cell list"   and ra :: "cell list"   and sb :: "nat"   and lb :: "cell list"   and rb :: "cell list" assumes "tinres l l'"   and "step0 (ss, l, r) t = (sa, la, ra)"   and "step0 (ss, l', r) t = (sb, lb, rb)" shows "tinres la lb \<and> ra = rb \<and> sa = sb"
preserves_seq: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "C.VV.seq g f" shows "C.seq (fst g \<star>\<^sub>C snd g) (fst f \<star>\<^sub>C snd f)"
assms: shows "\<forall>i\<le>n. f i = 1"
struct_specD4: fixes sel :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "punit.struct_spec sel ap ab compl" shows "punit.compl_struct compl"
right_total_mod_ring_rel32: shows "right_total mod_ring_rel32"
rm_set_lookup_def: fixes rm :: "('c, 'd list) RBT.rbt" shows "rm_set_lookup rm = (\<lambda>a. case rm.\<alpha> rm a of None \<Rightarrow> [] | Some rules \<Rightarrow> rules)"
spaceN_add: fixes x :: "'b"   and N :: "'b quasinorm"   and y :: "'b" assumes "x \<in> space\<^sub>N N"   and "y \<in> space\<^sub>N N" shows "x + y \<in> space\<^sub>N N"
distinct_79: fixes pi1 :: "pi"   and pi2 :: "pi"   and pi1' :: "pi"   and pi2' :: "pi" shows "pi1 \<oplus> pi2 \<noteq> pi1' \<parallel> pi2'"
composable_implies_arr_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "arr \<mu>"
len: shows "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|"
lformula_norm_ACI: fixes \<phi> :: "('a, 'k) aformula" shows "lformula \<langle>\<phi>\<rangle> = lformula \<phi>"
cartesian_monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and \<delta> :: "'b \<Rightarrow> 'b"   and \<tau> :: "'b \<Rightarrow> 'b" shows "cartesian_monoidal_category C T \<alpha> \<iota> \<delta> \<tau> \<equiv> (monoidal_category C T \<alpha> \<iota> \<and> constant_functor C C (monoidal_category.unity C \<iota>) \<and> CartesianMonoidalCategory.diagonal_functor C) \<and> natural_transformation C C (identity_functor.map C) (constant_functor.map C C (monoidal_category.unity C \<iota>)) \<tau> \<and> natural_transformation C C (identity_functor.map C) (T \<circ> (\<lambda>f. if partial_magma.arr C f then (f, f) else partial_magma.null (product_category.comp C C))) \<delta> \<and> cartesian_monoidal_category_axioms C T \<alpha> \<iota> \<delta> \<tau>"
satC_map_gL: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" shows "GE.satC \<xi> (map gL c) = Ik.satC \<xi> c"
lr_fps_denominator'_def: fixes cs :: "'b list" shows "lr_fps_denominator' cs = Poly cs"
coprime_abs_left_iff: fixes k :: "int"   and l :: "int" shows "algebraic_semidom_class.coprime \<bar>k\<bar> l = algebraic_semidom_class.coprime k l"
callee_of_rest_def: fixes rest :: "('a, 'e, 'b, 'c, 'd, 'f, 'g) rest_scheme" shows "callee_of_rest rest = rfunc_adv rest \<oplus>\<^sub>O rfunc_usr rest"
IH: assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "set Es = E"   and "distinct Es" shows "all_security_requirements_fulfilled M (generate_valid_topology_some M Es \<lparr>nodes = V, edges = E\<rparr>)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "Core.term" assumes "\<And>inc n i. P inc n (Bv i)"   and "\<And>inc n T body. P inc (n + 1) body \<Longrightarrow> P inc n (Abs T body)"   and "\<And>inc n f t. \<lbrakk>P inc n f; P inc n t\<rbrakk> \<Longrightarrow> P inc n (f $ t)"   and "\<And>uu_ uv_ v va. P uu_ uv_ (Ct v va)"   and "\<And>uu_ uv_ v va. P uu_ uv_ (Fv v va)" shows "P a0 a1 a2"
has_ticket_simps_2: fixes k :: "nat"   and k' :: "nat" shows "has_ticket (WAIT k) k' = (k' = k)"
spanning_forest_def: fixes f :: "'a"   and g :: "'a" shows "spanning_forest f g \<equiv> forest f \<and> f \<le> - - g \<and> components g \<le> forest_components f \<and> regular f"
lemma1': fixes ts :: "lam list"   and ts' :: "lam list"   and r :: "lam"   and r' :: "lam" assumes "ts [\<rightarrow>\<^sub>s] ts'"   and "r \<rightarrow>\<^sub>s r'" shows "r \<degree>\<degree> ts \<rightarrow>\<^sub>s r' \<degree>\<degree> ts'"
Red_F_\<G>_def: fixes N :: "'f set" shows "local.wf_lift.Red_F_\<G> N = {C. \<forall>D\<in>\<G>_F_q q C. D \<in> Red_F_q q (wf_lift.\<G>_Fset N) \<or> (\<exists>E\<in>N. Prec_F_g D E C \<and> D \<in> \<G>_F_q q E)}"
l2_inv6I: fixes x :: "l2_state" assumes "\<And>Ra K. \<lbrakk>role (guessed_runs Ra) = Init; run_ended (progress x Ra); guessed_frame Ra xsk = Some K\<rbrakk> \<Longrightarrow> Aenc K (epubKF (Ra $ kE)) \<in> extr (bad x) (ik x) (chan x)" shows "x \<in> l2_inv6"
upower_inductr: fixes R :: "('b, 'c) urel"   and Q :: "('b, 'c) urel"   and P :: "'c hrel"   and n :: "nat" assumes "Q \<sqsubseteq> R \<sqinter> (Q ;; P)" shows "Q \<sqsubseteq> R ;; power.power II (;;) P n"
induct: fixes P1 :: "Term.var \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and P3 :: "stmt \<Rightarrow> bool"   and P4 :: "expr list \<Rightarrow> bool"   and expr_list :: "expr list" assumes "\<And>x. P1 (LVar x)"   and "\<And>x1 x2 x3 x4 x5. P2 x4 \<Longrightarrow> P1 ({x1,x2,x3}x4..x5)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P1 (x1.[x2])"   and "\<And>x1 x2. \<lbrakk>P3 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (InsInitV x1 x2)"   and "\<And>x. P2 (NewC x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (New x1[x2])"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1 InstOf x2)"   and "\<And>x. P2 (Lit x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (UnOp x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (BinOp x1 x2 x3)"   and "P2 Super"   and "\<And>x. P1 x \<Longrightarrow> P2 (Acc x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1:=x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (x1 ? x2 : x3)"   and "\<And>x1 x2 x3 x4 x5 x6 x7. \<lbrakk>P2 x4; P4 x7\<rbrakk> \<Longrightarrow> P2 ({x1,x2,x3}x4\<cdot>x5( {x6}x7))"   and "\<And>x1 x2. P2 (Methd x1 x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P2 (Body x1 x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (InsInitE x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Callee x1 x2)"   and "P3 Skip"   and "\<And>x. P2 x \<Longrightarrow> P3 (Expr x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (x1\<bullet> x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P3 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (If(x1) x2 Else x3)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (x1\<bullet> While(x2) x3)"   and "\<And>x. P3 (Jmp x)"   and "\<And>x. P2 x \<Longrightarrow> P3 (Throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P3 x1; P3 x4\<rbrakk> \<Longrightarrow> P3 (Try x1 Catch(x2 x3) x4)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 Finally x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (FinA x1 x2)"   and "\<And>x. P3 (Init x)"   and "P4 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1 # x2)" shows "P4 expr_list"
substitution_lemma: fixes x :: "name"   and y :: "name"   and L :: "lam"   and N :: "lam"   and M :: "lam" assumes "x \<noteq> y"   and "x \<sharp> L" shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"