hcju/isabelleps huggingface dataset:
TopoS_modelLibrary_yields_new_configured_SecurityInvariant: fixes m :: "('a, 'b) TopoS_packed"   and sinvar_spec :: "'a graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and nP :: "'a \<Rightarrow> 'b"   and C :: "('a, 'b) TopoS_Params"   and Spec :: "'a SecurityInvariant_configured" assumes "TopoS_modelLibrary m sinvar_spec"   and "nP = nm_node_props m C"   and "Spec = \<lparr>c_sinvar = \<lambda>G. sinvar_spec G nP, c_offending_flows = \<lambda>G. SecurityInvariant_withOffendingFlows.set_offending_flows sinvar_spec G nP, c_isIFS = nm_receiver_violation m\<rparr>" shows "new_configured_SecurityInvariant (sinvar_spec, nm_default m, nm_receiver_violation m, nP) = Some Spec"
diff_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
at_prm_eq_refl: fixes pi :: "('a \<times> 'a) list" shows " pi \<triangleq> pi "
push_forward_inverse: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "X \<in> src.tangent_space p"   and "p \<in> src.carrier" shows "inv.push_forward (push_forward X) = X"
3: shows "llength (lselect s w) = eSuc (llength (lselect (s - {least s}) w))"
entropy_distr: fixes MX :: "'e measure"   and X :: "'a \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> real" assumes "distributed M MX X (\<lambda>x. ennreal (f x))"   and "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> f x" shows "entropy b MX X = - (LINT x|MX. f x * log b (f x))"
prob_space: fixes M :: "'d measure" assumes "prob_space M" shows "Sigma_Algebra.measure M (space M) = 1"
omega_sup_id: fixes y :: "'a"   and x :: "'a" assumes "(1::'a) \<le> y" shows "x\<^sup>\<omega> \<cdot> y = x\<^sup>\<omega>"
is_order_B: shows "partial_order \<Y>\<^bsub>G\<^esub>"
step_invariantI: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool" assumes "\<And>s a s'. \<lbrakk>s \<in> reachable A I; (s, a, s') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P (s, a, s')" shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"
assms_1: shows "diff_invariant (\<lambda>s. \<nu> s < \<mu> s) f U S t\<^sub>0 G"
map_add_right_eq: fixes h :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" assumes "h ++ h\<^sub>0 = h ++ h\<^sub>1"   and "h\<^sub>0 \<bottom> h"   and "h\<^sub>1 \<bottom> h" shows "h\<^sub>0 = h\<^sub>1"
\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>_\<AA>\<rr>\<ii>\<tt>\<yy>: fixes opubx :: "'g" shows "\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> (\<BB>\<oo>\<xx> opubx)) = length (fst (\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> opubx))"
not_in_path_image_join: fixes x :: "'a"   and g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "x \<notin> path_image g1"   and "x \<notin> path_image g2" shows "x \<notin> path_image (g1 +++ g2)"
put_det: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d"   and \<rho> :: "'e" assumes "pbij_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v"
filterlim_iff_le_filtercomap: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and G :: "'a filter" shows "filterlim f F G = (G \<le> filtercomap f F)"
eq_poly_def: fixes p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" shows "p =p q \<equiv> \<forall>\<alpha>. eval_poly \<alpha> p = eval_poly \<alpha> q"
max_Var_ode_fa: fixes xs :: "floatarith list"   and z :: "nat" assumes "max_Var_floatariths ode_e \<le> length xs"   and "max_Var_floatariths xs \<le> z" shows "max_Var_floatariths (ode_fa xs) \<le> z"
nat_term_order_eq_DEG_dg': fixes to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order"   and ps :: "bool" shows "nat_term_order_eq to1 (DEG to2) True ps = nat_term_order_eq to1 to2 True ps"
simps_4: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and tst :: "'b"   and c :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_WbisT tval aval indis (while tst do c) = (if PL_Indis.compatTst tval indis tst then PL_Indis.SC_WbisT tval aval indis c else noWhile (while tst do c) \<and> PL_Indis.SC_discr aval indis (while tst do c) \<or> PL_Indis.SC_siso tval aval indis (while tst do c))"
simps: fixes uu :: "val"   and v :: "nat"   and uv :: "val"   and u :: "nat"   and r :: "nat" shows "vadd ((uu, v), uv, u) r = v + u + r"
inverse_octo_1_sym: fixes x :: "octo" assumes "x \<noteq> 0" shows "inverse x * x = 1"
finite_funas_term: fixes t :: "('c, 'd) Term.term" shows "finite (funas_term t)"
simps: fixes atom_ordering :: "('a \<times> 'a) set"   and I :: "'a set"   and S :: "'a Literal set set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.validate_formula I S = (\<forall>C. C \<in> S \<longrightarrow> propositional_atoms.validate_clause I C)"
fcomp_def: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "fcomp f g = (\<lambda>x. g (f x))"
B_eq_17: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"   and "2 \<le> i"   and "i \<le> k" shows "subresultant_prs_locale.h \<delta> f i = (- 1) ^ (n 1 + n i + i + 1) / f i * (\<Prod>l\<leftarrow>[3..<Suc (Suc i)]. subresultant_prs_locale.\<alpha> \<delta> f l / \<beta> l)"
subtype_transitivity: fixes \<Gamma> :: "binding list"   and S :: "ty"   and Q :: "ty"   and T :: "ty" assumes "\<Gamma>\<turnstile>S<:Q"   and "\<Gamma>\<turnstile>Q<:T" shows "\<Gamma>\<turnstile>S<:T"
hyps_1: fixes s :: "'a"   and n :: "nat"   and t :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile> \<langle>merge_guards c1,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> =n\<Rightarrow> t"
simps: fixes s :: "('i, 'a) state" shows "check s = (if \<U> s then s else let x\<^sub>i' = min_lvar_not_in_bounds s in case x\<^sub>i' of None \<Rightarrow> s | Some x\<^sub>i \<Rightarrow> let dir = if \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i then Positive else Negative in check (check' dir x\<^sub>i s))"
map_vector_def: fixes f :: "'d \<Rightarrow> 'b"   and v :: "('d, 'c) Finite_Cartesian_Product.vec" shows "map_vector f v \<equiv> \<chi>i. f (v $h i)"
sort_of_atom_eq: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and atom :: "'a \<Rightarrow> atom"   and a :: "'a"   and b :: "'a" assumes "class.at permute atom" shows "sort_of (atom a) = sort_of (atom b)"
empty_collapse_empty: fixes s :: "nat \<Rightarrow> 'b" assumes "emptyseq s" shows "emptyseq (\<natural> s)"
strip_comb_empty: fixes t :: "'a" assumes "snd (strip_comb t) = []" shows "fst (strip_comb t) = t"
height_Unbal_l: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and l :: "'a tree"   and l' :: "'a tree"   and y :: "'a"   and r :: "'a tree" assumes "ins n (d + 1) x l = Unbal l'"   and "bal_i n (height \<langle>l, y, r\<rangle> + d)" shows "height r < height l'"
indaut_fixes_fundfacet: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "fixespointwise (OpposedThinChamberComplexFoldings.induced_automorphism X f g) (C0 \<inter> g ` C0)"
induced_induce: fixes vs :: "'a set" assumes "vs \<subseteq> verts (G \<restriction> {v. u \<rightarrow>\<^sup>* v})" shows "induced_subgraph ((G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<restriction> vs) (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"
bisim_can_sync_preserve1: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2"   and LT :: "('l + 't + 't) set" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "t \<turnstile> \<langle>x1, m1\<rangle> LT \<wrong>1" shows "t \<turnstile> \<langle>x2, m2\<rangle> LT \<wrong>2"
translation_diff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and s :: "'c set"   and t :: "'c set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (s - t) = plus a ` s - plus a ` t"
naturality: fixes f :: "'b \<times> 'b" assumes "B.VV.arr f" shows "\<Phi>\<^sub>F (B.VV.cod f) \<cdot>\<^sub>C F.H\<^sub>DoFF.map f = F.FoH\<^sub>C.map f \<cdot>\<^sub>C \<Phi>\<^sub>F (B.VV.dom f)"
\<omega>consistent_implies_consistent: assumes "B.\<omega>consistent" shows "B.consistent"
ideal_deg_shifts_pp: fixes xs :: "'x list"   and d :: "nat"   and fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "ideal (set (deg_shifts_pp xs d fs)) = ideal (set fs)"
rel_spmf_measureD: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'d spmf"   and A :: "'c set" assumes "rel_spmf R p q" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> Sigma_Algebra.measure (measure_spmf q) {y. \<exists>x\<in>A. R x y}"
exponent_normal_of_Float: fixes x :: "Float.float" assumes "is_normal_Float TYPE(('a, 'b) IEEE.float) x" shows "IEEE.exponent (normal_of_Float x) = nat (Float.exponent x + int (bias TYPE(('a, 'b) IEEE.float)) + bitlen \<bar>mantissa x\<bar> - 1)"
stable_pair_on_allocation: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "inj_on Xd (match XD_XH)"
Zorn_Lemma2: fixes A :: "'b set set" assumes "\<forall>C\<in>chains A. \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U" shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
simps_1: fixes p :: "(real, 'a) vec" shows "invar (Leaf p) = True"
plus_absorb2: fixes g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "g \<in> o[F](f)" shows "o[F](\<lambda>x. f x + g x) = o[F](f)"
CONS_def: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" shows "CONS M N = In1 (Scons M N)"
is_empty_default: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "\<not> is_empty (Mapping.default k v m)"
liftM_def: fixes f :: "'b \<Rightarrow> 'a"   and ma :: "'b Heap" shows "liftM f ma = ma \<bind> (\<lambda>a. return (f a))"
pos_norm_vec_def: fixes n :: "nat"   and D :: "'b vec set"   and x :: "'b vec" shows "gram_schmidt.pos_norm_vec n D x = (let c' = gram_schmidt.normal_vector n D; c = if (0::'b) < c' \<bullet> x then c' else - c' in c)"
low_eq_strong_def: fixes s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" shows "low_eq_strong s s' \<equiv> s ''high'' \<noteq> s' ''high'' \<and> low_eq s s'"
distinct_5: fixes x2 :: "'b"   and x3 :: "'c" shows "In2 x2 \<noteq> In3 x3"
nos_inc_simp: fixes \<xi> :: "state"   and \<xi>' :: "state" shows "nos_inc \<xi> \<xi>' = (no \<xi> \<le> no \<xi>')"
harry_meas_False_True: fixes xs :: "bool list" shows "harry_meas (False # xs @ [True]) = harry_meas xs + 2 * length xs + 3"
alpha_Tree_eqvt_aux: fixes t :: "('a, 'b, 'c) Tree"   and p :: "perm" assumes "\<And>a b. ((a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> t) = (p \<bullet> (a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> p \<bullet> t)" shows "p \<bullet> supp_rel (=\<^sub>\<alpha>) t = supp_rel (=\<^sub>\<alpha>) (p \<bullet> t)"
assms_2: shows "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"
delete_edge_is_path: fixes G :: "('c, 'd) graph"   and x :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and y :: "'c"   and a :: "'c"   and b :: "'d"   and c :: "'c" assumes "valid_graph G"   and "is_path_undir G x p y"   and "(a, b, c) \<notin> set p"   and "(c, b, a) \<notin> set p" shows "is_path_undir (delete_edge a b c G) x p y"
squarefree_part_unit: fixes n :: "'a" assumes "is_unit n" shows "squarefree_part n = n"
BSD_implies_SD_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>' :: "'b V_rec" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "BSD \<V>' Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "SD \<V> Tr\<^bsub>ES\<^esub>"
freshAbs_asAbs_qFreshAbs: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and xs :: "'varSort"   and x :: "'var" assumes "qGoodAbs qA" shows "freshAbs xs x (asAbs qA) = qFreshAbs xs x qA"
lunit'_in_hom_2: fixes f :: "'b" assumes "D.ide f" shows "\<guillemotleft>\<l>\<^sub>D\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>D trg\<^sub>D f \<star>\<^sub>D f\<guillemotright>"
append_take_drop_id: fixes n :: "nat"   and xs :: "'a list" shows "take n xs @ drop n xs = xs"
extensible_def: fixes R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph"   and G :: "('d, 'f) labeled_graph"   and f :: "('e \<times> 'f) set" shows "extensible R G f \<equiv> \<exists>g. graph_homomorphism (snd R) G g \<and> agree_on (fst R) f g"
disjoint_effects_no_effects: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "\<forall>a. ListMem a as \<longrightarrow> fmdom' (fmrestrict_set vs (snd a)) = {}" shows "fmrestrict_set vs (exec_plan s as) = fmrestrict_set vs s"
degree_eq_univariate_degree: fixes p :: "'a mpoly"   and v :: "nat" shows "MPoly_Type.degree p v = (if p = 0 then 0 else Polynomial.degree (mpoly_to_nested_poly p v))"
map_filter_def: fixes f :: "'b \<Rightarrow> 'a option"   and xs :: "'b list" shows "List.map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
star_class_defs_6: shows "(*) \<equiv> *f2* (*)"
l3_inv2_def: shows "l3_inv2 \<equiv> {s. Keys_bad (ik s) (l3_state.bad s)}"
UN_lessThan_UNIV: shows "\<Union> (range lessThan) = UNIV"
snocs_is_append: fixes xs :: "'b list"   and ys :: "'b list" shows "snocs xs ys = xs \<cdot> ys"
mtf2_moves_to_frontm1: fixes xs :: "'b list"   and q :: "'b" assumes "distinct xs"   and "q \<in> set xs" shows "index (mtf2 (length xs - 1) q xs) q = 0"
le_cases3: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.linorder less_eq less" obtains "less_eq x y" and "less_eq y z"   | "less_eq y x" and "less_eq x z"   | "less_eq x z" and "less_eq z y"   | "less_eq z y" and "less_eq y x"   | "less_eq y z" and "less_eq z x"   | "less_eq z x" and "less_eq x y"
R1_idem: shows "R1 \<circ> R1 = R1"
trss_drop_context: fixes s :: "'a list"   and c :: "'a list multiset"   and ce :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c + ce), w, s', c' + ce) \<in> trcl (trss fg)" shows "((s, c), w, s', c') \<in> trcl (trss fg) \<and> mon_w fg w \<inter> mon_c fg ce = {}"
af_respectfulness_2: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and \<nu> :: "'b set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "af_letter \<phi> \<nu> \<equiv>\<^sub>P af_letter \<psi> \<nu>"
assms_1: shows "\<forall>f\<in>set fs. irreducible G f"
compat_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubseteq> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubset> z"
substs_app: fixes t :: "trm"   and k :: "nat"   and ts :: "trm list"   and us :: "trm list" shows "t[k \<mapsto>\<^sub>s ts @ us] = t[k + \<parallel>us\<parallel> \<mapsto>\<^sub>s ts][k \<mapsto>\<^sub>s us]"
numeral_le_\<omega>: fixes n :: "num" shows "numeral n \<le> \<omega>"
nat_not_less: fixes m :: "nat"   and n :: "nat" assumes "\<not> m < n" shows "n \<le> m"
assms_2: shows "set (bvars\<^sub>s\<^sub>t\<^sub>p x) \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
neg_invariant_iff_compl_pos_invariant: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.negatively_invariant f X M = auto_ll_on_open.positively_invariant f X (X - M)"
fin_dim_2: shows "vectorspace.fin_dim class_ring (vs (local.span {vec n f})\<lparr>carrier := local.span (set (cols A))\<rparr>)"
changes_gt_smods_ext_def: fixes a :: "real"   and p :: "real poly"   and q :: "real poly" shows "changes_gt_smods_ext a p q = (let ps = smods_ext p q in changes_poly_at ps a - changes_poly_pos_inf ps)"
freshEnv_idEnv: fixes xs :: "'a"   and x :: "'b" shows "freshEnv xs x idEnv"
simps_2: fixes m :: "nat"   and a :: "'a" shows "repeat (Suc m) a = a # repeat m a"
set_fmap_inv2: fixes x :: "'g \<times> 'h"   and F :: "'g \<Rightarrow> 'h option" assumes "fst x \<notin> dom F" shows "insert x (set_fmap F) = set_fmap (F(fst x \<mapsto> snd x))"
sum_closure: fixes a :: "i"   and b :: "i" shows "\<exists>z. \<forall>w. O w z = (O w a \<or> O w b)"
l_set_tag_name_get_tag_name_axioms_def: fixes 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"   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" shows "l_set_tag_name_get_tag_name_axioms get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs \<equiv> (\<forall>h element_ptr tag h'. h \<turnstile> set_tag_name element_ptr tag \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_tag_name element_ptr \<rightarrow>\<^sub>r tag) \<and> (\<forall>ptr ptr' w h h' r. ptr \<noteq> ptr' \<longrightarrow> w \<in> set_tag_name_locs ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_tag_name_locs ptr' \<longrightarrow> r h h')"
Qg_one: fixes N :: "'a set" assumes "G \<triangleright> N" shows "\<one>\<^bsub>G / N\<^esub> = N"
push_precondI_nz: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and v :: "nat" assumes "discharge_invar c s t u fo lo no f l n"   and "Network.excess c f u \<noteq> (0::'a)"   and "(u, v) \<in> Graph.E (residualGraph c f)"   and "l u = l v + 1" shows "Network.push_precond c f l (u, v)"
intro: fixes \<gamma> :: "'b \<Rightarrow> int set"   and num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Val_abs \<gamma> num' plus'" shows "Gamma num' plus' \<gamma>"
reachable_mode_states_def: fixes gc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "reachable_mode_states gc \<equiv> {mdss. \<exists>cms' mem' sched. gc \<rightarrow>\<^bsub>sched\<^esub> (cms', mem') \<and> map snd cms' = mdss}"
flatten_sigma_2_intersection: fixes M :: "'b ltln set"   and \<phi> :: "'b ltln"   and S :: "'b ltln set" assumes "M \<inter> subformulas\<^sub>\<mu> \<phi> \<subseteq> S" shows "\<phi>[M \<inter> S]\<^sub>\<Sigma>\<^sub>2 = \<phi>[M]\<^sub>\<Sigma>\<^sub>2"
finite_valuations: fixes V :: "nat set"   and t0 :: "(nat \<times> linear_poly) list"   and v0 :: "nat \<Rightarrow> 'a"   and bl0 :: "nat \<Rightarrow> 'a"   and bu0 :: "nat \<Rightarrow> 'a" assumes "finite V" shows "finite {v. check_valuation v v0 bl0 bu0 t0 V}"
kruskal_exists_minimal_spanning: fixes g :: "'a" assumes "symmetric g" shows "\<exists>f. minimum_spanning_forest f g"
zipWithS_defined: fixes a :: "('b, 'c) Stream"   and b :: "('d, 'e) Stream"   and f :: "'b \<rightarrow> 'd \<rightarrow> 'f" assumes "a \<noteq> \<bottom>"   and "b \<noteq> \<bottom>" shows "zipWithS\<cdot>f\<cdot>a\<cdot>b \<noteq> \<bottom>"
PP_deff: fixes t :: "'b" shows "PP t = subst P t xx"
ins3_ins2: fixes n :: "nat"   and t :: "'a tree"   and d :: "nat"   and x :: "'a" assumes "bal_i n (height t + d)" shows "ins3 n d x t = ins2 n d x t"
f': shows "f' = of_int_poly_i ff_ops (Mp f)"
assms: shows "x \<sharp> \<alpha> \<prec> P"
spr_repRels_def: fixes envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'e \<times> 'd option" shows "SPRViewDet.spr_repRels envObs \<equiv> \<lambda>a. rel_ext (\<lambda>((u, v), u', v'). envObs a u = envObs a u' \<and> envObs a v = envObs a v')"
liveVal'D: fixes g :: "'g"   and vs :: "'val list"   and v :: "'val"   and vs' :: "'val list" assumes "liveVal' g vs"   and "vs = v # vs'" shows "liveVal g v"
prefix_butlastD: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs (butlast ys)" shows "prefix xs ys"
nlex_add: fixes a :: "real \<times> real"   and b :: "real \<times> real" assumes "Counterclockwise_2D_Arbitrary.lex a 0"   and "Counterclockwise_2D_Arbitrary.lex b 0" shows "Counterclockwise_2D_Arbitrary.lex (a + b) 0"
intro: fixes S :: "('a \<times> 'a) set" assumes "SN S" shows "SN_ars S"
scale_right_distrib: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
zip_with_swap: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'d list" shows "zip_with (\<lambda>x y. f y x) xs ys = zip_with f ys xs"
left_congruent: shows "(\<lambda>U. msgrel `` {freeleft U}) respects msgrel"
member_product: fixes x :: "'a \<times> 'b"   and A :: "'a set"   and B :: "'b set" shows "(x \<in> Product_Type.product A B) = (x \<in> A \<times> B)"
scaleR_one: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR 1 x = x"
non_terminating_refine_CHAOS: fixes P :: "'b process" shows "non_terminating P = CHAOS UNIV \<sqsubseteq>\<^sub>T P"
rep_eq: fixes x :: "nat"   and xa :: "uint16" shows "Rep_uint16 (push_bit x xa) = push_bit x (Rep_uint16 xa)"
has_density_cong: fixes N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_density M N f = has_density M N g"
itypeI: fixes c :: "'c"   and I :: "interface" shows "c ::\<^sub>i I"
assms: shows "[a \<^bold>= (\<^bold>\<iota>y. \<lparr>G,y\<^sup>P\<rparr>) in v]"
geodesic_segment_dist: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and a :: "'b" assumes "geodesic_segment_between G x y"   and "a \<in> G" shows "dist x a + dist a y = dist x y"
elims: fixes x :: "'a le_constraint"   and xa :: "'a le_constraint"   and y :: "'a le_constraint" assumes "x + xa = y" obtains r1 :: "le_rel" and p1 :: "linear_poly" and c1 :: "'a" and r2 :: "le_rel" and p2 :: "linear_poly" and c2 :: "'a" where "x = Le_Constraint r1 p1 c1" and "xa = Le_Constraint r2 p2 c2" and "y = Le_Constraint (r1 + r2) (p1 + p2) (c1 + c2)"
IH: fixes p :: "'a poly" shows "poly ((pderiv ^^ n) ([:- c, 1::'a:] ^ n * p)) c = fact n * poly p c"
top_dres_def: shows "dFAIL \<equiv> dFAILi"
negligible_subset: fixes s :: "'c set"   and t :: "'c set" assumes "negligible s"   and "t \<subseteq> s" shows "negligible t"
fold_graph_Or\<^sub>n: fixes \<Phi> :: "'b ltln set" assumes "finite \<Phi>" shows "fold_graph Or_ltln false\<^sub>n \<Phi> (Or\<^sub>n \<Phi>)"
h: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "N #>\<^bsub>G\<^esub> x = N #>\<^bsub>G\<^esub> y" shows "h x = h y"
sums: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "X sums a" shows "(\<lambda>n. f (X n)) sums f a"
diff_delta_arctan_upper_42_def: shows "diff_delta_arctan_upper_42 \<equiv> \<lambda>x. - 16384 * x ^ 16 / ((35 * x ^ 8 + 1260 * x ^ 6 + 6930 * x ^ 4 + 12012 * x\<^sup>2 + 6435)\<^sup>2 * (1 + x\<^sup>2))"
col_rot_CW: fixes P :: "Points"   and Q :: "Points"   and R :: "Points" assumes "col P Q R" shows "col R P Q"