hcju/isabelleps huggingface dataset:
wls_templateInduct_fresh: fixes vars :: "'varSort \<Rightarrow> 'var set"   and terms :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set set"   and abs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set set"   and envs :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option) set"   and rel :: "'sort \<Rightarrow> (('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set) set"   and phi :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "\<And>xs us s. (|vars xs| <o |UNIV| \<or> finite (vars xs)) \<and> (|terms s| <o |UNIV| \<or> finite (terms s)) \<and> (|abs (us, s)| <o |UNIV| \<or> finite (abs (us, s))) \<and> (\<forall>X\<in>terms s. wls s X) \<and> (\<forall>A\<in>abs (us, s). wlsAbs (us, s) A) \<and> (|envs| <o |UNIV| \<or> finite envs) \<and> Ball envs wlsEnv"   and "\<And>s X Y. \<lbrakk>wls s X; (X, Y) \<in> rel s\<rbrakk> \<Longrightarrow> wls s Y \<and> skel Y = skel X"   and "\<And>xs x. phi (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; wlsBinp delta binp; liftAll2 phi (arOf delta) inp; liftAll2 (\<lambda>(us, s). phiAbs (us, s)) (barOf delta) binp\<rbrakk> \<Longrightarrow> phi (stOf delta) (Op delta inp binp)"   and "\<And>s xs x X. \<lbrakk>isInBar (xs, s); wls s X; x \<notin> vars xs; \<And>s' Y. Y \<in> terms s' \<Longrightarrow> fresh xs x Y; \<And>xs' s' A. A \<in> abs (xs', s') \<Longrightarrow> freshAbs xs x A; \<And>rho. rho \<in> envs \<Longrightarrow> freshEnv xs x rho; \<And>Y. (X, Y) \<in> rel s \<Longrightarrow> phi s Y\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (Abs xs x X)" shows "(wls s X \<longrightarrow> phi s X) \<and> (wlsAbs (xs, s') A \<longrightarrow> phiAbs (xs, s') A)"
nat_to_string_simps_2: fixes n :: "nat" assumes "\<not> is_digit n" shows "nat_to_string n = nat_to_string (n div 10) @ [digit_to_char (n mod 10)]"
subo_no_path: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C' :: "char list"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and Ds :: "char list list" assumes "Subobjs P C' (Cs @ C # Cs')"   and "wf_prog wf_md P"   and "Cs' \<noteq> []" shows "\<not> P \<turnstile> Path last Cs' to C via Ds "
lm053: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "argmax f A \<subseteq> f -` {Max (f ` A)}"
FP: fixes f :: "'a \<Rightarrow> 'a" assumes "continuous_on S f"   and "f ` S \<subseteq> S" shows "\<exists>x\<in>S. f x = x"
simps_1: fixes a :: "'a" shows "local.plus_oracle (Inl a) = map_spmf (apfst Inl) (left s a)"
pred_entailment_trans: fixes P :: "'BExp set"   and P' :: "'BExp set"   and P'' :: "'BExp set" assumes "P \<turnstile> P'"   and "P' \<turnstile> P''" shows "P \<turnstile> P''"
nat_geSucSuc0_conv: fixes n :: "nat" shows "(Suc (Suc 0) \<le> n) = (n \<noteq> 0 \<and> n \<noteq> Suc 0)"
SN_on_subset2: fixes A :: "'b set"   and B :: "'b set"   and r :: "('b \<times> 'b) set" assumes "A \<subseteq> B"   and "SN_on r B" shows "SN_on r A"
even_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
is_module: fixes W :: "'c set" assumes "subspace K W V" shows "LinearCombinations.submodule K W V"
wfE_min: fixes R :: "('e \<times> 'e) set"   and x :: "'e"   and Q :: "'e set" assumes "wf R"   and "x \<in> Q" obtains z :: "'e" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
separated_insert: fixes g :: "graph"   and a :: "nat"   and V :: "nat set" assumes "minGraphProps g"   and "a \<in> \<V> g"   and "V \<subseteq> \<V> g"   and "separated g V"   and "\<And>f. f \<in> set (facesAt g a) \<Longrightarrow> f \<bullet> a \<notin> V"   and "\<And>f. \<lbrakk>f \<in> set (facesAt g a); |vertices f| \<le> 4\<rbrakk> \<Longrightarrow> \<V> f \<inter> V \<subseteq> {a}" shows "separated g (insert a V)"
cnj_diff: fixes x :: "octo"   and y :: "octo" shows "Octonions.cnj (x - y) = Octonions.cnj x - Octonions.cnj y"
finite_UNIV_uprod_def: shows "finite_UNIV = Phantom('a uprod) (of_phantom finite_UNIV)"
L2P_P2L: fixes p :: "proj2" shows "L2P (P2L p) = p"
m_comm: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y = y \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x"
sum_distrib_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.semiring_0 plus zero times" shows "times r (comm_monoid_add.sum plus zero f A) = comm_monoid_add.sum plus zero (\<lambda>n. times r (f n)) A"
simps: fixes P :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and A :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and Q :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" shows "diagonalize_2x2_aux (P, A, Q) = (let a = A $h 0 $h 0; b = A $h 0 $h 1; c = A $h 1 $h 0; d = A $h 1 $h 1 in if a \<noteq> (0::'b) \<and> \<not> a dvd b then let bezout_mat = Finite_Cartesian_Product.transpose (bezout_matrix (Finite_Cartesian_Product.transpose A) 0 1 0 euclid_ext2) in diagonalize_2x2_aux (P, A ** bezout_mat, Q ** bezout_mat) else if a \<noteq> (0::'b) \<and> \<not> a dvd c then let bezout_mat = bezout_matrix A 0 1 0 euclid_ext2 in diagonalize_2x2_aux (bezout_mat ** P, bezout_mat ** A, Q) else let Q' = column_add (Finite_Cartesian_Product.mat (1::'b)) 1 0 (- (b div a)); P' = row_add (Finite_Cartesian_Product.mat (1::'b)) 1 0 (- (c div a)) in (P' ** P, P' ** A ** Q', Q ** Q'))"
Compl_eq: fixes A :: "'m set" shows "- A = {x. x \<notin> A}"
N: shows "\<bar>lnorm (lebesgue_on S) p ((\<lambda>x. max (- real N) (min (real N) (f x))) - f)\<bar> < e"
GrD1: fixes x :: "'e"   and fx :: "'f"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "(x, fx) \<in> BNF_Def.Gr A f" shows "x \<in> A"
all_interval_int_def: fixes P :: "int \<Rightarrow> bool"   and i :: "int"   and j :: "int" shows "List.all_interval_int P i j = (\<forall>k\<in>{i..j}. P k)"
intro: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool" assumes "FWdelay_bisimulation_final_base final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2"   and "FWdelay_bisimulation_obs_axioms final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"
JF1set_simps: fixes a :: "'d JF1" shows "JF1set a = F1set1 (dtor1 a) \<union> (\<Union> (JF1set ` F1set2 (dtor1 a)) \<union> \<Union> (JF2set ` F1set3 (dtor1 a)))"
divisor_sum_def: fixes n :: "nat" shows "divisor_sum n = \<Sum> {d. d dvd n}"
cod_simps_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "identity_arrow_of_spans C \<mu>" shows "Leg0 (Cod \<mu>) = Leg0 (Dom \<mu>)"
Mp_shrink_modulus: fixes m :: "int"   and k :: "int"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod.Mp (m * k) f = poly_mod.Mp (m * k) g"   and "k \<noteq> 0" shows "poly_mod.Mp m f = poly_mod.Mp m g"
strong_induct: fixes P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a"   and lam :: "lam" assumes "\<And>name z. P z (Var name)"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z (lam1 \<degree> lam2)"   and "\<And>name lam z. \<lbrakk>name \<sharp> z; \<And>z. P z lam\<rbrakk> \<Longrightarrow> P z (Lam [name].lam)" shows "P z lam"
Minkowski_inequality_le_1_1: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "0 < p"   and "p \<le> 1"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"   and "integrable M (\<lambda>x. \<bar>g x\<bar> powr p)" shows "integrable M (\<lambda>x. \<bar>f x + g x\<bar> powr p)"
assms_7: shows "quicksort_post R h lo (p - Suc 0) xs' xs''"
prf_encrypt_oracle_def: fixes x :: "unit"   and plain :: "bool list" shows "prf_encrypt_oracle x plain = lift_spmf (spmf_of_set (nlists UNIV len)) \<bind> (\<lambda>r. Generative_Probabilistic_Value.Pause r (\<lambda>pad. Generative_Probabilistic_Value.Done ((r, plain [\<oplus>] pad), ())))"
fds_abs_converges: fixes f :: "'b fds"   and s :: "'b" assumes "abs_conv_abscissa f < ereal (s \<bullet> (1::'b))" shows "fds_abs_converges f s"
mcontI: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and luba :: "'c set \<Rightarrow> 'c"   and lubb :: "'d set \<Rightarrow> 'd" assumes "monotone orda ordb f"   and "cont luba orda lubb ordb f" shows "mcont luba orda lubb ordb f"
qAFresh_qSwap_compose: fixes zs :: "'d"   and y :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and z :: "'e"   and x :: "'e" assumes "qAFresh zs y X"   and "qAFresh zs z X" shows "X #[[y \<and> x]]_zs #[[z \<and> y]]_zs = X #[[z \<and> x]]_zs"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(SKIP {x1} = SKIP {y1}) = (x1 = y1)"
divides_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a divides b"   and "b divides c"   and "a \<in> carrier G" shows "a divides c"
cases: fixes a1 :: "(char list \<times> trm) list"   and a2 :: "(char list \<times> trm) list" assumes "a1 [\<longmapsto>] a2" obtains t :: "trm" and t' :: "trm" and l :: "char list" and fs :: "(char list \<times> trm) list" where "a1 = (la__, t) \<Colon> fsb__" and "a2 = (la__, t') \<Colon> fsb__" and "t \<longmapsto> t'"   | v :: "trm" and fs :: "(char list \<times> trm) list" and fs' :: "(char list \<times> trm) list" and l :: "char list" where "a1 = (l, v) \<Colon> fs" and "a2 = (l, v) \<Colon> fs'" and "v \<in> value" and "fs [\<longmapsto>] fs'"
monoid_mult_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "monoid_mult_hom hom \<equiv> one_hom hom \<and> semigroup_mult_hom hom"
path_nret_ex_nipd: fixes \<pi> :: "nat \<Rightarrow> 'n" assumes "is_path \<pi>"   and "\<forall>i. \<pi> i \<noteq> local.return" shows "\<forall>i. \<exists>j\<ge>i. \<forall>k>j. \<pi> k \<noteq> ipd (\<pi> j)"
refl: fixes S :: "('d, 'e) eq_object_scheme"   and x :: "'d" assumes "equivalence S"   and "x \<in> carrier S" shows "x .=\<^bsub>S\<^esub> x"
dense: fixes x :: "'c"   and y :: "'c" assumes "x < y" shows "\<exists>z>x. z < y"
inj: fixes f :: "'c \<Rightarrow> 'c" assumes "bijection f" shows "inj f"
assms_3: shows "\<forall>p. prop_invs (prop_config c0 p)"
crename_cfresh: fixes a :: "coname"   and b :: "coname"   and M :: "trm" assumes "a \<sharp> b" shows "a \<sharp> M[a\<turnstile>c>b]"
rep_eq: fixes x :: "complex"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "cblinfun_apply (x *\<^sub>C xa) = (\<lambda>xb. x *\<^sub>C cblinfun_apply xa xb)"
simps_6: fixes var :: "nat"   and \<phi> :: "atom fm"   and \<Gamma> :: "real list" shows "convert_fm var (Neg \<phi>) \<Gamma> = None"
comp_char: fixes g :: "'a"   and f :: "'a" shows "g \<cdot>\<^sub>S f = (if S.arr f \<and> S.arr g \<and> seq g f then g \<cdot> f else null)"
mset_single_iff: fixes xs :: "'c list"   and x :: "'c" shows "(mset xs = {#x#}) = (xs = [x])"
sgn_zero: shows "sgn (0::'a) = (0::'a)"
surj_hom_normal_subgroup: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and F :: "('e, 'f) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e" assumes "H \<lhd> G"   and "group_hom G F \<phi>"   and "\<phi> ` carrier G = carrier F" shows "\<phi> ` H \<lhd> F"
exhaust_sel: fixes msstream :: "'a msstream" obtains "msstream = MSSCons (un_MSSCons1 msstream) (un_MSSCons2 msstream)"
positively_invariant_left_inter: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.positively_invariant f X C"   and "\<forall>x\<in>C \<inter> D. auto_ll_on_open.trapped_forward f X x D" shows "auto_ll_on_open.positively_invariant f X (C \<inter> D)"
atom_map_minus_pres: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot (minus x y) = bounded_lattice.atom_map less_eq less bot x - bounded_lattice.atom_map less_eq less bot y"
strict_def: fixes f :: "'c \<Rightarrow> 'd" shows "strict f \<equiv> f bot = bot"
compliant_stateful_ACS_no_side_effects': fixes \<T> :: "'a stateful_policy"   and G :: "'a graph"   and M :: "'a SecurityInvariant_configured list" assumes "stateful_policy_compliance \<T> G M" shows "\<forall>E\<subseteq>backflows (flows_state \<T>). \<forall>F\<in>get_offending_flows (get_ACS M) \<lparr>nodes = hosts \<T>, edges = flows_fix \<T> \<union> flows_state \<T> \<union> E\<rparr>. F \<subseteq> E"
assms_2: shows "a \<in>\<^sub>\<circ> x"
cyclic_group_def: fixes G :: "('c, 'd) cyclic_group_scheme" shows "cyclic_group G \<equiv> Group.group G \<and> cyclic_group_axioms G"
ncols_le: fixes A :: "'a matrix"   and n :: "nat" shows "(ncols A \<le> n) = (\<forall>j i. n \<le> i \<longrightarrow> Rep_matrix A j i = (0::'a))"
init_bounds_2: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "\<B>\<^sub>i\<^sub>u (init t) = Mapping.empty"
beukers_aux_by_parts_aux: fixes x :: "real"   and z :: "real"   and k :: "nat" assumes "x \<in> {0<..<1}"   and "z \<in> {0<..<1}"   and "k \<le> n" shows "LBINT y=0..1. Q n y * (1 / (1 - (1 - x * y) * z)) = LBINT y=0..1. Q (n - k) y * (fact k * (x * z) ^ k / (1 - (1 - x * y) * z) ^ (k + 1))"
lincomb_smult: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and c :: "'a" assumes "A \<subseteq> carrier V"   and "a \<in> A \<rightarrow> carrier K"   and "c \<in> carrier K" shows "lincomb (\<lambda>w. c \<otimes>\<^bsub>K\<^esub> a w) A = c \<odot>\<^bsub>V\<^esub> lincomb a A"
cases: fixes a :: "'a stream" assumes "run a" obtains x :: "'a" and y :: "'a" and xs :: "'a stream" where "a = x ## y ## xs" and "E x y" and "run (y ## xs)"
prog_3: shows "let f = multi_arith_prog k start steps in \<forall>c\<in>{0..<k} \<rightarrow> {0..l}. \<forall>s<k. (\<forall>j\<le>s. c j < l) \<longrightarrow> col (f c) = col (f (\<lambda>i. if i \<le> s then 0 else c i))"
widen_null: fixes P :: "'a prog"   and C :: "String.literal" shows "P \<turnstile> NT \<le> Class C"
eq_cpayload: fixes x :: "cmsg"   and y :: "cmsg" assumes "eq x y"   and "x \<in> cpayload" shows "y \<in> cpayload"
exi_\<sigma>: shows "prv (exi out (subst \<sigma> (Var zz) inp))"
rhs_step_bind_RES: fixes x' :: "'b"   and X' :: "'b set"   and m :: "'c nres"   and R :: "('c \<times> 'd) set"   and f' :: "'b \<Rightarrow> 'd nres" assumes "x' \<in> X'"   and "m \<le> \<Down> R (f' x')" shows "m \<le> \<Down> R (RES X' \<bind> f')"
geodesic_segment_param_2: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "geodesic_segment_param G x (dist x y) = y"
chamber_is_singleton: fixes S :: "'b set"   and x :: "'b set set" assumes "CoxeterComplex S"   and "SimplicialComplex.maxsimp (CoxeterComplex.TheComplex S) x" shows "\<exists>w\<in>\<langle>S\<rangle>. dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w} = x"
cobounded2: fixes b :: "'a"   and a :: "'a" shows "b \<le> a \<squnion>1 b"
reachable_tot_imp_reachable: fixes s :: "'b"   and F :: "'b program" assumes "s \<in> reachable (totalize F)" shows "s \<in> reachable F"
ASSERT_leI: fixes \<Phi> :: "bool"   and M :: "'b nres"   and M' :: "'b nres" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> M \<le> M'" shows "ASSERT \<Phi> \<bind> (\<lambda>_. M) \<le> M'"
preserves_arr: fixes f :: "'a1" assumes "A1.arr f" shows "A2_B.arr (F f)"
subgraph_def: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" shows "subgraph H G \<equiv> verts H \<subseteq> verts G \<and> arcs H \<subseteq> arcs G \<and> wf_digraph G \<and> wf_digraph H \<and> compatible G H"
V_def: shows "V \<equiv> eq_onp (\<lambda>x. x \<in> vars e)"
T2_axiom_implies_T1_axiom: fixes T :: "'b set set" assumes "\<forall>x\<in>carrier\<^bsub>T\<^esub>. \<forall>y\<in>carrier\<^bsub>T\<^esub>. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds\<^bsub>T\<^esub> x. \<exists>v\<in>nhds\<^bsub>T\<^esub> y. u \<inter> v = {})" shows "T1_axioms T"
cases: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.discr aval tval cval indis a" obtains c :: "('d, 'c, 'a) cmd" where "a = c" and "\<And>s i. i < PL.brn c \<longrightarrow> (s, PL.eff aval tval cval c s i) \<in> indis \<and> PL_Indis.discr aval tval cval indis (PL.cont aval tval cval c s i)"
elims_2: fixes x :: "'a simple_match"   and xa :: "('a, 'b) simple_packet_scheme" assumes "simple_matches x xa" obtains m :: "'a simple_match" and p :: "('a, 'b) simple_packet_scheme" where "x = m" and "xa = p" and "match_iface (iiface m) (p_iiface p) \<and> match_iface (oiface m) (p_oiface p) \<and> simple_match_ip (src m) (p_src p) \<and> simple_match_ip (dst m) (p_dst p) \<and> match_proto (proto m) (p_proto p) \<and> simple_match_port (sports m) (p_sport p) \<and> simple_match_port (dports m) (p_dport p)"
simps_984: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x301 :: "t"   and x302 :: "relop_f" shows "(case Relop_f x301 x302 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f30 x301 x302"
monotone_const: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'a" shows "monotone ord (\<le>) (\<lambda>_. c)"
inj_segment: fixes a :: "'b"   and b :: "'b"   and I :: "real set" assumes "a \<noteq> b" shows "inj_on (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) I"
simps_1: fixes I :: "interp"   and x :: "variable" shows "term_sem I (Var x) = (\<lambda>\<omega>. \<omega> x)"
simps_2: shows "Aodv_Basic.inv \<noteq> val"
Lambda_app: fixes x :: "ZF"   and A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" assumes "Elem x A" shows "Lambda A f \<acute> x = f x"
assms: shows "is_subobj P (C, C' # Cs @ [C''])"
U_3: shows "S = U \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>"
internalize_ordLess: fixes r' :: "('a \<times> 'a) set"   and r :: "('b \<times> 'b) set" shows "(r' <o r) = (\<exists>p. Field p \<subset> Field r \<and> r' =o p \<and> p <o r)"
simps_2: fixes l :: "'a tree"   and uu :: "'a"   and r :: "'a tree" shows "ipl \<langle>l, uu, r\<rangle> = ipl l + size l + ipl r + size r"
split_sel: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (option = Some (the option) \<longrightarrow> P (f2 (the option))))"
when_simps_2: fixes zero :: "'c"   and a :: "'c" shows "zero.when zero a False = zero"
card_surjective_functions_range_permutation: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // range_permutation A B) = Stirling (card A) (card B)"
of_bool_eq_2: fixes one :: "'d"   and zero :: "'d" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero True = one"
bc: shows "(b, c) \<in> (NS \<union> S)\<^sup>*"