hcju/isabelleps huggingface dataset:
typeof_Void: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Void\<rfloor>" shows "v = Unit"
arrE: fixes f :: "'a" assumes "L.arr f" shows "left a f"
need_second_arg_CondAnd: fixes b :: "bool" shows "need_second_arg CondAnd (Bool b) = b"
cond_idem: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> b \<triangleright>  P) = P"
simps_10: fixes x1 :: "'a" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<I>"
cases: fixes x :: "'a list" obtains "x = []"   | s :: "'a" and ss :: "'a list" where "x = s # ss"
ereal_le_add_self2: fixes y :: "ereal"   and x :: "ereal" assumes "0 \<le> y" shows "x \<le> y + x"
exhaust: fixes y :: "ExtendedBool" obtains "y = TRUE"   | "y = FALSE"   | "y = UNDEF"
degree_def: fixes x :: "'a pdevs" shows "degree x = (LEAST i. \<forall>j\<ge>i. pdevs_apply x j = (0::'a))"
inf_right_idem: fixes x :: "'a upred"   and y :: "'a upred" shows "((x \<and> y) \<and> y) = (x \<and> y)"
assms_2: shows "\<rho> f|` S = \<rho>' f|` S"
AE_pair_measure: fixes P :: "'c \<times> 'd \<Rightarrow> bool" assumes "{x \<in> space (T \<Otimes>\<^sub>M P). P x} \<in> sets (T \<Otimes>\<^sub>M P)"   and "AE x in T. AE y in P. P (x, y)" shows "almost_everywhere (T \<Otimes>\<^sub>M P) P"
rel_intros_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" assumes "R x2 y2" shows "rel_option R (pure x2) (pure y2)"
is_weak_ranking_iff: fixes xs :: "'c set list" shows "is_weak_ranking xs = (distinct xs \<and> disjoint (set xs) \<and> {} \<notin> set xs)"
inc: shows "supp q \<subseteq> as \<inter> supp x \<union> q \<bullet> (as \<inter> supp x)"
gcd_exists: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "\<exists>x\<in>carrier G. x = somegcd G a b"
order_antisym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "order r"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> x" shows "x = y"
divide_eq_0_iff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "b \<noteq> \<zero>" shows "(a \<oslash> b = \<zero>) = (a = \<zero>)"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "QA a"   and "QB b" shows "FA a = FB b"
\<psi>_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'c \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'c"   and \<phi> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and \<psi> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'b"   and g :: "'c"   and y :: "'c" assumes "meta_adjunction C D F G \<phi> \<psi>"   and "partial_magma.ide C x"   and "partial_magma.in_hom D g y (G x)" shows "partial_magma.in_hom C (\<psi> x g) (F y) x"
nAct_same_not_active: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and n' :: "enat"   and t :: "nat \<Rightarrow> cnf"   and n :: "enat" assumes "dynamic_component.nAct tCMP active c n' (inf_llist t) = dynamic_component.nAct tCMP active c n (inf_llist t)" shows "\<forall>k. n \<le> enat k \<longrightarrow> enat k < n' \<longrightarrow> \<not> active c (t k)"
can_lunit: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a" shows "free_monoidal_category.can C (free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C a)"
join: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" assumes "h a = h b"   and "a \<in> A" shows "\<exists>ab. m a b = Some ab \<and> bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u)"
lunit'_simps_5: fixes f :: "'b" assumes "D.ide f" shows "D.cod \<l>\<^sub>D\<^sup>-\<^sup>1[f] = trg\<^sub>D f \<star>\<^sub>D f"
product_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "\<forall>w. P w z = (P w x \<and> P w y)" shows "x \<otimes> y = z"
simps: fixes a1 :: "s_context"   and a2 :: "cl"   and a3 :: "tf" shows "cl_typing a1 a2 a3 = ((\<exists>i \<S> \<C> tf t1s t2s ts es. a1 = \<S> \<and> a2 = Func_native i tf ts es \<and> a3 = t1s _> t2s \<and> i < length (s_inst \<S>) \<and> s_inst \<S> ! i = \<C> \<and> tf = t1s _> t2s \<and> \<C>\<lparr>local := local \<C> @ t1s @ ts, label := [t2s] @ label \<C>, return := Some t2s\<rparr> \<turnstile> es : [] _> t2s) \<or> (\<exists>\<S> tf h. a1 = \<S> \<and> a2 = Func_host tf h \<and> a3 = tf))"
quotientfun_def: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set" shows "quotientfun f X = (THE b. b \<in> f ` X)"
product_is_first_summand: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "CEM"   and "\<not> O y z" shows "(x \<oplus> y) \<otimes> (x \<oplus> z) = x"
ReachImplAgree: shows "\<forall>cfg. reachable (last cfgList) cfg \<longrightarrow> agreement cfg"
adds_lcs_fun: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" shows "s adds lcs s t"
distinct_1: fixes x21 :: "bool"   and x22 :: "trie \<times> trie" shows "Lf \<noteq> Nd x21 x22"
indifferent_prefD: fixes x :: "'c"   and r :: "('c \<times> 'c) set"   and y :: "'c" assumes "x \<^bsub>r\<^esub>\<approx> y" shows "x \<^bsub>r\<^esub>\<preceq> y \<and> y \<^bsub>r\<^esub>\<preceq> x"
sorted_domain_def: fixes ko :: "'o"   and xs :: "'x" shows "sorted_domain ko xs = sorted_domain_raw ko (list_of_oalist xs)"
rep_eq: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and xa :: "'b" shows "mapping_of (monom x xa) = Poly_Mapping.single x xa"
with_bot_UNIV: shows "UNIV\<^sub>\<bottom> = UNIV"
prv_rawpsubst_scnj: fixes F :: "'fmla set"   and txs :: "('trm \<times> 'var) list" assumes "F \<subseteq> fmla"   and "finite F"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "prv (eqv (rawpsubst (scnj F) txs) (scnj ((\<lambda>\<phi>. rawpsubst \<phi> txs) ` F)))"
norm_eq_sqrt_inner: fixes x :: "'a" shows "norm x = sqrt (inner x x)"
eq_subst_type_scheme_eq_free: fixes S1 :: "nat \<Rightarrow> typ"   and sch :: "type_scheme"   and S2 :: "nat \<Rightarrow> typ"   and n :: "nat" assumes "$ S1 sch = $ S2 sch"   and "n \<in> free_tv sch" shows "S1 n = S2 n"
cobounded1: fixes a :: "'d multiset"   and b :: "'d multiset" shows "a \<subseteq># a \<union># b"
inorder_tree: fixes t :: "'b bro" shows "inorder (tree t) = inorder t"
jkbpCn_jkbpC_inc: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview"   and "t \<in> PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n" shows "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"
iszero_def: fixes z :: "'a" shows "iszero z = (z = (0::'a))"
min: shows "\<forall>y\<in>A. m \<le> y"
unit_counit_G: fixes x :: "'c" assumes "C.ide x" shows "G x = G (\<epsilon>o x) \<cdot>\<^sub>D \<eta>o (G x)"
satL_o: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and l :: "('fsym, 'psym) lit" shows "satL (int \<xi> \<circ> \<rho>) l = satL \<xi> (substL \<rho> l)"
cases: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" assumes "ev \<phi> a" obtains xs :: "'a stream" where "a = xs__" and "\<phi> xs__"   | xs :: "'a stream" where "a = xs" and "ev \<phi> (stl xs)"
lemma2: fixes K :: "key"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<in> analz H" shows "insert (Crypt K X) (analz (insert X H)) \<subseteq> analz (insert (Crypt K X) H)"
locally_diff_closed: fixes P :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and t :: "'a set" assumes "locally P S"   and "closedin (top_of_set S) t" shows "locally P (S - t)"
abs_leI: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq a b"   and "less_eq (uminus a) b" shows "less_eq (abs a) b"
T_opt_def: fixes s :: "'a list"   and rs :: "'a list" shows "T\<^sub>p_opt s rs = \<Sqinter> {T\<^sub>p s rs as |as. length as = length rs}"
c_pair_inj: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "c_pair x1 y1 = c_pair x2 y2" shows "x1 = x2 \<and> y1 = y2"
emeasure_space: fixes M :: "'a measure"   and A :: "'a set" shows "emeasure M A \<le> emeasure M (space M)"
sign_swap_id: fixes a :: "'c"   and b :: "'c" shows "sign (Fun.swap a b id) = (if a = b then 1 else - 1)"
always_lft_rel_comp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'd \<times> 'e"   and r' :: "'e \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<box> lft_rel_st r) (fst \<circ> u) OO (\<box> lft_rel_st r') (snd \<circ> u) = (\<box> lft_rel_st (\<lambda>(u, v) (u', v'). r u u' OO r' v v')) u"
pm_pthi_def: fixes m :: "('b, 'c) pointermap_impl_scheme"   and p :: "nat" shows "pm_pthi m p \<equiv> arl_nth (entriesi m) p"
gf_U_def: fixes x :: "'s"   and y :: "'s"   and z :: "'a" shows "gf_U x y z = (\<Sum>n. u x y n *\<^sub>R z ^ Suc n)"
has_field_derivative_rGamma_no_nonpos_int: fixes z :: "'a"   and A :: "'a set" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(rGamma has_field_derivative - rGamma z * Digamma z) (at z within A)"
assms_1: shows "Y \<subseteq> f -` {0::'b}"
mod_add_cong: fixes divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and euclidean_size :: "'k \<Rightarrow> nat"   and a :: "'k"   and c :: "'k"   and a' :: "'k"   and b :: "'k"   and b' :: "'k" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (plus a b) c = modulo (plus a' b') c"
s2: shows "sets M = sets (Pi\<^sub>M J (\<lambda>_. N))"
arc_pos_plus: fixes x :: "'a"   and y :: "'a" assumes "arc_pos x" shows "arc_pos (x + y)"
simps_3: fixes dt :: "loc \<Rightarrow> ty option"   and i :: "int" shows "typeof dt (Intg i) = Some (PrimT Integer)"
card_multisets: fixes A :: "'c set"   and k :: "nat" assumes "finite A" shows "card {M. set_mset M \<subseteq> A \<and> size M = k} = card A + k - 1 choose k"
assms_1: shows "recfn (Suc n) (Pr n f g)"
matches_ipassmt_iface_replace_dstip_mexpr_case_Iface: fixes ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and p :: "('c, 'd) tagged_packet_scheme"   and p_ips :: "('c word \<times> nat) list"   and \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ifce :: "iface"   and a :: "action" assumes "ipassmt_sanity_nowildcards ipassmt"   and "ipassmt_sanity_disjoint ipassmt"   and "ipassmt (Iface (p_oiface p)) = Some p_ips \<and> p_dst p \<in> ipcidr_union_set (set p_ips)" shows "matches (common_matcher, \<alpha>) (ipassmt_iface_replace_dstip_mexpr ipassmt ifce) a p = matches (common_matcher, \<alpha>) (Match (OIface ifce)) a p"
exhaust: fixes y :: "('b, 'a) com" obtains x1 :: "'a" where "y = Atm x1"   | x21 :: "('b, 'a) com" and x22 :: "('b, 'a) com" where "y = x21 ;; x22"   | x31 :: "'b" and x32 :: "('b, 'a) com" and x33 :: "('b, 'a) com" where "y = if x31 then x32 else x33"   | x41 :: "'b" and x42 :: "('b, 'a) com" where "y = while x41 do x42"   | x51 :: "('b, 'a) com" and x52 :: "('b, 'a) com" where "y = x51 | x52"
acyclic_2_def: fixes x :: "'a" shows "acyclic_2 x \<equiv> \<forall>y. y \<le> x \<and> asymmetric y \<longrightarrow> pd_kleene_allegory_class.acyclic y"
absorb_iff1: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c" shows "less_eq_prod b a = (sup_prod a b = a)"
min_dist_succ: fixes v :: "nat"   and v' :: "nat"   and v'' :: "nat" assumes "local.connected v v'"   and "(v', v'') \<in> E" shows "min_dist v v'' \<le> Suc (min_dist v v')"
rcis_div_cis: fixes n :: "real"   and a :: "real"   and b :: "real" shows "rcis n a / cis b = rcis n (a - b)"
obj_def: fixes a :: "'a arrow_of_spans_data" shows "obj a \<equiv> arr a \<and> src a = a"
ltrm_inv_simp: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "monom P (monom P (f (deg R f)) (deg R f) (deg R (monom P (f (deg R f)) (deg R f)))) (deg R (monom P (f (deg R f)) (deg R f))) = monom P (f (deg R f)) (deg R f)"
flow_avoids_surface_eventually_at_right: fixes x :: "'a"   and S :: "'a set"   and s :: "'a \<Rightarrow> real"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" assumes "x \<notin> S \<or> s x \<noteq> 0 \<or> blinfun_apply (Ds x) (f x) \<noteq> 0"   and "x \<in> X"   and "closed S"   and "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"   and "\<And>x. isCont Ds x" shows "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> {x \<in> S. s x = 0}"
preserves_reflects_arr: fixes f :: "'c" shows "D.arr (G f) = C.arr f"
pcorresD: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and Q :: "'d \<Rightarrow> real" assumes "pcorres \<phi> G A B"   and "unitary Q" shows "\<guillemotleft> G \<guillemotright> && (wp A Q \<circ> \<phi>) = \<guillemotleft> G \<guillemotright> && wp B (Q \<circ> \<phi>)"
c_tl_aux1: fixes u :: "nat" assumes "c_len u = 1" shows "c_tl u = 0"
vector_derivative_chain_at_general: fixes f :: "real \<Rightarrow> 'b"   and x :: "real"   and g :: "'b \<Rightarrow> 'b" assumes "f differentiable at x"   and "g field_differentiable at (f x)" shows "vector_derivative (g \<circ> f) (at x) = vector_derivative f (at x) * deriv g (f x)"
sg_i_closed: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and x :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "x \<in> H" shows "\<rho>\<^bsub>G\<^esub> x \<in> H"
rl_lpoly_coeff_nth_non_empty: fixes i :: "nat"   and cs :: "rat list" assumes "i < length cs"   and "cs \<noteq> []" shows "Abstract_Linear_Poly.coeff (list_to_lpoly cs) i = cs ! i"
boundr: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Minimal_Solutions"   and "(x, y) \<notin> Special_Solutions" shows "boundr x y"
gen_kleene_lfp: fixes f :: "'g \<Rightarrow> 'g"   and m :: "'g" assumes "cont f" shows "lfp (\<lambda>x. sup m (f x)) = (SUP i. (f ^^ i) m)"
minus_flow_def: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and g :: "'v \<times> 'v \<Rightarrow> ennreal" shows "f \<ominus> g = (\<lambda>(x, y). if edge \<Delta> x y then f (x, y) - g (x, y) else if edge \<Delta> y x then g (y, x) - f (y, x) else 0)"
assms_2: shows "f \<in> Ar\<^bsub>AA\<^esub>"
conjugate_neg: fixes conjugate :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (uminus a) = uminus (conjugate a)"
remap_polys_def: fixes spec :: "int mpoly" shows "remap_polys spec = (\<lambda>\<V> A. SPEC (\<lambda>dom. set_mset (dom_m A) \<subseteq> dom \<and> finite dom) \<bind> (\<lambda>dom. SPEC (\<lambda>_. True) \<bind> (\<lambda>failed. if failed then RETURN (FAILED, \<V>, fmempty) else FOREACH dom (\<lambda>i (b, \<V>, A'). if i \<in># dom_m A then SPEC (\<lambda>p. the (fmlookup A i) - p \<in> More_Modules.ideal polynomial_bool \<and> vars p \<subseteq> vars (the (fmlookup A i))) \<bind> (\<lambda>p. SPEC (\<lambda>eqa. eqa \<longrightarrow> p = spec) \<bind> (\<lambda>eq. SPEC ((\<subseteq>) (\<V> \<union> vars (the (fmlookup A i)))) \<bind> (\<lambda>\<V>. RETURN (b \<or> eq, \<V>, fmupd i p A')))) else RETURN (b, \<V>, A')) (False, \<V>, fmempty) \<bind> (\<lambda>(b, N). RETURN (if b then FOUND else SUCCESS, N)))))"
linorder_permutation_exists: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and R' :: "('b \<times> 'b) set" assumes "finite A"   and "linorder_on A R"   and "linorder_on A R'" obtains \<pi> :: "'b \<Rightarrow> 'b" where "\<pi> permutes A" and "R' = map_relation A \<pi> R"
p\<^sub>1_simps: shows "trg p\<^sub>1 = src r\<^sub>0"
emeasure_eq_measure: fixes M :: "'a measure"   and A :: "'a set" assumes "finite_measure M" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
linear_injective_contraction: fixes f :: "'a \<Rightarrow> 'a"   and c :: "real" assumes "linear f"   and "c < 1"   and "\<And>x. norm (f x - x) \<le> c * norm x" shows "inj f"
singleton_matrix_zero: fixes j :: "nat"   and i :: "nat" shows "singleton_matrix j i (0::'b) = 0"
inv_Node: fixes l :: "('a \<times> 'b) tree"   and a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) tree" assumes "inv \<langle>l, (a, b), r\<rangle>" shows "inv l \<and> inv r"
dsj_commute_imp: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla" shows "H \<turnstile> B OR A IMP A OR B"
fls_divide_shift_numer_nonzero: fixes f :: "'a fls"   and m :: "int"   and g :: "'a fls" assumes "f \<noteq> 0" shows "fls_shift m f / g = fls_shift m (f / g)"
PSQ: shows "(P, S, Q) = Smith_mxn_HMA Smith_1x2 Smith_2x2 div_op A"
2: fixes x :: "real" assumes "0 < x"   and "x * 2 < pi" shows "isCont (\<lambda>x. 1 + (tan x)\<^sup>2) x"
WT_converter_mono: fixes \<I>1 :: "('a, 'b) \<I>"   and \<I>2 :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and \<I>1' :: "('a, 'b) \<I>"   and \<I>2' :: "('c, 'd) \<I>" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv \<surd>"   and "\<I>1' \<le> \<I>1"   and "\<I>2 \<le> \<I>2'" shows "\<I>1', \<I>2' \<turnstile>\<^sub>C conv \<surd>"
point_reachable_converse: fixes v :: "'a"   and r :: "'a"   and x :: "'a" assumes "is_vector v"   and "v \<noteq> 0"   and "point r"   and "v \<le> x\<^sup>T\<^sup>+ ; r" shows "r \<le> x\<^sup>+ ; v"
rel_F_mono: fixes L1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L1' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L3 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and L3' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co1' :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co2 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Co2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Co3 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Co3' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra1' :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra1 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra2' :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra2 :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra3' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra3 :: "'af \<Rightarrow> 'ag \<Rightarrow> bool" assumes "L1 \<le> L1'"   and "L2 \<le> L2'"   and "L3 \<le> L3'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Co3 \<le> Co3'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2"   and "Contra3' \<le> Contra3" shows "rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 \<le> rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3'"
complement_disjointness: fixes x :: "i" assumes "CMC"   and "\<exists>v. \<not> O v x" shows "\<not> O x (\<midarrow> x)"
fls_of_nat: fixes n :: "nat" shows "of_nat n = fls_const (of_nat n)"
alw_HLD_iff_streams: fixes X :: "'a set"   and \<omega> :: "'a stream" shows "alw (HLD X) \<omega> = (\<omega> \<in> streams X)"
fun_upd_apply: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a"   and z :: "'b" shows "(f(x := y)) z = (if z = x then y else f z)"
plus_flow_def: fixes G :: "('a, 'b) graph_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" shows "f \<oplus>\<^bsub>G\<^esub> g = (\<lambda>(x, y). if edge G x y then f (x, y) + g (x, y) - g (y, x) else 0)"
fst_eq_Domain: fixes R :: "('h \<times> 'i) set" shows "fst ` R = Domain R"
mono_mprefix_FD: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and P' :: "'a \<Rightarrow> 'a process" assumes "\<forall>x\<in>A. P x \<le> P' x" shows "Mprefix A P \<le> Mprefix A P'"
ext_inject: fixes nodes :: "'b set"   and edges :: "('b \<times> 'b) set"   and more :: "'c"   and nodes' :: "'b set"   and edges' :: "('b \<times> 'b) set"   and more' :: "'c" shows "(\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes', edges = edges', \<dots> = more'\<rparr>) = (nodes = nodes' \<and> edges = edges' \<and> more = more')"
lset_intros_1: fixes x :: "'c"   and xs :: "'c llist" shows "x \<in> lset (LCons x xs)"
distinct_1: fixes x1 :: "nat"   and x2 :: "nat" shows "Const x1 \<noteq> Intv x2"
of_int_floor_cancel: fixes x :: "'d" shows "(of_int \<lfloor>x\<rfloor> = x) = (\<exists>n. x = of_int n)"
Card_order_csum: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" shows "Card_order (r1 +c r2)"
nonsat_push_action_bound': fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and p :: "pr_operation' list"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts'" shows "length (filter is_NONSAT_PUSH' p) \<le> 18 * (card V)\<^sup>2 * card E"
preserves_cod: fixes f :: "'b" assumes "F.B.arr f" shows "G.B.cod (G f) = G (F.B.cod f)"
assms: fixes x :: "'a"   and y :: "'b" assumes "fmlookup m x = Some y"   and "P x y" shows "Q x y"
assms_2: shows "\<forall>x\<in>cbox a b. norm (f x) \<le> e"
is_renaming_list_inv_renaming_cancel_cls_list: fixes Cs :: "'a literal multiset list"   and \<rho>s :: "'s list" assumes "length Cs = length \<rho>s"   and "is_renaming_list \<rho>s" shows "Cs \<cdot>\<cdot>cl \<rho>s \<cdot>\<cdot>cl map inv_renaming \<rho>s = Cs"
comm_neutral: fixes a :: "'e" shows "a * (1::'e) = a"
set_evs_rev: fixes evs :: "'a list" shows "set evs = set (rev evs)"
heap_top_geq: fixes a :: "'b"   and t :: "'b Tree" assumes "a \<in># Heap.multiset t"   and "is_heap t" shows "a \<le> val t"
ffunion_insert: fixes a :: "'c fset"   and B :: "'c fset fset" shows "ffUnion (finsert a B) = a |\<union>| ffUnion B"
full_pmdl_closed_monom_mult: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and K :: "'k set"   and c :: "'c"   and t :: "'a" assumes "p \<in> full_pmdl K" shows "monom_mult c t p \<in> full_pmdl K"
raw_ports_compress_dst_CannotMatch: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and pss :: "ipt_l4_ports list"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "l4_ports_compress pss = CannotMatch" shows "\<not> matches (\<beta>, \<alpha>) (alist_and (map (Pos \<circ> Dst_Ports) pss)) a p"
h_def: fixes n :: "nat" shows "h n = (\<lambda>(x, y). if x \<in> A_n n \<and> y \<in> Y n then enn2real (h' n (Some x) y) else 0)"
floor_add2: fixes x :: "'b"   and y :: "'b" assumes "x \<in> \<int> \<or> y \<in> \<int>" shows "\<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>"
MM_def: shows "MM \<equiv> M \<and> MM_axioms"