hcju/isabelleps huggingface dataset:
float_defs_5: fixes a :: "('a, 'b) IEEE.float" shows "is_normal a = (0 < IEEE.exponent a \<and> IEEE.exponent a < emax TYPE(('a, 'b) IEEE.float))"
compat_equiv_prec: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Q :: "'c set"   and entails_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G_q :: "'c \<Rightarrow> 'b inference set"   and Red_I_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b inference set"   and Red_F_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set"   and \<G>_F_q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and \<G>_I_q :: "'c \<Rightarrow> 'a inference \<Rightarrow> 'b inference set option"   and Inf_FL :: "('a \<times> 'd) inference set"   and Equiv_F :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Prec_F :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Prec_L :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and active :: "'d"   and C1 :: "'a"   and D1 :: "'a"   and C2 :: "'a"   and D2 :: "'a" assumes "given_clause_basis Bot_F Inf_F Bot_G Q entails_q Inf_G_q Red_I_q Red_F_q \<G>_F_q \<G>_I_q Inf_FL Equiv_F Prec_F Prec_L active"   and "Equiv_F C1 D1"   and "Equiv_F C2 D2"   and "Prec_F C1 C2" shows "Prec_F D1 D2"
Amin_ge: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant"   and z :: "ant" assumes "\<forall>j\<le>n. f j \<in> Z\<^sub>\<infinity>"   and "\<forall>j\<le>n. z \<le> f j" shows "z \<le> Amin n f"
1: shows "continuous_on (UNIV - \<real>\<^sub>\<ge>\<^sub>0) Arg2pi"
simps_51: fixes x31 :: "poscheckvariant"   and x32 :: "('b, 'a) Term.term"   and x33 :: "('b, 'a) Term.term" shows "vars\<^sub>s\<^sub>t\<^sub>p \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t = fv x32 \<union> fv x33"
i_drop_o: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'e" shows "(f \<circ> g) \<Up> n = f \<circ> g \<Up> n"
assms: shows "a_consistent (ae, a, as) (\<Gamma>, e, Upd x # S)"
mat_hom_mult: fixes hom :: "'a \<Rightarrow> 'b"   and A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat" assumes "semiring_hom hom"   and "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "map_mat hom (A * B) = map_mat hom A * map_mat hom B"
order_trans_rules_27: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "b = c" shows "a < c"
analz_insert_freshK3: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and authK' :: "nat" assumes "AKcryptSK authK servK evs"   and "authK' \<noteq> authK"   and "authK' \<notin> range shrK"   and "evs \<in> kerbIV" shows "(Key servK \<in> analz (insert (Key authK') (knows Spy evs))) = (servK = authK' \<or> Key servK \<in> analz (knows Spy evs))"
simps_3: fixes las :: "lock_action list" shows "must_acquire_lock (Unlock # las) = False"
dflt_cmp_def: fixes le :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and lt :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" shows "dflt_cmp le lt a b \<equiv> if lt a b then LESS else if le a b then EQUAL else GREATER"
weak_ring_morphismI: fixes I :: "'d set"   and R :: "('d, 'e) ring_scheme"   and f :: "'d \<Rightarrow> 'f" assumes "ideal I R"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R\<rbrakk> \<Longrightarrow> (f a = f b) = (a \<ominus>\<^bsub>R\<^esub> b \<in> I)" shows "weak_ring_morphism f I R"
l9_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet A B P Q" shows "Tarski_neutral_dimensionless.TS Bet A B Q P"
from_list_def: fixes l :: "('b \<times> 'b \<times> nat) list" shows "from_list l \<equiv> fold (\<lambda>(u, v, d). add_edge (u, v) d) l empty_graph"
regular_in_p_image_iff: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "(x = uminus (uminus x)) = (\<exists>y. x = uminus y)"
right_neutral: fixes a :: "'b option" shows "lift_option (+) a (Some (0::'b)) = a"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "(\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
simple_cipher'_def: fixes prf_key_gen :: "nat \<Rightarrow> 'a spmf"   and prf_fun :: "nat \<Rightarrow> 'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_domain :: "nat \<Rightarrow> bool list set"   and prf_dlen :: "nat \<Rightarrow> nat"   and prf_clen :: "nat \<Rightarrow> nat"   and upf_key_gen :: "nat \<Rightarrow> 'b spmf" shows "simple_cipher' prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen \<equiv> \<forall>\<eta>. simple_cipher (prf_key_gen \<eta>) (prf_fun \<eta>) (prf_domain \<eta>) (prf_dlen \<eta>) (prf_clen \<eta>) (upf_key_gen \<eta>)"
altc_stateT_def: shows "altc = altc_state altc"
add_diff_eq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.group_add minus plus zero uminus" shows "plus a (minus b c) = minus (plus a b) c"
fsmult_Gmult_comm: fixes g :: "'g"   and v :: "'v"   and a :: "'f" assumes "g \<in> G"   and "v \<in> V" shows "a \<sharp>\<cdot> g *\<cdot> v = g *\<cdot> a \<sharp>\<cdot> v"
routing_ipassmt_wi_disjoint: fixes tbl :: "'i routing_rule list"   and a1 :: "char list"   and a2 :: "char list"   and b1 :: "'i wordinterval"   and b2 :: "'i wordinterval" assumes "valid_prefixes tbl"   and "a1 \<noteq> a2"   and "(a1, b1) \<in> set (routing_ipassmt_wi tbl)"   and "(a2, b2) \<in> set (routing_ipassmt_wi tbl)" shows "wordinterval_to_set b1 \<inter> wordinterval_to_set b2 = {}"
r_def: shows "r \<equiv> JVM_SemiType.le P mxs mxl"
assms_3: fixes c :: "'a option acom" shows "strip (f c) = strip c"
relcomp_empty2: fixes R :: "('a \<times> 'c) set" shows "R O {} = {}"
get_lower_bound_2: fixes U :: "'a set"   and V :: "'a set" assumes "U \<in> I"   and "V \<in> I" shows "get_lower_bound U V \<subseteq> U"
atom_property: fixes A :: "'a"   and L :: "'a Literal" assumes "A = atom L" shows "L = Pos A \<or> L = Neg A"
shadow_root_eq_h3: fixes ptr' :: "(_) element_ptr"   and children :: "(_) shadow_root_ptr option" assumes "element_ptr \<noteq> ptr'" shows "h3 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r children"
red_chainTr0m1_1: fixes m :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "tw_cmpser G m f" shows "(SOME g. g \<in> {h. red_ch_cd G f m h}) \<in> {h. red_ch_cd G f m h}"
ladder_early_stage_1: fixes index :: "nat" assumes "index + 1 < length L" shows "ladder_\<gamma> (\<alpha> @ \<delta>) D L index = ladder_\<gamma> \<alpha> E L' index @ \<delta>"
assms_3: shows "\<forall>X\<in>Q0. wf_atable n X"
fcard_finsert: fixes x :: "'b"   and A :: "'b fset" shows "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
vunion_vdoubleton: fixes a :: "V"   and b :: "V"   and A :: "V" shows "set {a, b} \<union>\<^sub>\<circ> A = vinsert a (vinsert b A)"
wAbsU: fixes \<theta>\<^sub>1 :: "trm"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([Abs \<theta>\<^sub>1]<>\<nu>) = (let (l1, u1) = [\<theta>\<^sub>1]<>\<nu> in (wmax l1 (wneg u1), wmax u1 (wneg l1)))"
assms_1: shows "hn_refine \<Gamma> c \<Gamma>' R m"
Union_empty_eq: fixes P :: "'a set \<Rightarrow> bool" shows "\<Union> {D. D = {} \<and> P D} = {}"
ortho_scalprod0: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0"   and "z2 \<noteq> 0" shows "(\<angle> z1 z2 = pi / 2 \<or> \<angle> z1 z2 = - pi / 2) = (scalprod z1 z2 = 0)"
top_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times top x = top"
L7_Controller_One: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l t = One"   and "y 0 = [500]" shows "last (fin_make_untimed (inf_truncate z t)) = One"
nodes_edges_consistent_def: fixes V :: "'a set"   and E :: "('a \<times> 'a) set" shows "nodes_edges_consistent V E \<equiv> finite V \<and> irrefl E \<and> sym E \<and> E \<subseteq> V \<times> V"
tagged_fun_relD_rhs: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'd"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set"   and x :: "'a"   and x' :: "'c" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, Let x' f') \<in> B"
p_star_def: fixes X :: "'a list set" shows "p_star X \<equiv> \<Union> (range (path_dioid.power X))"
rtrancl_cases: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> r\<^sup>*" obtains "a = b"   | "(a, b) \<in> r\<^sup>+"
normNum: fixes x :: "int \<times> int" shows "INum (normNum x) = INum x"
DERIV_sum: fixes S :: "'a set"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'c"   and F :: "'b filter" assumes "\<And>n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative f' x n) F" shows "((\<lambda>x. sum (f x) S) has_field_derivative sum (f' x) S) F"
increment_def: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and h :: "real star" shows "increment f x h = (SOME inc. f NSdifferentiable x \<and> inc = (*f* f) (hypreal_of_real x + h) - hypreal_of_real (f x))"
residue_mult_assoc: fixes p :: "int"   and k :: "nat"   and x :: "int"   and y :: "int"   and z :: "int" assumes "padic_integers p" shows "x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> z = x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> (y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> z)"
nomatch: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and a :: "action" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"
prems_2: fixes d :: "nat" assumes "d \<in> set (d__ # ds)" shows "m \<le> d"
member_frontier_pos_zmset: fixes t :: "'a"   and M :: "'a zmultiset" assumes "t \<in>\<^sub>A frontier M" shows "0 < zcount M t"
le_neq_trans: fixes a :: "'t"   and b :: "'t" assumes "a \<preceq>\<^sub>t b"   and "a \<noteq> b" shows "a \<prec>\<^sub>t b"
simps: fixes m :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme" shows "simple_matches m p = (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))"
assms_2: fixes r :: "nat" assumes "r \<in> set rs" shows "0 < r"
U'_neq_zero: fixes x :: "'s"   and y :: "'s" shows "U' x y \<noteq> 0"
gamma2_def: shows "gamma2 \<equiv> $pc2 = #g \<and> pc2$ = #a \<and> sem$ = Suc<$sem> \<and> unchanged (x, y, pc1)"
ide_dom: fixes f :: "'a" assumes "Right.arr f" shows "Right.ide (Right.dom f)"
vcard_add: fixes x :: "V"   and y :: "V" shows "vcard (x + y) = vcard x \<oplus> vcard y"
less_le_not_le: fixes x :: "'b list"   and y :: "'b list" shows "strict_border x y = (border x y \<and> \<not> border y x)"
induces_equivalence: shows "\<exists>F \<eta> \<epsilon>. equivalence_of_categories (\<cdot>) (\<cdot>) F ((\<otimes>) \<I>) \<eta> \<epsilon>"
convergent_imp_Zp_cauchy: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier Zp"   and "Zp_converges_to s a" shows "is_Zp_cauchy s"
mset_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "mset (linorder.sort_key less_eq k xs) = mset xs"
facetrel_conv_insert: fixes y :: "'b set"   and x :: "'b set"   and v :: "'b" assumes "y \<lhd> x"   and "v \<in> x - y" shows "x = insert v y"
less_int_def: fixes z :: "Quotient_Int.int"   and w :: "Quotient_Int.int" shows "(z < w) = (z \<le> w \<and> z \<noteq> w)"
admissible_eqI: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "cont luba orda lub ord f"   and "cont luba orda lub ord g" shows "ccpo.admissible luba orda (\<lambda>x. f x = g x)"
that: fixes R :: "('a \<times> 'a) set" assumes "wf R"   and "\<And>s s'. g s = Skip s' \<Longrightarrow> (s', s) \<in> R"   and "\<And>s a s'. g s = Yield a s' \<Longrightarrow> (s', s) \<in> R" shows "thesis"
treewidth_complete_graph: fixes G :: "('a, 'b) Graph_scheme" assumes "Graph G"   and "\<And>v w. \<lbrakk>v \<in> V\<^bsub>G\<^esub>; w \<in> V\<^bsub>G\<^esub>; v \<noteq> w\<rbrakk> \<Longrightarrow> v \<rightarrow>\<^bsub>G\<^esub> w" shows "Graph.treewidth G = card V\<^bsub>G\<^esub> - 1"
enc: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"
neq_LNil_conv: fixes xs :: "'b llist" shows "(xs \<noteq> LNil) = (\<exists>x xs'. xs = x ## xs')"
Dom_map: fixes g :: "'c"   and f :: "'c" assumes "C.arr g"   and "C.arr f" shows "S.Dom (Hom.map (g, f)) = Hom.set (C.cod g, C.dom f)"
id_o: fixes g :: "'c \<Rightarrow> 'd" shows "id \<circ> g = g"
in_face_cycle_setsD: fixes x :: "'b set" assumes "x \<in> H.face_cycle_sets" shows "x \<subseteq> arcs H"
unit_vec_to_col: fixes A :: "complex mat"   and n :: "nat"   and i :: "nat" assumes "dim_col A = n"   and "i < n" shows "col A i = col_fst (A * |unit_vec n i\<rangle>)"
foo29: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "dom (Cp a) \<noteq> {}"   and "dom (Cp a) \<inter> dom (Cp b) = {}" shows "a \<noteq> b"
sup_Pair_Pair: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and d :: "'b" shows "sup (a, b) (c, d) = (sup a c, sup b d)"
snd2: fixes B :: "'a trm"   and B' :: "'a trm"   and A :: "'a trm" assumes "B >> B'" shows "Snd (SimplyTyped.Pair A B) >> B'"
hom_induced_reduced: fixes c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and p :: "int"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "c \<in> carrier (reduced_homology_group p X)" shows "hom_induced p X {} Y {} f c \<in> carrier (reduced_homology_group p Y)"
simps: fixes uu :: "'b"   and hs :: "'b heap list" shows "hps (Hp uu hs) = hs"
is_ub_def: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d set"   and x :: "'d" assumes "class.po below" shows "po.is_ub below S x = (\<forall>y\<in>S. below y x)"
category_with_equalizers_axioms: shows "category_with_equalizers local.comp"
i_set_cont_IN: fixes n :: "nat"   and d :: "nat" shows "i_set_cont_minp [n\<dots>,d]"
de_morgan1: fixes x :: "'a"   and y :: "'a" shows "n (n x \<cdot> n y) = t x \<oplus> t y"
isomorphic_reflexive: fixes f :: "'c \<times> 'c" assumes "C.VV.ide f" shows "C.VV.isomorphic f f"
preserves_assoc_1: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "H \<a>\<^sub>C[f, g, h] = \<Phi>\<^sub>H (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (H f \<star>\<^sub>D \<Phi>\<^sub>H (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[H f, H g, H h] \<cdot>\<^sub>D (D.inv (\<Phi>\<^sub>H (f, g)) \<star>\<^sub>D H h) \<cdot>\<^sub>D D.inv (\<Phi>\<^sub>H (f \<star>\<^sub>C g, h))"
\<alpha>_eq: fixes x :: "'a" assumes "x \<in> S" shows "\<alpha>' x = \<alpha> x"
convex_convex_hull: fixes s :: "'a set" shows "convex (convex hull s)"
Setcompr_eq_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "{f x |x. x \<in> A} = \<P> f A"
supp_alpha_supp_rel: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "supp t\<^sub>\<alpha> = supp_rel (=\<^sub>\<alpha>) (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
S_def: shows "S = {\<mu> c s u + ereal (c e) |u e. arc e (u, v)}"
elims_3: fixes x :: "('a, 'e, 'f, 'g) SLin_state_scheme \<times> 'h" assumes "\<not> P15 x" obtains s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme" and s2 :: "'h" where "x = (s1, s2)" and "\<not> Ball (abortVals s1) ((\<preceq>) (dstate s1))"
ftv_tyS_subst: fixes X :: "W.tvar"   and S :: "tyS"   and Y :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "X \<in> set (ftv S)"   and "Y \<in> set (ftv (lookup \<theta> X))" shows "Y \<in> set (ftv (\<theta><S>))"
cod_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "partial_magma.cod (product_category.comp C1 C2) f = (if partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f) then (partial_magma.cod C1 (fst f), partial_magma.cod C2 (snd f)) else (partial_magma.null C1, partial_magma.null C2))"
inv_push_forward_inverse: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and f' :: "'e \<Rightarrow> 'c"   and X :: "('e \<Rightarrow> real) \<Rightarrow> real"   and p :: "'e" assumes "Differentiable_Manifold.diffeomorphism k charts1 charts2 f f'"   and "X \<in> c_manifold.tangent_space charts2 k p"   and "p \<in> manifold.carrier charts2" shows "diff.push_forward k charts1 charts2 f (diff.push_forward k charts2 charts1 f' X) = X"
drop_bit_minus_one: fixes n :: "nat" shows "drop_bit n (- 1) = - 1"
thetaWhileZOT_ZObisT: shows "thetaWhileZOT \<subseteq> ZObisT"
Ord: fixes x :: "hf" assumes "\<exists>i. x = ord_of i" shows "coding_hf x"
simps_4: fixes b1 :: "bool"   and v :: "'a"   and va :: "value list" shows "local.less_gexp_aux (Bc b1) (In v va) = True"
cod_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)" shows "cod (mkArr A B F) = mkIde B"
ovalidNF_wp_comb3: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and P' :: "'c \<Rightarrow> bool"   and Q' :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "ovalidNF P f Q"   and "ovalidNF P' f Q'" shows "ovalidNF (\<lambda>s. P s \<and> P' s) f (\<lambda>r s. Q r s \<and> Q' r s)"
out_arcs_add_arc_iff: fixes a :: "'a \<times> 'a"   and u :: "'a" shows "out_arcs (G.add_arc a) u = (if fst a = u then insert a (out_arcs (with_proj G) u) else out_arcs (with_proj G) u)"
plane_of_def: fixes sctn :: "'c sctn" shows "plane_of sctn = plane (normal sctn) (pstn sctn)"
lm044: fixes a :: "('c \<times> 'd set) set" shows "pseudoAllocation a = \<Union> {{(fst pair, {y}) |y. y \<in> snd pair} |pair. pair \<in> a}"
aval_take: fixes x :: "vname aexp"   and a :: "nat"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "AExp.max_input x < Some a" shows "aval x (join_ir i r) = aval x (join_ir (take a i) r)"
prems_3: shows "older_seniors x n = older_seniors x (n + Suc m)"
poss_int_neq_0: fixes z :: "int" assumes "0 < z" shows "z \<noteq> 0"
ltl_prop_unfold_equiv_implies_ltl_lang_equiv: fixes \<phi> :: "'d ltln"   and \<psi> :: "'d ltln" assumes "\<phi> \<sim>\<^sub>Q \<psi>" shows "\<phi> \<sim>\<^sub>L \<psi>"
intros_4: fixes A :: "nat \<Rightarrow> 'c set"   and \<Omega> :: "'c set"   and M :: "'c set set" assumes "range A \<in> Pow (smallest_ccdi_sets \<Omega> M)"   and "disjoint_family A" shows "\<Union> (range A) \<in> smallest_ccdi_sets \<Omega> M"
preserves_ide: fixes a :: "'a" assumes "C.ide a" shows "D.ide (G a)"
nn_expl_cond_expect_pos: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> real"   and Y :: "'c \<Rightarrow> 'd" assumes "\<forall>w\<in>space M. 0 \<le> X w" shows "\<forall>w\<in>space M. 0 \<le> expl_cond_expect M Y X w"
list_emb_Nil2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "list_emb P xs []" shows "xs = []"
lhs_step_bind: fixes m' :: "'a nres"   and m :: "'b nres"   and f :: "'b \<Rightarrow> 'a nres" assumes "nofail m' \<Longrightarrow> nofail m"   and "\<And>x. nf_inres m x \<Longrightarrow> f x \<le> m'" shows "m \<bind> f \<le> m'"
tendstoI: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and F :: "'d filter" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F x in F. dist (f x) l < e" shows "filterlim f (topological_space.nhds open l) F"
collapse_4: fixes "term" :: "Core.term" assumes "is_Abs term" shows "Abs (un_Abs1 term) (un_Abs2 term) = term"
swapSyms_id: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t" shows "swapSyms t a a = t"
normal_subgrp_intersection_normal: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and S :: "'c set" assumes "second_isomorphism_grp H G S" shows "S \<inter> H \<lhd> G\<lparr>carrier := S\<rparr>"
K2_abs: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "(proj2_abs v \<in> hyp2) = (v \<bullet> (M *v v) < 0)"
has_integral_empty: fixes f :: "'a \<Rightarrow> 'b" shows "(f has_integral (0::'b)) {}"
not_in_netmap: fixes i :: "nat"   and ns :: "'c net_state" assumes "i \<notin> net_ips ns" shows "netmap ns i = None"
Basis: fixes s :: "'d set"   and S :: "'d set set" assumes "s \<in> S" shows "generate_topology S s"
CombineStrip_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/{}\<^esub> P strip_guards (- F) c UNIV,UNIV" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A"
bind_eq_return_pmf_None: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "(p \<bind> f = return_pmf None) = (\<forall>x\<in>set_spmf p. f x = return_pmf None)"
is_group: shows "Group.group (G\<lparr>carrier := J\<rparr>)"
modulo_stream_def: fixes x :: "'a stream"   and y :: "'a stream" shows "x mod y = pure (mod) \<diamondop> x \<diamondop> y"
is_conjugation_sym: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "is_conjugation f g = is_conjugation g f"
IH_2: shows "singular_subdivision p (chain_map p g b) = chain_map p g (singular_subdivision p b)"
that: fixes P' :: "ccs" assumes "P \<longmapsto>\<alpha> \<prec> P'"   and "(P', Q') \<in> Rel" shows "thesis"
is_redI: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and a :: "'a \<Rightarrow>\<^sub>0 'c"   and b :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F a b" shows "punit.is_red F a"
monom_mult_eq_zero_iff: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(local.punit.monom_mult c t p = 0) = (c = (0::'c) \<or> p = 0)"