hcju/isabelleps huggingface dataset:
iMOD_icard: fixes m :: "nat"   and r :: "nat" assumes "0 < m" shows "icard [ r, mod m ] = \<infinity>"
b_rel_def: fixes R :: "('a \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" shows "b_rel R P \<equiv> R \<inter> UNIV \<times> Collect P"
insert_position_element_technical: fixes as :: "('c \<times> 'd \<times> bool) list"   and a :: "'c"   and cs :: "('c \<times> 'd \<times> bool) list"   and ds :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool"   and aa :: "'d"   and b :: "bool" assumes "\<forall>x\<in>set as. a \<noteq> fst x"   and "insert_body (cs @ ds) e = cs @ e # ds" shows "insert (as @ (a, aa, b) # cs @ ds) e (Some a) = Some (as @ (a, aa, b) # cs @ e # ds)"
induct: fixes P1 :: "expr \<Rightarrow> bool"   and P2 :: "expr list \<Rightarrow> bool"   and expr :: "expr" assumes "\<And>x. P1 (new x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Cast x1 x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lparr>x1\<rparr>x2)"   and "\<And>x. P1 (Val x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 \<guillemotleft>x2\<guillemotright> x3)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (x1:=x2)"   and "\<And>x1 x2 x3. P1 x1 \<Longrightarrow> P1 (x1\<bullet>x2{x3})"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P1 x4\<rbrakk> \<Longrightarrow> P1 (x1\<bullet>x2{x3} := x4)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x1; P2 x4\<rbrakk> \<Longrightarrow> P1 (Call x1 x2 x3 x4)"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P1 {x1:x2; x3}"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (if (x1) x2 else x3)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (while (x1) x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 (throw x)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 expr"
parts_mpair_pwd_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and n :: "nat" assumes "s \<turnstile> s'"   and "\<lbrace>X, Y\<rbrace> \<in> parts (used s')"   and "\<lbrace>X, Y\<rbrace> \<notin> parts (used s)"   and "X = Pwd n \<or> Y = Pwd n" shows "{X, Y} \<subseteq> spied s'"
isCont_real_sqrt: fixes x :: "real" shows "isCont sqrt x"
outstanding_volatile_writes_unowned_by_others: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'p"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'p memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'p"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'p memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>i = {}"
is_sig_GB_uptD2: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and v :: "'t" assumes "is_sig_GB_upt d G u"   and "v \<prec>\<^sub>t u" shows "is_sig_GB_in d G v"
tendsto_blinfun_of_matrix: fixes b :: "'f \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> real"   and a :: "'d \<Rightarrow> 'e \<Rightarrow> real"   and F :: "'f filter" assumes "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk> \<Longrightarrow> ((\<lambda>n. b n i j) \<longlongrightarrow> a i j) F" shows "((\<lambda>n. blinfun_of_matrix (b n)) \<longlongrightarrow> blinfun_of_matrix a) F"
lhs_def: shows "lhs \<equiv> \<lambda>plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1. linear_ow (Collect (Domainp B)) plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 (Collect (Domainp B))"
hyps_1: shows "z \<turnstile> p[App d []/0]"
SAStates_ChiPlus2: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and SA :: "('a, 'b, 'c) seqauto" assumes "(S, T) \<in> ChiPlus A"   and "SA \<in> SAs A"   and "S \<in> States SA" shows "T \<notin> States SA"
TS_index_1: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<in> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs" shows "Suc (length xs) \<le> i"
terminates_Normal_elim_cases_6: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and b :: "'b set"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "'b" assumes "\<Gamma>\<turnstile>Cond b c1 c2 \<down> Normal s" obtains "s \<in> b" and "\<Gamma>\<turnstile>c1 \<down> Normal s"   | "s \<notin> b" and "\<Gamma>\<turnstile>c2 \<down> Normal s"
hk: fixes x :: "'a" shows "ord_spmf (=) (h x) (k x)"
intros_3: fixes P\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>1 :: "com"   and P\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and P\<^sub>3 :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1 {P\<^sub>2}"   and "\<turnstile>\<^sub>t {P\<^sub>2} c\<^sub>2 {P\<^sub>3}" shows "\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1;; c\<^sub>2 {P\<^sub>3}"
owner_document': shows "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r owner_document'"
3: shows "A n \<in> lambda_system \<Omega> M f"
t1_space_derived_set_of_infinite_openin: fixes X :: "'b topology" shows "t1_space X = (\<forall>S. X derived_set_of S = {x \<in> topspace X. \<forall>U. x \<in> U \<and> openin X U \<longrightarrow> infinite (S \<inter> U)})"
hequiv_sym': fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and j :: "'b" assumes "Hintikka A H"   and "i \<in> A"   and "hequiv H i j" shows "hequiv H j i"
matchesI: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b"   and env :: "(name, 'b) fmap"   and u :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "subst t env = u" shows "term.matches subst t u"
connected_def: fixes G :: "('c, 'd) pre_digraph" shows "Digraph_Component.connected G \<equiv> strongly_connected (with_proj (mk_symmetric G))"
range_composition: fixes f :: "'c::type \<Rightarrow> 'a::type"   and g :: "'b::type \<Rightarrow> 'c::type" shows "range (\<lambda>x::'b::type. (f::'c::type \<Rightarrow> 'a::type) ((g::'b::type \<Rightarrow> 'c::type) x)) = f ` range g"
lappend_code_1: fixes ys :: "'a llist" shows "lappend LNil ys = ys"
in_cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and h :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> \<Theta>[F](h)) = (g \<in> \<Theta>[F](h))"
least_non_zero_position_of_vector_def: fixes A :: "'a iarray" shows "least_non_zero_position_of_vector A = least_non_zero_position_of_vector_from_index A 0"
subcls'_def: fixes G :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "subcls' G = (subcls1p G)\<^sup>*\<^sup>*"
assms_2: fixes x :: "'a \<Rightarrow>\<^sub>0 int"   and y :: "'a \<Rightarrow>\<^sub>0 int" assumes "P x"   and "P y" shows "P (x - y)"
grid_interval_notempty: fixes n :: "nat" shows "t 0 \<le> t n"
that: fixes p :: "'a" assumes "p \<in> S"   and "p \<in> T" shows "thesis"
runit_char_3: fixes f :: "'a" assumes "Right.ide f" shows "\<exists>!\<mu>. Right.in_hom \<mu> (Right.R f) f \<and> Right.R \<mu> = Right.comp (f \<star> \<i>[src f]) \<a>[f, src f, src f]"
sum_map_hom: fixes M1 :: "('c \<times> 'c) set"   and M2 :: "('c \<times> 'c) set" assumes "LinearCombinations.submodule R M1 (direct_sum (md N1) (md N2))"   and "LinearCombinations.submodule R M2 (direct_sum (md N1) (md N2))" shows "mod_hom R (direct_sum (l.md M1) (l.md M2)) (direct_sum (md N1) (md N2)) (\<lambda>v. fst v \<oplus>\<^bsub>direct_sum (md N1) (md N2)\<^esub> snd v)"
OneRootState_Source: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> Source TS"   and "S \<in> States (HARoot (HA ST))" shows "S = RootState ST"
countable_support: fixes M :: "'b measure" assumes "finite_measure M" shows "countable {x. Sigma_Algebra.measure M {x} \<noteq> 0}"
set_integrable_scaleR_right: fixes a :: "real"   and M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f" shows "set_integrable M A (\<lambda>t. a *\<^sub>R f t)"
eventually_at_right: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder_topology less_eq less open"   and "less x y" shows "eventually P (topological_space.at_within open x (ord.greaterThan less x)) = (\<exists>b. less x b \<and> (\<forall>y. less x y \<longrightarrow> less y b \<longrightarrow> P y))"
ex_unique_reduced_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "\<exists>!G. G \<subseteq> dgrad_p_set d m \<and> finite G \<and> is_reduced_GB G \<and> pmdl G = pmdl F"
valid_cases: fixes X :: "msg" assumes "X \<in> valid" obtains A :: "agent" and B :: "agent" and M :: "msg" where "X = implInsec A__ B__ M__" and "M__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implConfid Aa__ Ba__ Ma__" and "Ma__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implAuth Ab__ Bb__ Mb__" and "Mb__ \<in> payload"   | A :: "agent" and B :: "agent" and M :: "msg" where "X = implSecure A B M" and "M \<in> payload"
isSome_eq: fixes x :: "'b option" shows "isSome x = (\<exists>y. x = Some y)"
S5Basic_2: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<diamond>\<^bold>\<box>\<phi> in v]"
submodule_subset1: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and h :: "'e" assumes "R module M"   and "submodule R M H"   and "h \<in> H" shows "h \<in> carrier M"
pospos: fixes A :: "'b"   and LL :: "'b literal multiset" shows "{#Pos (atm_of x). x \<in># {#L \<in># LL. L = Pos A#}#} = {#L \<in># LL. L = Pos A#}"
wp_def: fixes x :: "'b"   and p :: "'b Assertion" shows "wp x p = abs_wpt (x * {\<cdot> p })"
trg_in_targets: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "trg \<mu> \<in> targets \<mu>"
lookupEvalHeap': fixes \<Gamma> :: "('b \<times> 'a) list"   and v :: "'b"   and e :: "'a"   and f :: "'a \<Rightarrow> 'c" assumes "map_of \<Gamma> v = Some e" shows "evalHeap \<Gamma> f v = f e"
inferences_between_def: fixes CC :: "'a literal multiset set"   and C :: "'a literal multiset" shows "ord_FO_resolution.inferences_between CC C = {\<gamma> \<in> ord_FO_\<Gamma>. infer_from (CC \<union> {C}) \<gamma> \<and> C \<in># prems_of \<gamma>}"
elims_2: fixes x :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "invc x" obtains "x = \<langle>\<rangle>"   | l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and a :: "'c \<times> 'd" and c :: "tcolor" and uu_ :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (a, c, uu_), r\<rangle>" and "invc l \<and> invc r \<and> (c = Red \<longrightarrow> color l = Black \<and> color r = Black)"
inext_cut_less_conv: fixes n :: "nat"   and I :: "nat set"   and t :: "nat" assumes "inext n I < t" shows "inext n (I \<down>< t) = inext n I"
quantale_closed_maps_iff: fixes f :: "'c \<Rightarrow> 'd" shows "(f \<in> quantale_closed_maps) = (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y))"
id_eq_prop_prop_8: fixes v :: "i"   and p :: "\<o>"   and q :: "\<o>" shows "[p \<^bold>= q \<^bold>\<rightarrow> q \<^bold>= p in v]"
set_iterator_abs_genord_trivial: shows "set_iterator_abs_genord id (\<lambda>_. True) = set_iterator_genord"
simps_2: fixes N :: "nat"   and n :: "nat"   and j :: "nat"   and i :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_shift_n N ((n, j, i) # L) = (n - N, j, i) # ladder_shift_n N L"
intro: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y. R x y \<longrightarrow> R y x \<longrightarrow> x = y" shows "Quickcheck_Examples.antisym R"
2: shows "range (flow0 y) = \<omega>_limit_set x"
obj_is_equivalence_map: fixes a :: "'a" assumes "obj a" shows "equivalence_map a"
f_def: fixes A :: "V set" shows "f A \<equiv> SOME i. \<exists>x y. A = {x, y} \<and> x < y \<and> (\<pi> x < \<pi> y \<and> i = 0 \<or> \<pi> y < \<pi> x \<and> i = 1)"
Ea_1: shows "Ea \<subseteq> Pow (space A)"
ring_simprules_17: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> (\<ominus> x \<oplus> y) = y"
list_all_lemma: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and Q :: "'a \<Rightarrow> bool" assumes "list_all P xs"   and "\<And>x. P x \<Longrightarrow> Q x" shows "list_all Q xs"
map_simp_ide: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "cmp\<^sub>U\<^sub>P a = \<Phi>\<^sub>o a"
measurable_comp: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and g :: "'e \<Rightarrow> 'f"   and L :: "'f measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "g \<circ> f \<in> M \<rightarrow>\<^sub>M L"
integrals_eq: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and x :: "real" assumes "f 0 = g 0"   and "\<And>x. ((\<lambda>x. f x - g x) has_real_derivative 0) (at x)" shows "f x = g x"
contour_integral_add: fixes f1 :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and f2 :: "complex \<Rightarrow> complex" assumes "f1 contour_integrable_on g"   and "f2 contour_integrable_on g" shows "contour_integral g (\<lambda>x. f1 x + f2 x) = contour_integral g f1 + contour_integral g f2"
subst_preserves_fresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "fresh zs z X"   and "fresh zs z Y" shows "fresh zs z (X #[Y / y]_ys)"
diff_Suc_Suc: fixes m :: "nat"   and n :: "nat" shows "Suc m - Suc n = m - n"
funpow_decreasing: fixes m :: "nat"   and n :: "nat"   and f :: "'b \<Rightarrow> 'b" assumes "m \<le> n"   and "mono f" shows "(f ^^ m) bot \<le> (f ^^ n) bot"
mem_rel_interior_cball: fixes x :: "'a"   and S :: "'a set" shows "(x \<in> rel_interior S) = (x \<in> S \<and> (\<exists>e>0. cball x e \<inter> affine hull S \<subseteq> S))"
surj_on_add_mult: fixes x :: "nat"   and q :: "nat"   and y :: "nat" assumes "coprime x q"   and "inj_on (\<lambda>b. (y + x * b) mod q) {..<q}" shows "(\<lambda>b. (y + x * b) mod q) ` {..<q} = {..<q}"
frac_frac: fixes x :: "'a" shows "frac (frac x) = frac x"
exists_def: fixes p :: "'d \<Rightarrow> bool"   and as :: "'d iarray" shows "IArray.exists p as = (\<exists>a\<in>set (IArray.list_of as). p a)"
connected_open_path_connected: fixes S :: "'b set" assumes "open S"   and "connected S" shows "path_connected S"
IH_1: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "F, F, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
bisim': shows "P,es,h' \<turnstile> (es'', xs'') [\<leftrightarrow>] (stk'', loc', pc' - length (compE2 e), xcp')"
assms: shows "f \<in> M \<rightarrow>\<^sub>M N"
cbv_type_preservation: fixes t :: "lam"   and t' :: "lam"   and \<Gamma> :: "(name \<times> ty) list"   and T :: "ty" assumes "t \<longrightarrow>cbv t'"   and "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile> t' : T"
wt_ge_arity_head_if_\<delta>_gt_0: fixes s :: "('s, 'v) tm" assumes "0 < \<delta>" shows "arity_hd (head s) \<le> enat (wt s)"
ncols_iarray_def: fixes A :: "'d iarray iarray" shows "ncols_iarray A = IArray.length (A !! 0)"
add: fixes x :: "'c"   and y :: "'c"   and ns :: "('c \<times> 'c) set"   and X :: "'c multiset"   and Y :: "'c multiset" assumes "(x, y) \<in> ns"   and "(X, Y) \<in> multpw ns" shows "(add_mset x X, add_mset y Y) \<in> multpw ns"
SpanD_lincomb_arb_len_coeffs: fixes ms :: "'v list" assumes "set ms \<subseteq> V" shows "RSpan ms = {rs \<bullet>\<cdot> ms |rs. set rs \<subseteq> UNIV}"
assms_2: shows "u \<cdot> r \<^sup>@ m = r \<^sup>@ m \<cdot> u"
openin_subtopology_alt: fixes X :: "'a topology"   and U :: "'a set"   and S :: "'a set" shows "openin (subtopology X U) S = (S \<in> (\<inter>) U ` Collect (openin X))"
arr_cod: fixes f :: "'a \<times> 'a" assumes "H.VV.arr f" shows "H.VV.arr (H.VV.cod f)"
absorb2: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "a \<cdot> b = b"
knows2know_neg_k: fixes A :: "specID"   and m :: "Keys" assumes "\<not> knows A [kE m]" shows "\<not> know A (kKS m)"
prefixeq_iprefix_trans: fixes xs :: "'d list"   and ys :: "'d list"   and f :: "nat \<Rightarrow> 'd" assumes "prefix xs ys"   and "ys \<sqsubseteq> f" shows "xs \<sqsubseteq> f"
false_not_true: fixes d :: "'dyn" assumes "is_false d" shows "\<not> is_true d"
simps_3: fixes p :: "pf"   and q :: "pf"   and xs :: "nat list" shows "eval_pf (And p q) xs = (eval_pf p xs \<and> eval_pf q xs)"
CollectI: fixes P :: "'k \<Rightarrow> bool"   and a :: "'k" assumes "P a" shows "a \<in> {x. P x}"
not_Cons_self: fixes xs :: "'e list"   and x :: "'e" shows "xs \<noteq> x # xs"
eq_free_eq_subst_scheme_list: fixes A :: "type_scheme list"   and S1 :: "nat \<Rightarrow> typ"   and S2 :: "nat \<Rightarrow> typ" assumes "\<forall>n. n \<in> free_tv A \<longrightarrow> S1 n = S2 n" shows "$ S1 A = $ S2 A"
horner_sum_of_bool_2_concat: fixes ws :: "'b word list" shows "horner_sum of_bool 2 (concat (map (\<lambda>x. map (bit x) [0..<LENGTH('b)]) ws)) = horner_sum uint (2 ^ LENGTH('b)) ws"
real_of_integer_def: shows "real_of_integer = Real.real_of_int \<circ> int_of_integer"
process_coind: fixes \<phi> :: "'a process \<Rightarrow> 'a process \<Rightarrow> bool"   and p :: "'a process"   and p' :: "'a process" assumes "\<phi> p p'"   and "\<And>p p'. \<phi> p p' \<Longrightarrow> isAction p = isAction p' \<and> isChoice p = isChoice p'"   and "\<And>a a' p p'. \<phi> (Action a p) (Action a' p') \<Longrightarrow> a = a' \<and> \<phi> p p'"   and "\<And>p q p' q'. \<phi> (Choice p q) (Choice p' q') \<Longrightarrow> \<phi> p p' \<and> \<phi> q q'" shows "p = p'"
columns_transpose: fixes A :: "(('b, 'c) vec, 'd) vec" shows "columns (Finite_Cartesian_Product.transpose A) = rows A"
abs_eq: fixes y :: "'c" shows "\<guillemotleft>y\<guillemotright> = mk\<^sub>e (\<lambda>b. y)"
quo_div: fixes x :: "nat"   and y :: "nat" shows "quo [x, y] = x div y"
AssertionStatEqSym: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" assumes "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>' \<simeq> \<Psi>"
Suc_eq_plus1_left: fixes n :: "nat" shows "Suc n = 1 + n"
hyps: fixes A :: "(('a, 'cols) vec, 'rows) vec" assumes "i \<le> nrows A" shows "matrix_to_iarray (Hermite_of_upt_row_i A i ass res) = Hermite_of_upt_row_i_iarrays (matrix_to_iarray A) i ass res"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Apply (VE (Lambda x e)) (VE v)])"
image_reduced_homology_group: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "topspace X \<inter> S \<noteq> {}" shows "hom_induced p X {} X S id ` carrier (reduced_homology_group p X) = hom_induced p X {} X S id ` carrier (homology_group p X)"
euler_approx_aux_Suc: fixes m :: "nat" shows "euler_approx_aux (Suc m) = 1 + Suc m * euler_approx_aux m"
entails_formula_def: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" shows "entails_formula S1 S2 = (\<forall>C\<in>S2. entails S1 C)"
auxhyp_prenum: fixes e :: "nat" shows "\<phi> (the (auxhyp e)) = prenum e"
6: shows "val_Zp (pderiv f \<bullet> \<one>) \<le> 1"
right_ideal_def: fixes I :: "'a set" shows "right_ideal I = (subgroup I \<and> (\<forall>x\<in>I. \<forall>r. x * r \<in> I))"
cstate_poss_def: fixes p :: "lambda" shows "cstate_poss p = calls p \<times> maps_over (labels p) UNIV \<times> smaps_over (vars p \<times> UNIV) (proc_poss p) \<times> UNIV"
of_char_Char: fixes b0 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and b3 :: "bool"   and b4 :: "bool"   and b5 :: "bool"   and b6 :: "bool"   and b7 :: "bool" shows "of_char (Char b0 b1 b2 b3 b4 b5 b6 b7) = horner_sum of_bool (2::'a) [b0, b1, b2, b3, b4, b5, b6, b7]"
InvariantCEntailedAfterLearn: fixes conflictFlag :: "bool"   and F :: "Literal list list"   and C :: "Literal list"   and F' :: "Literal list list" assumes "InvariantCEntailed conflictFlag F C"   and "F' = F @ [C]" shows "InvariantCEntailed conflictFlag F' C"
Units_l_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> Units G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(x \<otimes> y = x \<otimes> z) = (y = z)"
l_minus: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "\<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
wrapH_def: shows "wrapH \<equiv> \<Lambda> f xs. H2list\<cdot>(f\<cdot>xs)"
neutral_ow_def: fixes U :: "'a set"   and z :: "'a" shows "neutral_ow U z \<equiv> z \<in> U"
path_component_of_set: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "path_component_of X x y = (\<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y)"
M_U_completeness: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "UM_Model wtFsym wtPsym arOf resOf parOf \<Phi> D intF intP" shows "CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> (\<lambda>\<sigma>. D) intF intP"
fconst_on_eq_ftimes: fixes A :: "V"   and c :: "V" shows "fconst_on A c = A \<times>\<^sub>\<bullet> set {c}"
left_lang_def: fixes q :: "hf" shows "left_lang q \<equiv> {u. nextl (init M) u = q}"
value_cases: fixes t :: "sterm" obtains name :: "name" and vs :: "sterm list" where "list_all is_value vs" and "t = name $$ vs" and "name |\<in>| C"   | cs :: "(Term_Class.term \<times> sterm) list" where "t = Sabs cs"   | "\<not> is_value t"
closure_def: fixes S :: "'a set" shows "closure S = S \<union> {x. x islimpt S}"
"2_1": shows "list_all2 (\<in>) (stake (length as) xs) (stake (length as) (cycle as))"
Vr_mem_f_mem: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)" shows "x \<in> carrier K"