hcju/isabelleps huggingface dataset:
Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c\<^sub>1' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')" shows "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, s) \<rightarrow> (Seq c\<^sub>1' c\<^sub>2, s')"
pmf_positive: fixes x :: "'a"   and p :: "'a pmf" assumes "x \<in> set_pmf p" shows "0 < pmf p x"
smallstepsP_generates_proofstream: fixes \<pi>\<^sub>1 :: "Syntax.term list"   and eP :: "Syntax.term"   and i :: "nat"   and \<pi>\<^sub>2 :: "Syntax.term list"   and eP' :: "Syntax.term" assumes "\<lless>\<pi>\<^sub>1, eP\<ggreater> P\<rightarrow>i \<lless>\<pi>\<^sub>2, eP'\<ggreater>" obtains \<pi> :: "Syntax.term list" where "\<pi>\<^sub>2 = \<pi>\<^sub>1 @ \<pi>"
carrier_Gp: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "G \<guillemotright> H " shows "carrier (\<natural>\<^bsub>G\<^esub>H) = H"
in_carrier_3: shows "a [^] length q' \<in> carrier R"
N_zero_iff: fixes n :: "int" shows "(N n = 0) = (n = 0)"
qtable_project_fv: fixes n :: "nat"   and \<phi> :: "'b MFOTL.formula"   and R :: "'a list set"   and P :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" assumes "qtable (Suc n) (fv \<phi>) (mem_restr (lift_envs R)) P X" shows "qtable n (MFOTL.fvi (Suc 0) \<phi>) (mem_restr R) (\<lambda>v. \<exists>x. P ((if 0 \<in> fv \<phi> then Some x else None) # v)) (tl ` X)"
isomorphic_implies_hpar_3: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong> f'" shows "src f = src f'"
\<alpha>\<beta>_trans: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s'' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "s \<approx> s'"   and "s' \<approx> s''" shows "s \<approx> s''"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Lt_Rel of Leq_Rel \<Rightarrow> f1 | Lt_Rel \<Rightarrow> f2) = f2"
complex_mult_cnj: fixes z :: "complex" shows "z * cnj z = cor ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
assms_1: shows "x \<^sup>@ a \<cdot> y \<^sup>@ b = z \<^sup>@ c"
homeomorphic_empty_1: fixes S :: "'c set" shows "S homeomorphic {} = (S = {})"
\<phi>_in_hom_1: shows "\<guillemotleft>\<phi> : src p\<^sub>0 \<rightarrow> trg h\<guillemotright>"
subtract_mask_2: fixes p :: "'a word"   and n :: "nat" shows "p - (p AND NOT (mask n)) = p AND mask n"
main_ineq: shows "AE x in M. q (real_cond_exp M F X x) + psi (real_cond_exp M F X x) * (X x - real_cond_exp M F X x) \<le> q (X x)"
verit_minus_simplify_4: fixes b :: "'e" shows "- (- b) = b"
ds_ord_antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
partition_between_ref_correct: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and hi :: "nat"   and xs :: "'c list"   and lo :: "nat" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. R (h x) (h y) \<or> R (h y) (h x)"   and "hi < length xs"   and "lo \<le> hi" shows "partition_between_ref R h lo hi xs \<le> SPEC (uncurry (partition_spec R h xs lo hi))"
le_list_map: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "xs \<preceq> ys" shows "map f xs \<preceq> map f ys"
i_expand_i_take_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" shows "f \<odot> k \<Down> (n * k) = f \<Down> n \<odot> k"
eq_iff_11: fixes "term" :: "Syntax.term"   and terma :: "Syntax.term" shows "(Prj1 term = Prj1 terma) = (term = terma)"
domsMT_notND_DD: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set" assumes "dom (Cp (DenyAllFromTo a b)) \<inter> dom (Cp (DenyAllFromTo c d)) \<noteq> {}" shows "\<not> netsDistinct a c"
finsum_empty: fixes f :: "'d \<Rightarrow> 'a" shows "finsum R f {} = \<zero>"
vbex_cong: fixes A :: "V"   and B :: "V"   and P :: "V \<Rightarrow> bool"   and Q :: "V \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in>\<^sub>\<circ> B \<Longrightarrow> P x = Q x" shows "(\<exists>x\<in>elts A. P x) = (\<exists>x\<in>elts B. Q x)"
dom_eval: fixes f :: "'c local.term" assumes "Arr f" shows "local.dom \<lbrace>f\<rbrace> = \<lbrace>Dom f\<rbrace>"
simps_2: fixes v :: "'b" shows "headn (v)\<^sub>p = (\<lambda>m. (v)\<^sub>p)"
min_eint_simps_1: fixes m :: "int"   and n :: "int" shows "min (eint m) (eint n) = eint (min m n)"
nn: shows "AE x in M. \<forall>i. 0 \<le> f i x"
delete: shows "grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St) - grounding_of_clss (N_of_state St' \<union> P_of_state St' \<union> Q_of_state St') \<subseteq> sr.Rf (grounding_of_clss (N_of_state St' \<union> P_of_state St' \<union> Q_of_state St'))"
distinct_reorder: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "distinct (xs @ y # ys) = distinct (y # xs @ ys)"
SGCs: fixes \<phi>' :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "List.member (close SG \<phi>) \<phi>'" shows "seq_sem I \<phi>' = UNIV"
finite_set_rel_transfer_back: fixes s :: "'c set"   and s' :: "'d set"   and R :: "('c \<times> 'd) set" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "single_valued (R\<inverse>)"   and "finite s'" shows "finite s"
comp_def: fixes g :: "'a"   and f :: "'a" shows "S.comp g f = (if (arr f \<and> src f = a \<and> trg f = a) \<and> (arr g \<and> src g = a \<and> trg g = a) \<and> cod f = local.dom g then g \<cdot> f else null)"
tickFree_Cons: fixes a :: "'a event"   and t :: "'a event list" shows "tickFree (a # t) = (a \<noteq> tick \<and> tickFree t)"
shared_le_map_le: fixes m\<^sub>1 :: "nat \<Rightarrow> bool option"   and m\<^sub>2 :: "nat \<Rightarrow> bool option" assumes "m\<^sub>1 \<subseteq>\<^sub>s m\<^sub>2" shows "m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2"
denote_partial_density_operator: fixes dims :: "nat list"   and S :: "com"   and \<rho> :: "complex mat" assumes "state_sig.well_com dims S"   and "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)" shows "partial_density_operator (state_sig.denote dims S \<rho>)"
simplePath_length_less_V: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "Finite_Graph c"   and "u \<in> Graph.V c"   and "Graph.isSimplePath c u p v" shows "length p < card (Graph.V c)"
comp_inv_arr: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.inverse_arrows f g" shows "VV.comp g f = VV.dom f"
splitFace_distinct1: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVertexList :: "nat list" assumes "pre_splitFace g ram1 ram2 oldF newVertexList" shows "distinct (vertices (fst (snd (splitFace g ram1 ram2 oldF newVertexList))))"
partition_on_in_the_unique_part: fixes A :: "'c set"   and P :: "'c set set"   and x :: "'c" assumes "partition_on A P"   and "x \<in> A" shows "x \<in> (THE X. x \<in> X \<and> X \<in> P)"
carrier_FBinds: fixes Texp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var ttree"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var" assumes "TTreeAnalysisCarrier Texp Aexp" shows "carrier ((ExpAnalysis.AnalBinds Texp \<Gamma>\<cdot>ae) x) \<subseteq> fv \<Gamma>"
simps_20: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a acom"   and ve :: "'a acom" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> vd;; ve) = False"
n_c_def: fixes C :: "'av st option acom" shows "n\<^sub>c C = (\<Sum>a\<leftarrow>annos C. n\<^sub>o a (vars C))"
snd_Gauss_Jordan_in_ij_iarrays_PA: fixes P :: "'b iarray iarray"   and A :: "'b iarray iarray"   and i :: "nat"   and j :: "nat" shows "snd (Gauss_Jordan_in_ij_iarrays_PA (P, A) i j) = Gauss_Jordan_in_ij_iarrays A i j"
lift_gi_def: fixes f :: "'a \<Rightarrow> int"   and gi :: "bool \<times> 'a" shows "int.lift_gi f gi = (if fst gi then inv\<^bsub>int_group\<^esub> f (snd gi) else f (snd gi))"
ccw'_not_coll_5: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' a b c" shows "det3 c a b \<noteq> 0"
visit_subset_dfs: fixes ys :: "node list"   and g :: "(node \<times> node) list"   and xs :: "node list" shows "set ys \<subseteq> set (dfs g xs ys)"
measure_zero_union: fixes s :: "real set"   and t :: "real set" assumes "s \<in> sets M"   and "t \<in> sets M"   and "Sigma_Algebra.measure M t = 0" shows "Sigma_Algebra.measure M (s \<union> t) = Sigma_Algebra.measure M s"
simps_1: fixes Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" shows "valid_path Pi [] = True"
m_c_def: fixes C :: "'av st option acom" shows "m\<^sub>c C = (\<Sum>a\<leftarrow>annos C. m\<^sub>o a (vars C))"
prems: shows "del x \<langle>t1, x2, t2\<rangle> = Some t'"
2: shows "integrable (lebesgue_on {- pi..pi}) (\<lambda>x. f x * (sin (x + x * real n) * sin (t + t * real n)))"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_trms_subset: fixes G :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)" shows "trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G \<subseteq> pair ` set F \<union> pair ` set D"
fold_mset_add_mset: fixes s :: "'a"   and x :: "'a"   and M :: "'a multiset" shows "fold_mset (\<^bold>*) s (add_mset x M) = x \<^bold>* fold_mset (\<^bold>*) s M"
Assoc: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" shows "(\<Psi>', S \<parallel> (T \<parallel> U), S \<parallel> T \<parallel> U) \<in> Rel"
vcard_vintersection_left: fixes A :: "V"   and B :: "V" shows "vcard (A \<inter>\<^sub>\<circ> B) \<subseteq>\<^sub>\<circ> vcard A"
singleton_iff: fixes b :: "'x"   and a :: "'x" shows "(b \<in> {a}) = (b = a)"
starfun_add: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d star"   and g :: "'d \<Rightarrow> 'c" shows "(*f* f) x + (*f* g) x = (*f* (\<lambda>x. f x + g x)) x"
Limsup_filtermap_le: fixes f :: "'b \<Rightarrow> 'd"   and F :: "'b filter"   and g :: "'d \<Rightarrow> 'e" assumes "inj f" shows "Limsup (filtermap f F) g \<le> Limsup F (\<lambda>x. g (f x))"
intros: fixes x :: "('a \<times> 'b, 'c \<times> 'd, 'e) Vardi.var"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "set1_var x \<subseteq> {(x, y). R x y}"   and "set2_var x \<subseteq> {(x, y). S x y}" shows "rel_var R S (map_var fst fst x) (map_var snd snd x)"
prognosis_Alts: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "prognosis ae as a (\<Gamma>, if b then e1 else e2, S) \<sqsubseteq> prognosis ae (a # as) 0 (\<Gamma>, Bool b, Alts e1 e2 # S)"
Ex1_unfold: fixes P :: "'a \<Rightarrow> bool" shows "Ex1 P \<equiv> \<exists>x. {x. P x} = {x}"
in_domains_char: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "('a, 'b) concrete_category.arr"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp" shows "(a \<in> partial_magma.domains (concrete_category.COMP Obj Hom Comp) f) = ((f \<noteq> concrete_category.Null \<and> concrete_category.Dom f \<in> Obj \<and> concrete_category.Cod f \<in> Obj \<and> concrete_category.Map f \<in> Hom (concrete_category.Dom f) (concrete_category.Cod f)) \<and> a = concrete_category.MkArr (concrete_category.Dom f) (concrete_category.Dom f) (Id (concrete_category.Dom f)))"
closure_of_eq: fixes X :: "'b topology"   and S :: "'b set" shows "(X closure_of S = S) = closedin X S"
Limsup_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" shows "Limsup F f = (INF P\<in>{P. eventually P F}. Sup (f ` {x. P x}))"
sff: shows "square_free_factorization p (c, bs1 @ (a, i) # bs2)"
less_x_plus_1: fixes x :: "'a word"   and y :: "'a word" assumes "x \<noteq> - 1" shows "(y < x + 1) = (y < x \<or> y = x)"
dens_f: fixes x :: "'a" assumes "x \<in> space M" shows "has_density (f x) R (\<delta>f x)"
FF_def: shows "I.FF \<equiv> \<lambda>\<mu>\<nu>. if B.VV.arr \<mu>\<nu> then (local.map (fst \<mu>\<nu>), local.map (snd \<mu>\<nu>)) else B.VV.null"
exe_trap_wr_pc_result: fixes s :: "'b sparc_state" shows "snd (exe_trap_wr_pc () s) = False"
assms: fixes i :: "nat" shows "(0::'a) \<le> nth_default (0::'a) xs i"
frontier_cball: fixes a :: "'a"   and e :: "real" shows "frontier (cball a e) = sphere a e"
qe_eq_repeat_eval: fixes xs :: "real list"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "length xs = var" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (qe_eq_repeat var L F) (xs @ x # \<Gamma>))"
CACH_INJECT: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and n :: "nat" assumes "deriv G C m l (Cachera n)" shows "derivAssum G C m l (Cachera n)"
composable_char\<^sub>P\<^sub>B\<^sub>H: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "prebicategory_with_homs V H \<a> src trg" shows "(H \<nu> \<mu> \<noteq> partial_magma.null V) = (partial_magma.arr V \<mu> \<and> partial_magma.arr V \<nu> \<and> src \<nu> = trg \<mu>)"
pickBinp_def: fixes binp :: "'i \<Rightarrow> 'j set option" shows "pickBinp binp \<equiv> lift pick binp"
div_eq_minus1: fixes b :: "int" assumes "0 < b" shows "- 1 div b = - 1"
num_not0: fixes v :: "'a" assumes "v \<in> verts G"   and "v \<noteq> s"   and "num v \<noteq> \<infinity>" shows "num v \<noteq> enat 0"
dec_seq_mem: fixes a :: "int"   and A :: "int set"   and n :: "nat" assumes "a \<in> A"   and "A \<subseteq> Zset"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "dec_seq A a n \<in> A"
pointermap_getmk_def: fixes a :: "'a"   and m :: "'a pointermap" shows "pointermap_getmk a m \<equiv> case getentry m a of None \<Rightarrow> let u = pointermap_insert a m in (the (getentry u a), u) | Some p \<Rightarrow> (p, m)"
main: shows "scalar_prod r v * a + scalar_prod r (map (\<lambda>k. scalar_prod (tm ! k) cc) [0..<length r]) = scalar_prod r (map (\<lambda>k. v ! k * a + scalar_prod (tm ! k) cc) [0..<length r])"
blank_def: fixes P :: "'a prog"   and C :: "char list" shows "blank P C \<equiv> (C, init_fields (ifields P C))"
Says_Tgs_AKcryptSK: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and X :: "msg"   and evs :: "event list"   and authK' :: "nat" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, X\<rbrace>) \<in> set evs"   and "authK' \<noteq> authK"   and "evs \<in> kerbIV" shows "\<not> AKcryptSK authK' servK evs"
INF_superset_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<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 B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "B \<subseteq> A"   and "\<And>x. x \<in> B \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
pr_comp1_2': fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec1'"   and "pr_conv_2_to_1 g \<in> PrimRec1'" shows "pr_conv_2_to_1 (\<lambda>x y. f (g x y)) \<in> PrimRec1'"
weak_Local: fixes G :: "'b triple set"   and P :: "'b assn"   and c :: "com"   and Q :: "'b assn"   and Y :: "loc"   and a :: "state \<Rightarrow> nat" assumes "G|-{P}. c .{Q}"   and "\<forall>k Z s. Q Z s \<longrightarrow> Q Z (s[Loc Y::=k])" shows "G|-{\<lambda>Z s. P Z (s[Loc Y::=a s])}. LOCAL Y:=a IN c .{Q}"
prems_1: shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta>"
filtermap_bot_iff: fixes f :: "'d \<Rightarrow> 'c"   and F :: "'d filter" shows "(filtermap f F = bot) = (F = bot)"
return_bind: fixes x :: "int"   and f :: "int \<Rightarrow> 'm" shows "bind (return x) f = f x"
rep_eq: fixes x :: "'c stone_phi_pair" shows "Rep_stone_phi_pair (- x) = stone_phi.pairs_uminus (Rep_stone_phi_pair x)"
fmempty_eqvt: fixes p :: "perm" shows "p \<bullet> {$$} = {$$}"
straight_path_diffrentiable_x: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and b :: "real"   and y2 :: "real"   and y1 :: "real" assumes "\<gamma> = (\<lambda>x. (b, y2 + y1 * x))" shows "\<forall>x. \<gamma> differentiable at x"
measurable_coordinatewise_then_product: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd"   and M :: "'c measure" assumes "\<And>i. (\<lambda>x. f x i) \<in> borel_measurable M" shows "f \<in> borel_measurable M"
chain_with_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.chain_with betw x y z X \<equiv> betw x y z \<and> x \<in> X \<and> y \<in> X \<and> z \<in> X \<and> (\<exists>f. TernaryOrdering.ordering f betw X)"
monom_subst_inv: fixes \<sigma> :: "'c \<Rightarrow> ('d monom \<times> 'e) list"   and m :: "'c monom" assumes "\<And>x. poly_inv (\<sigma> x)" shows "poly_inv (monom_subst \<sigma> m)"
FUadmit_def: fixes \<sigma> :: "('a, 'b, 'c) subst"   and \<theta> :: "('a, 'b, 'c) formula"   and U :: "('c + 'c) set" shows "FUadmit \<sigma> \<theta> U = (\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF \<theta>)) \<inter> U = {})"
rai: shows "local.select_correct_factor_int_poly init p = rai"
raw_induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and y :: "nat"   and xa :: "nat" assumes "\<And>test2a x xa. \<lbrakk>\<And>x. test2a x = test2a x; \<not> P x xa\<rbrakk> \<Longrightarrow> xa \<in> insert x (sup (\<lambda>x. {y. \<not> P x y}) test2 (Suc x))"   and "y \<notin> test2 xa" shows "P xa y"
FunctorMapsTo: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme"   and f :: "'f"   and X :: "'d"   and Y :: "'d" assumes "Ftor F : A \<longrightarrow> B"   and "f maps\<^bsub>A\<^esub> X to Y" shows "F ## f maps\<^bsub>B\<^esub> F @@ X to F @@ Y"
wfactors_exist: fixes a :: "'a" assumes "a \<in> carrier G" shows "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a"
lagrange_finite: fixes H :: "'a set" assumes "finite (carrier G)"   and "subgroup H (add_monoid G)" shows "card (rcosets\<^bsub>add_monoid G\<^esub> H) * card H = order (add_monoid G)"
cross7_add_right: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec"   and z :: "(real, 7) vec" shows "x \<times>\<^sub>7 (y + z) = x \<times>\<^sub>7 y + x \<times>\<^sub>7 z"
admissible_leI: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.ccpo Sup less_eq less"   and "mcont luba orda Sup less_eq f"   and "mcont luba orda Sup less_eq g" shows "ccpo.admissible luba orda (\<lambda>x. less_eq (f x) (g x))"
Im_i_times: fixes z :: "complex" shows "Im (\<i> * z) = Re z"
usubstappf_pred2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and p :: "char"   and r :: "fml"   and U :: "variable set"   and \<theta> :: "trm" assumes "SPreds \<sigma> p = Afml r"   and "FVF r \<inter> U \<noteq> {}" shows "usubstappf \<sigma> U (Pred p \<theta>) = undeff"
decl_acc_modi_def: fixes d :: "'a decl_scheme" shows "accmodi d = access d"
angle_actrans_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<langle>P\<rangle>_v \<equiv> \<not> _actrans (\<not> P) v"
cfExtendFile_def: fixes cfile :: "CFile"   and byteIndex :: "nat" shows "cfExtendFile cfile byteIndex \<equiv> cfile\<lparr>fileSize := Suc byteIndex, nextFreeBlock := Suc (byteIndex div blockSize)\<rparr>"
lookup_def: fixes F :: "'b Abs_State.st"   and x :: "char list" shows "lookup F x = (if x \<in> set (Abs_State.dom F) then fun F x else \<top>)"
l2_inv7_step3: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Nr :: "msg"   and gny :: "msg" shows "{l2_inv7 \<inter> l2_inv1 \<inter> l2_inv4 \<inter> l2_inv6'} l2_step3 Ra A B Nr gny {> l2_inv7}"
that: shows "y \<in> aform_err e ((X, xs), d)"
is_coext_def: fixes D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c"   and e :: "'c"   and x' :: "'b"   and f :: "'c"   and g :: "'b" assumes "arrow_from_functor D C F x y e" shows "arrow_from_functor.is_coext D C F x e x' f g \<equiv> partial_magma.in_hom D g x' x \<and> f = C e (F g)"
abs_eq: fixes x :: "((real, 'a) vec, 'c) vec" shows "blinfun_of_vmatrix x = Blinfun ((*v) x)"
mem_Units: fixes y :: "'a" shows "(y \<in> Units mk_monoid) = (y dvd (1::'a))"
image_image: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "f ` g ` A = {f (g x) |. x \<in> A}"
indRelSTEQ_impl_SRel_and_TRel_weakly_reflect_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelSTEQ SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs SRel SWB"
hyps_3: assumes "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "A\<^sub>F \<sharp>* P"   and "A\<^sub>G \<sharp>* P"   and "A\<^sub>P \<sharp>* P" shows "\<Psi>\<^sub>G \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
surj_const_mult: fixes a :: "'c" assumes "a \<in> carrier H" shows "(\<otimes>\<^bsub>H\<^esub>) a ` carrier H = carrier H"