hcju/isabelleps huggingface dataset:
new1_less_sum: fixes \<phi> :: "'d ltlr" shows "size_set (new1 \<phi>) < size_set {\<phi>}"
subset_implies_remove_empty: fixes M :: "'a multiset"   and S :: "'a set" assumes "set_mset M \<subseteq> S" shows "M -s S = {#}"
select_convs_1: fixes init :: "'a set"   and trans :: "('a \<times> 'a) set"   and more :: "'b" shows "init \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = init"
unit_not_fn: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b trm" shows "Unit \<noteq> Fn x T A"
inj_on_g: fixes H :: "'a set"   and a :: "'a" assumes "H \<subseteq> carrier G"   and "a \<in> carrier G" shows "inj_on (\<lambda>y. y \<otimes> a) H"
hyperplane_eq_empty: fixes a :: "'b"   and b :: "real" shows "({x. a \<bullet> x = b} = {}) = (a = (0::'b) \<and> b \<noteq> 0)"
assms_2: fixes s :: "'a stream" shows "P (shd s) s"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "semiring_hom hom" shows "ring_hom hom"
disjnt_inj_on_iff: fixes f :: "'b \<Rightarrow> 'c"   and \<A> :: "'b set set"   and X :: "'b set"   and Y :: "'b set" assumes "inj_on f (\<Union> \<A>)"   and "X \<in> \<A>"   and "Y \<in> \<A>" shows "disjnt (f ` X) (f ` Y) = disjnt X Y"
PO_rhoare_def: fixes pre :: "('a \<times> 'b) set"   and Ra :: "('a \<times> 'a) set"   and Rc :: "('b \<times> 'b) set"   and post :: "('a \<times> 'b) set" shows "{pre} Ra, Rc {> post} \<equiv> pre O Rc \<subseteq> Ra O post"
alg_dvd_uminus_right: fixes x :: "'a"   and y :: "'a" assumes "x alg_dvd y" shows "x alg_dvd - y"
smult_Poly: fixes c :: "'b"   and xs :: "'b list" shows "smult c (Poly xs) = Poly (map ((*) c) xs)"
ta_union_def: fixes TA :: "('a, 'b, 'c) tree_automaton_rec_scheme"   and TA' :: "('a, 'b, 'd) tree_automaton_rec_scheme" shows "ta_union TA TA' \<equiv> \<lparr>ta_initial = ta_initial TA \<union> ta_initial TA', ta_rules = ta_rules TA \<union> ta_rules TA'\<rparr>"
rtranclp_tranclp_tranclp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
simps_9: fixes v :: "('a, 'b) form" shows "compl (Exists v) = Neg (Exists v)"
span_same: fixes S :: "'a vec set" assumes "S \<subseteq> mat_kernel D" shows "KD.span S = KD.NC.span S"
assms_4: shows "net_ips s2 = net_tree_ips n2"
not_eq_complement: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "not a = minus (uminus a) one"
flatdom_strict2cont: fixes f :: "'d \<Rightarrow> 'c" assumes "f \<bottom> = \<bottom>" shows "cont f"
aform_errI: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and Xe :: "(real \<times> real pdevs) \<times> real" assumes "\<bar>x - aform_val e (fst Xe)\<bar> \<le> snd Xe" shows "x \<in> aform_err e Xe"
accs_is_ranked: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option"   and t :: "'b tree"   and q :: "'a" assumes "ranked_tree_automaton TA A"   and "accs (ta_rules TA) t q" shows "t \<in> ranked_trees A"
ik_append_subset_2: fixes S' :: "('a, 'b) strand_step list"   and S :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t S' \<subseteq> ik\<^sub>s\<^sub>t (S @ S')"
successive_split: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "xs = ys @ zs"   and "successive P xs" shows "successive P ys \<and> successive P zs"
finterI: fixes c :: "'e"   and A :: "'e fset"   and B :: "'e fset" assumes "c |\<in>| A"   and "c |\<in>| B" shows "c |\<in>| A |\<inter>| B"
convex_open_segment: fixes a :: "'a"   and b :: "'a" shows "convex (open_segment a b)"
unlr_inject: fixes x :: "'a admS"   and y :: "'a admS" shows "(unlr x = unlr y) = (x = y)"
rel_option_None1: fixes P :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "'m option" shows "rel_option P None x = (x = None)"
simps_5: fixes t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (delete\<langle>t,t'\<rangle>) = fv t \<union> fv t'"
1: shows "as \<in> valid_plans (snapshot PROB (fmrestrict_set vs s))"
pos: fixes l :: "nat" shows "0 < Re (\<Sum>i\<in>I. a i * Lam i ^ l)"
lvl_insert_aux: fixes x :: "'b"   and t :: "('b \<times> nat) tree" shows "lvl (insert x t) = lvl t \<or> lvl (insert x t) = lvl t + 1 \<and> sngl (insert x t)"
zip_nth_conv: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys" shows "zip xs ys = map (\<lambda>i. (xs ! i, ys ! i)) [0..<length ys]"
strict_monoI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>A B. A < B \<Longrightarrow> f A < f B" shows "OrdinalInduct.strict_mono f"
rep_eq: shows "lookup 1 = (\<lambda>k. (1::'b) when k = (0::'a))"
l_identity: fixes \<alpha> :: "'b"   and \<beta> :: "'b"   and \<phi> :: "'b \<Rightarrow> \<o>" shows "[[\<alpha> \<^bold>= \<beta> \<^bold>\<rightarrow> (\<phi> \<alpha> \<^bold>\<rightarrow> \<phi> \<beta>)]]"
trivial_limit_sequentially: shows "sequentially \<noteq> \<bottom>"
row_length_def: fixes xs :: "'a list list" shows "row_length xs \<equiv> if xs = [] then 0 else length (hd xs)"
real_of_real_def: fixes x :: "real" shows "real_of x = x"
1: shows "openin (top_of_set T) (T \<inter> g -` U)"
simps: fixes s :: "'b"   and h :: "'b \<rightarrow> ('c, 'b) Step" assumes "s \<noteq> \<bottom>" shows "unstream\<cdot>(Stream\<cdot>h\<cdot>s) = unfold\<cdot>h\<cdot>s"
Max_ge_iff: fixes A :: "complex set"   and x :: "complex" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le>\<^sub>\<complex> complex_lex.Max A) = (\<exists>a\<in>A. x \<le>\<^sub>\<complex> a)"
SREF: shows "(\<sigma>, \<sigma>') \<in> br \<alpha>s dinvarm"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subst_obtain_var: fixes x :: "'c"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)" shows "\<exists>y\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F. x \<in> fv (\<delta> y)"
simps: fixes am :: "nat list"   and n :: "nat"   and v :: "nat" shows "abc_lm_s am n v = (if n < length am then am[n := v] else am @ 0 \<up> (n - length am) @ [v])"
Abs_Int_axioms: shows "Abs_Int num' plus' \<gamma>"
cases: fixes r :: "ConvSchema" obtains cLengthF :: "rat" and cMassF :: "rat" and cTimeF :: "rat" and cCurrentF :: "rat" and cTemperatureF :: "rat" and cAmountF :: "rat" and cIntensityF :: "rat" where "r = \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF\<rparr>"
le_nat_any: fixes n :: "nat"   and v :: "val" assumes "VNat n \<sqsubseteq> v" shows "v = VNat n"
induct: fixes P1 :: "('e, 'f, 'd) hp \<Rightarrow> bool"   and P2 :: "('e, 'f, 'd) formula \<Rightarrow> bool"   and x1 :: "('e, 'f, 'd) hp"   and x2 :: "('e, 'f, 'd) formula" assumes "\<And>x. P1 ($\<alpha> x)"   and "\<And>e x. dsafe e \<Longrightarrow> P1 (x := e)"   and "\<And>e x. dsafe e \<Longrightarrow> P1 (DiffAssign x e)"   and "\<And>P. \<lbrakk>fsafe P; P2 P\<rbrakk> \<Longrightarrow> P1 (? P)"   and "\<And>ODE P. \<lbrakk>osafe ODE; fsafe P; P2 P\<rbrakk> \<Longrightarrow> P1 (EvolveODE ODE P)"   and "\<And>a b. \<lbrakk>hpsafe a; P1 a; hpsafe b; P1 b\<rbrakk> \<Longrightarrow> P1 (a \<union>\<union> b)"   and "\<And>a b. \<lbrakk>hpsafe a; P1 a; hpsafe b; P1 b\<rbrakk> \<Longrightarrow> P1 (a ;; b)"   and "\<And>a. \<lbrakk>hpsafe a; P1 a\<rbrakk> \<Longrightarrow> P1 a**"   and "\<And>t1 t2. \<lbrakk>dsafe t1; dsafe t2\<rbrakk> \<Longrightarrow> P2 (Geq t1 t2)"   and "\<And>args p. (\<And>i. dsafe (args i)) \<Longrightarrow> P2 ($\<phi> p args)"   and "\<And>p. \<lbrakk>fsafe p; P2 p\<rbrakk> \<Longrightarrow> P2 (! p)"   and "\<And>p q. \<lbrakk>fsafe p; P2 p; fsafe q; P2 q\<rbrakk> \<Longrightarrow> P2 (p && q)"   and "\<And>p x. \<lbrakk>fsafe p; P2 p\<rbrakk> \<Longrightarrow> P2 (Exists x p)"   and "\<And>a p. \<lbrakk>hpsafe a; P1 a; fsafe p; P2 p\<rbrakk> \<Longrightarrow> P2 (\<langle> a \<rangle> p)"   and "\<And>f C. \<lbrakk>fsafe f; P2 f\<rbrakk> \<Longrightarrow> P2 (InContext C f)" shows "(hpsafe x1 \<longrightarrow> P1 x1) \<and> (fsafe x2 \<longrightarrow> P2 x2)"
proj5_def: shows "\<pi>\<^sub>5 \<equiv> \<lambda>(uu_, uu_, uu_, hops, uu_). hops"
card_atLeastLessThan_int: fixes l :: "int"   and u :: "int" shows "card {l..<u} = nat (u - l)"
not_empty_imp_ex: fixes A :: "'b set" assumes "A \<noteq> {}" shows "\<exists>x. x \<in> A"
ls_lookup_map_of: fixes l :: "('a \<times> 'b) list list"   and k :: "'a" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l" shows "ls_lookup k (l ! bounded_hashcode_nat (length l) k) = map_of (concat l) k"
new_array_def: fixes a :: "'a"   and n :: "nat" shows "new_array a n = Array (replicate n a)"
complete_monitor: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a option list"   and i :: "nat" assumes "prefix_of \<pi> \<sigma>"   and "wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v"   and "\<And>\<sigma>. prefix_of \<pi> \<sigma> \<Longrightarrow> MFOTL.sat \<sigma> (map the v) i \<phi>" shows "\<exists>\<pi>'. prefix_of \<pi>' \<sigma> \<and> (i, v) \<in> joiner (\<lambda>k. M (splitter \<pi>' k))"
juncts_eq_set_juncts_list_2: fixes \<phi> :: "('a, 'b) aformula" shows "conjuncts \<phi> = set (conjuncts_list \<phi>)"
evalRR_monom: fixes a :: "int"   and x :: "int"   and d :: "nat" assumes "a \<in> carrier R"   and "x \<in> carrier R" shows "eval R R id x (monom (UP R) a d) = a \<otimes>\<^bsub>R\<^esub> x [^]\<^bsub>R\<^esub> d"
isPath_edgeset: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and e :: "nat \<times> nat" assumes "isPath u p v"   and "e \<in> set p" shows "e \<in> E"
less_type_def: shows "(<) \<equiv> (\<sqsubset>)\<^sup>+\<^sup>+"
bconf_FAcc: fixes P :: "'a prog"   and sh :: "sheap"   and e :: "'b exp"   and F :: "char list"   and D :: "char list"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (e\<bullet>F{D},b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e,b) \<surd>)"
length_append_singleton: fixes xs :: "'f list"   and x :: "'f" shows "length (xs @ [x]) = Suc (length xs)"
simps: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and a :: "('a, 'b) tm" shows "arity.wary arity_sym arity_var a = ((\<exists>\<zeta>. a = Hd \<zeta>) \<or> (\<exists>s t. a = App s t \<and> arity.wary arity_sym arity_var s \<and> arity.wary arity_sym arity_var t \<and> enat (num_args s) < arity.arity_hd arity_sym arity_var (head s)))"
Subspace_is_findim: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and U :: "'b set" assumes "FinDimVectorSpace smult V"   and "RModule UNIV smult U \<and> U \<subseteq> V" shows "\<exists>ms. set ms \<subseteq> U \<and> R_scalar_mult.RSpan UNIV smult ms = U"
assms: fixes i :: "nat" assumes "p i \<noteq> 0" shows "odd i"
good_item_simps_62: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xsa :: "'varSort"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and xa :: "'var" assumes "goodEnv rhoa" shows "freshEnv xsa (xa @xsa[y1 \<and> y2]_ys) (rhoa &[y1 \<and> y2]_ys) = freshEnv xsa xa rhoa"
linorder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.linorder less_eq less \<equiv> class.order less_eq less \<and> class.linorder_axioms less_eq"
term_less_def: fixes t :: "'a \<Rightarrow> 'b"   and w :: "'b" shows "term_less t w = {s. t s \<sqsubset> w}"
sigma_finite_measure_axioms: fixes s :: "'b" shows "sigma_finite_measure (K2.T s)"
order_refl: fixes less_eq :: "'y \<Rightarrow> 'y \<Rightarrow> bool"   and less :: "'y \<Rightarrow> 'y \<Rightarrow> bool"   and x :: "'y" assumes "class.preorder less_eq less" shows "less_eq x x"
periodic_set_def: fixes S :: "real set"   and \<delta> :: "real" shows "periodic_set S \<delta> = (\<exists>B. finite B \<and> (\<forall>x\<in>S. \<exists>b\<in>B. \<exists>xa. x = b + real_of_int xa * \<delta>))"
hb: fixes x :: "real" shows "b = insertion (nth_default 0 (xs' @ x # xs)) B"
partial_equivalence_axioms_def: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "partial_equivalence_axioms A equiv \<equiv> transitive A equiv"
allocate_Eps: fixes h'' :: "'heap"   and a'' :: "'addr"   and h :: "'heap"   and hT :: "htype"   and h' :: "'heap"   and a' :: "'addr" assumes "(h'', a'') \<in> allocate h hT"   and "(SOME ha. ha \<in> allocate h hT) = (h', a')" shows "(h', a') \<in> allocate h hT"
functor_axioms: shows "functor (\<cdot>\<^sup>o\<^sup>p) FG.Dop.comp F'"
Rats_minus_iff: fixes a :: "'b" shows "(- a \<in> \<rat>) = (a \<in> \<rat>)"
Inf_lower: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "Inf A \<le> x"
f: shows "f ` {..<card (Pow S)} = Pow S"
enumerate_step: fixes S :: "'b set"   and n :: "nat" assumes "infinite S" shows "wellorder_class.enumerate S n < wellorder_class.enumerate S (Suc n)"
lunit_simps_5: fixes f :: "'a" assumes "C.ide f" shows "C.cod \<l>\<^sub>C[f] = f"
faceCountMax_bound: fixes g :: "graph"   and v :: "nat" assumes "tame g"   and "v \<in> \<V> g" shows "tri g v + quad g v \<le> 7"
simps_18: fixes P :: "nat" shows "bound0 (Closed P) = True"
left_idem: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<union># (a \<union># b) = a \<union># b"
stl_decr: fixes js :: "'ord stream"   and m :: "'ord" assumes "alw (\<lambda>js. (shd (stl js), shd js) \<in> Ord) js"   and "alw (ev (\<lambda>js. shd (stl js) \<noteq> shd js)) js"   and "(shd js, m) \<in> Ord" shows "\<exists>js1 js2. js = js1 @- js2 \<and> set js1 \<subseteq> {m} \<and> (shd js2, m) \<in> Ord \<and> shd js2 \<noteq> m \<and> shd (decr m js) = shd js2 \<and> stl (decr m js) = decr (shd js2) js2"
simps_1: fixes r :: "real" shows "sgn (ereal r) = ereal (sgn r)"
edge_obs_intra_slice_eq: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "intra_kind (kind a)"   and "sourcenode a \<notin> SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)" shows "CFG.obs_intra sourcenode targetnode kind valid_edge (targetnode a) (SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)) = CFG.obs_intra sourcenode targetnode kind valid_edge (sourcenode a) (SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S))"
Dom_Ide: fixes t :: "'a local.term" assumes "Ide t" shows "Dom t = t"
less_dag_Node: fixes x :: "dag"   and l :: "dag"   and a :: "ref"   and r :: "dag" shows "(x < Node l a r) = (x \<le> l \<or> x \<le> r)"
assms_1: fixes i :: "nat" assumes "i \<le> Suc n" shows "ideal (I i) R"
mult2_alt_s_def: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "mult2_alt_s ns s = {(X, Y). \<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> Y2 \<noteq> {#} \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s))}"
is_acc_def: fixes r :: "nat \<Rightarrow> 'a node" shows "gba.is_acc r \<equiv> \<forall>A\<in>gba.F. \<exists>\<^sub>\<infinity>i. r i \<in> A"
SFAss: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sfs :: "sfields"   and sfs' :: "sfields"   and sh\<^sub>1' :: "sheap" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "sfs' = sfs(F \<mapsto> v)"   and "sh\<^sub>1' = sh\<^sub>1(D \<mapsto> (sfs', Done))" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1')\<rangle>"
hnr_RETURN_pass: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and x :: "'b"   and p :: "'a" shows "hn_refine (hn_ctxt R x p) (return p) (hn_invalid R x p) R (RETURN x)"
conj_left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<^bold>\<sqinter> a \<^bold>\<sqinter> c = a \<^bold>\<sqinter> b \<^bold>\<sqinter> c"
rel_pmf_top: shows "rel_pmf \<top> = \<top>"
deutsch_transform_dim_2: shows "dim_col U\<^sub>f = 4"
continuous_on_Ree: fixes s :: "'a set"   and g :: "'a \<Rightarrow> octo" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. Ree (g x))"
path_cons_simp: fixes v :: "'d"   and v' :: "'d"   and e :: "('d \<times> 'c) \<times> 'd \<times> 'b"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" shows "path v v' (e # pth) = (fst (fst e) = v \<and> e \<in> edges \<and> path (fst (snd e)) v' pth)"
term_variants_pred_inv'_1: fixes P :: "'a \<Rightarrow> 'a list"   and f :: "'a"   and T :: "('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term" assumes "term_variants_pred P (Fun f T) t" shows "is_Fun t"
HoHV_def: fixes \<mu> :: "'b \<times> 'b \<times> 'b" shows "HoHV \<mu> \<equiv> if VVV.arr \<mu> then (fst \<mu> \<star> fst (snd \<mu>)) \<star> snd (snd \<mu>) else null"
assms_4: shows "t \<le> v * v\<^sup>T"
emeasure_C: fixes m :: "nat"   and m' :: "nat"   and A :: "(nat \<Rightarrow> 'a) set"   and n :: "nat"   and x :: "nat \<Rightarrow> 'a" assumes "m \<le> m'"   and "A \<in> sets (Pi\<^sub>M {0..<n + m} M)"   and "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "emeasure (C n m' x) (prod_emb {0..<n + m'} M {0..<n + m} A) = emeasure (C n m x) A"
isInitialState_def: fixes state :: "State"   and F0 :: "Literal list list" shows "isInitialState state F0 \<equiv> getF state = F0 \<and> getM state = []"
Var_NF: fixes n :: "nat" shows "NF (Var n)"
simps_2: fixes v :: "'b"   and va :: "'b"   and l :: "'b set list" shows "smaller (DenyAllFromTo v va) DenyAll l = False"
is_basis_imp_full_rank: fixes A :: "(('a, 'c) vec, 'b) vec" assumes "is_basis (columns A)"   and "card (columns A) = ncols A" shows "rank A = ncols A"
LIMSEQ_INF: fixes X :: "nat \<Rightarrow> 'b" assumes "decseq X" shows "X \<longlonglongrightarrow> Inf (range X)"
es_def: shows "es = filter (\<lambda>y. y = x) xs"
that: fixes \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "thesis"
subst_not_free: fixes s :: "dB"   and i :: "nat"   and t :: "dB"   and u :: "dB" assumes "\<not> free s i" shows "s[t/i] = s[u/i]"
summable_LIMSEQ: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "(\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> suminf f"
is_ground_subst_def: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "is_ground_subst \<sigma> = (\<forall>A. is_ground_atm (subst_atm_abbrev A \<sigma>))"
shunt_mapping: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "coreflexive (z[[z]]) \<and> times_top_class.total z" shows "(x \<le> z * y) = (z[[x]] \<le> y)"
finite_measure_restrict_space: fixes M :: "'b measure"   and A :: "'b set" assumes "finite_measure M"   and "A \<in> sets M" shows "finite_measure (restrict_space M A)"
fmrel_on_fset_alt_def: fixes S :: "'a fset"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and n :: "('a, 'c) fmap" shows "fmrel_on_fset S P m n = fBall S (\<lambda>x. rel_option P (fmlookup m x) (fmlookup n x))"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "Module smult V" shows "VectorSpace smult V"
mirror_Leaf: fixes t :: "'b tree" shows "(mirror t = \<langle>\<rangle>) = (t = \<langle>\<rangle>)"
simps_1: fixes bs :: "'a list"   and c :: "int \<times> int" shows "\<lparr>C c\<rparr>\<^sub>p\<^bsup>bs\<^esup> = INum c"
assms: shows "Fun c [] \<in> subst_range (\<theta>1 \<circ>\<^sub>s \<theta>2)"
vals_update_in_Store: fixes l :: "Location"   and s :: "Store"   and y :: "Value" assumes "alive (ref l) s"   and "alive y s"   and "\<tau> y \<le> ltype l" shows "Rep_Store s\<lparr>vals := (vals (Rep_Store s))(l := y)\<rparr> \<in> Store"
lead_coeff_pCons_1: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "lead_coeff (pCons a p) = lead_coeff p"
dom_prob_proj: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set" shows "prob_dom (prob_proj PROB vs) \<subseteq> vs"
him: shows "h ` ({0..1} \<times> S) \<subseteq> U"
sum_list'_float: fixes p :: "nat"   and xs :: "real list" shows "sum_list' p xs \<in> float"
AND_tl: fixes p :: "('a set set, 'b) Combinators list" assumes "allNetsDistinct p" shows "allNetsDistinct (tl p)"
rbt_insert_rbt_sorted: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and k :: "'b"   and v :: "'c" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "ord.rbt_sorted less (ord.rbt_insert less k v t)"
ord_eq_le_trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a = b"   and "leq2 b c" shows "leq2 a c"
infinite_small_llength: fixes xs :: "'b llist"   and n :: "nat" assumes "\<not> lfinite xs" shows "enat n < llength xs"
filterof_elem: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X" shows "x \<in> filterof X"
rcs_tool0: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "a \<cdot> \<rho> b \<in> H" shows "b \<cdot> \<rho> a \<in> H"
semicomplete_iff_conditionally_complete_extremed: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<noteq> {}" shows "semicomplete A less_eq = (conditionally_complete A less_eq \<and> extremed A less_eq)"
ex1_eqvt: fixes p :: "perm"   and P :: "'a \<Rightarrow> bool" shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"
content_field: fixes p :: "'b poly" shows "content p = (if p = 0 then 0::'b else (1::'b))"
strip_while_idem_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(strip_while P xs = xs) = no_trailing P xs"
w1_3: shows "w1 \<subseteq> fst vy"
birkhoff_sum_integral_1: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "integrable M f" shows "integrable M (birkhoff_sum f n)"
take_Suc: fixes xs :: "'e list"   and n :: "nat" assumes "xs \<noteq> []" shows "take (Suc n) xs = hd xs # take n (tl xs)"
dom_absorp_2: fixes x :: "'a"   and y :: "'a" shows "ad (ad x) + ad (ad x) \<cdot> ad (ad y) = ad (ad x)"