hcju/isabelleps huggingface dataset:
bisim1_Val_\<tau>red1r: fixes P :: "'addr J1_prog"   and h :: "'heap"   and E :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list"   and v :: "'addr val"   and loc :: "'addr val list"   and n :: "nat"   and t :: "'thread_id" assumes "P,E,h \<turnstile> (e, xs) \<leftrightarrow> ([v], loc, length (compE2 E), None)"   and "n + max_vars e \<le> length xs"   and "\<B> E n" shows "\<tau>red1r P t h (e, xs) (Val v, loc)"
refine: fixes cmpk :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res"   and Rv :: "('b \<times> 'c) set"   and succi :: "'b \<Rightarrow> 'b list"   and E :: "('c \<times> 'c) set"   and vdi :: "'b"   and tgt :: "'c"   and v0i :: "'b"   and src :: "'c" assumes "(cmpk, dflt_cmp (\<le>) (<)) \<in> Rv \<rightarrow> Rv \<rightarrow> comp_res_rel"   and "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"   and "(vdi, tgt) \<in> Rv"   and "(v0i, src) \<in> Rv" shows "(RETURN (dfs_impl3 cmpk succi v0i vdi), dfs E src tgt) \<in> \<langle>bool_rel\<rangle>nres_rel"
keys_monom_multI: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'c"   and t :: "'a" assumes "v \<in> keys p"   and "c \<noteq> (0::'c)" shows "t \<oplus> v \<in> keys (monom_mult c t p)"
eval_tm_rename: fixes k' :: "name"   and t :: "tm"   and e :: "(name, hf) finfun"   and k :: "name"   and x :: "hf" assumes "atom k' \<sharp> t" shows "\<lbrakk>t\<rbrakk>(finfun_update e k x) = \<lbrakk>(k' \<leftrightarrow> k) \<bullet> t\<rbrakk>(finfun_update e k' x)"
FAcc: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list"   and v :: "val" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l)\<rangle>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "fs (F, D) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, l)\<rangle>"
add_zero_right: fixes x :: "'a" assumes "x \<in> E" shows "x + (0::'a) = x"
map_filter_append: fixes P :: "'d \<Rightarrow> 'c option"   and xs :: "'d list"   and ys :: "'d list" shows "List.map_filter P (xs @ ys) = List.map_filter P xs @ List.map_filter P ys"
tdghm_dghm_comp_is_tdghm': fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V"   and \<AA> :: "V"   and \<HH> :: "V"   and \<FF>' :: "V"   and \<GG>' :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<FF>' = \<FF> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>"   and "\<GG>' = \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>" shows "\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> : \<FF>' \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>' : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
ord_eq_less_trans: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'t \<Rightarrow>\<^sub>0 'c" assumes "a = b"   and "b \<prec>\<^sub>p c" shows "a \<prec>\<^sub>p c"
select_convs_6: fixes Obj :: "'e set"   and Mor :: "'d set"   and Dom :: "'d \<Rightarrow> 'e"   and Cod :: "'d \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'd"   and Comp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'f" shows "(;;\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub>) = Comp"
trimmed_pos_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "multiseries.trimmed_pos trimmed dominant_term F = (trimmed F \<and> 0 < fst (dominant_term F))"
intros_8: fixes iop :: "binop_i"   and c1 :: "i64"   and c2 :: "i64" assumes "app_binop_i iop c1 c2 = None" shows "\<lparr>[$C ConstInt64 c1, $C ConstInt64 c2, $Binop_i T_i64 iop]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
sfg: fixes x :: "'a" assumes "x \<in> B" shows "subspace {a. f x a = g x a}"
get_addr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "'c \<times> inst_operand list" assumes "low_equal s1 s2" shows "ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) = ucast (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) = ucast (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) = ucast (get_addr (snd instr) (snd (fst (get_curr_win () s2))))"
viewRn_list: fixes t :: "('a, 'b) FingerTreeStruc"   and nd :: "('a, 'b) Node"   and s :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.viewRn t = Some (nd, s)" shows "FingerTreeStruc.toList t = FingerTreeStruc.toList s @ FingerTreeStruc.nodeToList nd"
map_rbt_sorted: fixes f :: "'a \<Rightarrow> 'e \<Rightarrow> 'd"   and t :: "('a, 'e) rbt" shows "rbt_sorted (RBT_Impl.map f t) = rbt_sorted t"
int2: shows "integrable M (\<lambda>x. f (T x) * max (- g x) 0)"
maps_def: fixes f :: "'d \<Rightarrow> 'c list"   and xs :: "'d list" shows "List.maps f xs = concat (map f xs)"
pasting_lemma_exists: fixes X :: "'a topology"   and T :: "'b \<Rightarrow> 'a set"   and I :: "'b set"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "topspace X \<subseteq> \<Union> (T ` I)"   and "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"   and "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x" obtains g :: "'a \<Rightarrow> 'c" where "continuous_map X Y g" and "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
finite_insert: fixes a :: "'c"   and A :: "'c set" shows "finite (insert a A) = finite A"
tvs_bind_fv2: fixes v :: "variable"   and T :: "typ"   and lev :: "nat"   and t :: "Core.term" shows "tvs (bind_fv2 (v, T) lev t) \<union> tvsT T = tvs t \<union> tvsT T"
insert_in_traverse_pairs: fixes t :: "('c, 'd) tree"   and x :: "'c"   and v :: "'d" assumes "tree_sorted t" shows "in_traverse_pairs (tree_insert x v t) = ordered_insert_pairs x v (in_traverse_pairs t)"
lookup_make_fmap: fixes k :: "'c"   and ks :: "'c fset"   and v :: "'d" assumes "k \<in> fset ks" shows "lookup [ ks |=> v ] k = v"
dims_tensor_prod: fixes A :: "'a tensor"   and B :: "'a tensor" shows "dims (A \<otimes> B) = dims A @ dims B"
methd_member_of: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "wf_prog G"   and "is_class G C"   and "methd G C sig = Some m" shows "G \<turnstile>Methd sig m member_of C"
aux: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" shows "(poly_mapping.lookup f = (\<lambda>_. 0::'b)) = (f = 0)"
meet_compl_bot_left1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>- x \<sqinter> (x \<sqinter> y) = \<bottom>"
assms_3: fixes n :: "nat"   and i :: "nat" assumes "n < length cl"   and "i < brn (cl ! n)"   and "ii = brnL cl n + i" shows "phi"
wf_graph_finite_filterE: fixes G :: "'a graph"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wf_graph G" shows "finite {(e1, e2). (e1, e2) \<in> edges G \<and> P e1 e2}"
bij_swap_comp: fixes p :: "'b \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "bij p" shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p"
sqrt_int: fixes x :: "int" shows "set (sqrt_int x) = {y. y * y = x}"
Q_neg_def: shows "Q_neg = set (zip A_neg (drop (length A_pos) L))"
pderivs_lang_subset: fixes A :: "'a list set"   and B :: "'a list set"   and r :: "'a rexp" assumes "A \<subseteq> B" shows "pderivs_lang A r \<subseteq> pderivs_lang B r"
allBox: fixes F :: "'c \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" shows "(\<box>(\<forall>x. F x)) s = (\<forall>x. (\<box>F x) s)"
assms_1: shows "ifte (var i) x y = ifte (var i) x' y'"
distr_lift_RealPairVal: fixes f :: "real \<Rightarrow> real \<Rightarrow> real"   and M :: "val measure"   and \<delta> :: "val \<Rightarrow> ennreal"   and g :: "(real \<times> real \<Rightarrow> ennreal) \<Rightarrow> real \<Rightarrow> ennreal"   and f' :: "int \<Rightarrow> int \<Rightarrow> int" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable borel"   and "has_subprob_density M (stock_measure (PRODUCT REAL REAL)) \<delta>"   and "\<And>M \<delta>. has_subprob_density M lborel \<delta> \<Longrightarrow> has_density (distr M borel (\<lambda>(x, y). f x y)) lborel (g \<delta>)" shows "has_density (distr M (stock_measure REAL) (lift_RealIntVal2 f' f)) (stock_measure REAL) (g (\<lambda>x. \<delta> (RealPairVal x)) \<circ> extract_real)"
split_list_last: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> list.set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> list.set zs"
inj_on_one_iff': fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "inj_on h (carrier G) = (\<forall>x. x \<in> carrier G \<longrightarrow> h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>)"
lang_nderiv: fixes a :: "'a"   and r :: "'a rexp" shows "lang (nderiv a r) = Deriv a (lang r)"
simps_1: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'a trm"   and x :: "Preliminaries.var" shows "subst \<pi> (Var x) = \<pi> x"
assms_4: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> \<A>"   and "B \<in> \<A>" shows "A \<inter> B \<in> \<A>"
ennreal_divide_top: fixes x :: "ennreal" shows "x / top = 0"
sol_dim_in_sat_primal: fixes x :: "'a vec"   and A :: "'a mat"   and b :: "'a vec" assumes "x \<in> sat_primal A b" shows "dim_vec x = dim_col A"
simps: fixes args :: "Monitor.args"   and tp :: "nat"   and tss :: "nat queue"   and len :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and a1_map :: "('a option list, nat) mapping"   and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping"   and "done" :: "'a option list set list"   and done_length :: "nat" shows "length_mmuaux args (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) = len + done_length"
Suc_0: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>" shows "\<chi> (Suc 0) = 1"
etc_3: shows "dom h\<^sub>0 = dom h\<^sub>0'"
distinct_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (list_all2 A) (=) distinct distinct"
is1: shows "{xs. Ball X ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m \<and> xs ! e} = {xs. Ball (insert e X) ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m}"
sigma_finite_subalgebra_axioms: shows "sigma_finite_subalgebra M F"
less_eq_prod_def: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "(x \<le> y) = (fst x \<le> fst y \<and> snd x \<le> snd y)"
card_reachable_s_non_zero: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap" assumes "finite PROB"   and "s \<in> valid_states PROB" shows "0 < card (reachable_s PROB s)"
less_numeral_extra_3: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
card_of_Times_mono: fixes A :: "'e set"   and B :: "'f set"   and C :: "'g set"   and D :: "'h set" assumes "|A| \<le>o |B|"   and "|C| \<le>o |D|" shows "|A \<times> C| \<le>o |B \<times> D|"
iprod_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "\<langle>x # xs,y # ys\<rangle> = x * y + \<langle>xs,ys\<rangle>"
inverse_arrow_unique: fixes f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and g' :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.inverse_arrows f g"   and "S.inverse_arrows f g'" shows "g = g'"
space_prob_algebra: fixes M :: "'d measure" shows "space (prob_algebra M) = {N. sets N = sets M \<and> prob_space N}"
eint_ord_simps_1: fixes m :: "int"   and n :: "int" shows "(eint m \<le> eint n) = (m \<le> n)"
pow_subgroup_indicator: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> carrier G" shows "a [^] subgroup_indicator G H a \<in> H"
intro: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c"   and s :: "'c" assumes "UP_pre_univ_prop R S h"   and "UP_univ_prop_axioms S s" shows "UP_univ_prop R S h s"
rwof_step: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and s :: "'c" assumes "nofail (m0 \<bind> WHILE cond step)"   and "rwof m0 cond step s"   and "cond s" shows "step s \<le> SPEC (rwof m0 cond step)"
ex_fresh: fixes x :: "'a" shows "\<exists>c. c \<sharp> x"
append_assoc: fixes xs :: "'l list"   and ys :: "'l list"   and zs :: "'l list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
subst_pp_restrict_indets_subst: fixes t :: "'b option \<Rightarrow>\<^sub>0 nat" shows "subst_pp (\<lambda>x. monomial (1::'c) (restrict_indets_subst x)) t = monomial (1::'c) (restrict_indets_pp t)"
d: fixes x :: "'a"   and y :: "'a" assumes "x \<notin> B"   and "y \<in> A" shows "d < dist x y"
sum_list_upt_zip: fixes xs :: "'c list"   and ys :: "'d list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "length xs = length ys" shows "(\<Sum>i\<leftarrow>[0..<length ys]. f (xs ! i) (ys ! i)) = sum_list (map2 f xs ys)"
par_oner: fixes x :: "'a" shows "x * (1::'a) = x"
distinct_3: fixes x11 :: "trm"   and x12 :: "trm"   and x3 :: "fmla" shows "x11 EQ x12 \<noteq> Robinson_Arithmetic.neg x3"
fs_int_mu_d_Z_m_m: fixes j :: "nat"   and ii :: "nat" assumes "j < length fs"   and "ii < length fs" shows "rat_of_int (d fs (Suc j)) * gs.\<mu> ii j \<in> \<int>"
inorder_rot2: fixes B :: "('b \<times> nat) tree"   and A :: "('b \<times> nat) tree"   and a :: "'b"   and b :: "'b"   and C :: "('b \<times> nat) tree" assumes "B \<noteq> \<langle>\<rangle>" shows "Tree2.inorder (rot2 A a B b C) = Tree2.inorder A @ a # Tree2.inorder B @ b # Tree2.inorder C"
Alm_all_def: fixes P :: "'c \<Rightarrow> bool" shows "Alm_all P = (\<not> (\<exists>\<^sub>\<infinity>x. \<not> P x))"
semidecidable_def: fixes X :: "nat set" shows "semidecidable X \<equiv> \<exists>f. recfn 1 f \<and> (\<forall>x. eval f [x] = (if x \<in> X then Some 1 else None))"
attract_mono_imp_fp_qfp_complete: fixes CC :: "'a set set"   and P :: "'a set" assumes "attractive A (\<sqsubseteq>)"   and "CC-complete A (\<sqsubseteq>)"   and "\<forall>C\<subseteq>A. well_related_set C (\<sqsubseteq>) \<longrightarrow> C \<in> CC"   and "\<forall>X\<in>CC. \<forall>Y\<in>CC. X \<sqsubseteq>\<^sup>s Y \<longrightarrow> X \<union> Y \<in> CC"   and "monotone_on A (\<sqsubseteq>) (\<sqsubseteq>) f"   and "P \<subseteq> {x \<in> A. f x = x}" shows "CC-complete ({q \<in> A. f q \<sim> q} \<union> P) (\<sqsubseteq>)"
MPoly_monomial_power: fixes n :: "nat"   and x' :: "nat \<Rightarrow>\<^sub>0 nat" assumes "0 < n" shows "MPoly (monomial (1::'a) x') ^ n = MPoly (monomial (1::'a) (((\<lambda>x. x + x') ^^ (n - 1)) x'))"
icd_cs: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and l :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_icdi prob l \<pi> k" shows "IFC_def.cs prob \<pi> l = IFC_def.cs prob \<pi> k @ [\<pi> l]"
rel_ipurge_def: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b" shows "rel_ipurge P I D u \<equiv> {(xs, ys). xs \<in> traces P \<and> ys \<in> traces P \<and> ipurge_tr_rev I D u xs = ipurge_tr_rev I D u ys}"
measure_induct_rule: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
rgsound: fixes P :: "'b com"   and pre :: "'b set"   and rely :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and post :: "'b set" assumes "\<turnstile> P sat [pre, rely, guar, post]" shows "\<Turnstile> P sat [pre, rely, guar, post]"
intros: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "hindrance \<Gamma> f"   and "current \<Gamma> f"   and "wave \<Gamma> f" shows "hindered \<Gamma>"
subcls1_def: shows "subcls1 \<equiv> {(C, D). C \<noteq> Object \<and> (\<exists>c. class C = Some c \<and> super c = D)}"
E: fixes i :: "'a" assumes "i \<in> I" shows "E i \<in> sets (M i)"
sum_list_simps_1: shows "sum_list [] = (0::'e)"
alpha_sym: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "X #= X'" shows "X' #= X"
measurable_sigma_sets: fixes N :: "'d measure"   and \<Omega> :: "'d set"   and A :: "'d set set"   and f :: "'e \<Rightarrow> 'd"   and M :: "'e measure" assumes "sets N = sigma_sets \<Omega> A"   and "A \<subseteq> Pow \<Omega>"   and "f \<in> space M \<rightarrow> \<Omega>"   and "\<And>y. y \<in> A \<Longrightarrow> f -` y \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M N"
GSMP_SMP_mono: fixes N :: "('fun, 'var) Term.term set"   and M :: "('fun, 'var) Term.term set" assumes "SMP N \<subseteq> SMP M" shows "GSMP N \<subseteq> GSMP M"
and_zero_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and a zero = zero"
add_commute: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "x + y = y + x"
assms: shows "(h, {}) \<in> set ps \<union> set qs"
f6: shows "((ucast (get_CWP (cpu_reg_val PSR s1)), s1), False) = (fst (get_curr_win () s1), snd (get_curr_win () s1))"
1: shows "\<forall>S\<in>\<S>. wf\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t \<A>) (dual\<^sub>s\<^sub>t S)"
mtf2_forward_effect2: fixes q :: "'b"   and xs :: "'b list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q = i" shows "index (mtf2 n q xs) (xs ! i) = index xs q - n \<and> index xs q - n = index (mtf2 n q xs) (xs ! i)"
subset_ffilter: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a fset"   and Q :: "'a \<Rightarrow> bool" shows "(ffilter P A |\<subseteq>| ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
gWlsInp_def: fixes MOD :: "('d, 'e, 'f, 'sort, 'g, 'h, 'i, 'j, 'k) Recursion.model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'i option" shows "gWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (gWls MOD) (arOf delta) inp"
select_convs_2: fixes g_V :: "'a set"   and g_E :: "('a \<times> 'a) set"   and g_V0 :: "'a set"   and more :: "'b" shows "g_E \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_E"
ln_Gamma_complex_conv_fact: fixes n :: "nat" assumes "0 < n" shows "ln_Gamma (of_nat n) = Ln (fact (n - 1))"
IH: fixes s :: "char list \<Rightarrow> int"   and stk :: "int list"   and n :: "nat"   and t :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "ccomp c \<turnstile> (0, s, stk) \<rightarrow>^n (size (ccomp c), t, stk')" shows "(c, s) \<Rightarrow> t \<and> stk' = stk"
actCases: fixes \<alpha> :: "act"   and P :: "ccs"   and \<beta> :: "act"   and P' :: "ccs"   and Prop :: "act \<Rightarrow> ccs \<Rightarrow> bool" assumes "\<alpha>.P \<longmapsto>\<beta> \<prec> P'"   and "Prop \<alpha> P" shows "Prop \<beta> P'"
substNext_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g" shows "uninst.substNext (usesOf \<circ> u) (\<lambda>g. Mapping.lookup (p g)) g \<equiv> \<lambda>v. if v = uninst_code.chooseNext u p g then uninst.substitution (usesOf \<circ> u) (\<lambda>g. Mapping.lookup (p g)) g else v"
one_d_integral_integrable: shows "(\<lambda>x. integral UNIV (\<lambda>y. f (y, x))) integrable_on UNIV"
inter_aform_plane_refine_unoverloaded: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and d :: "nat"   and xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and si :: "real list sctn"   and s :: "real list sctn"   and optns :: "nat" assumes "class.enum enum enum_all enum_ex \<and> d = CARD('a)"   and "(xi, x) \<in> aforms_rel"   and "(si, s) \<in> \<langle>rl_rel\<rangle>sctn_rel"   and "length xi = d"   and "length (normal si) = d" shows "(nres_of (inter_aform_plane_lv (length xi) optns xi si), inter_sctn_specs d x s) \<in> \<langle>aforms_rel\<rangle>nres_rel"
execute_len: fixes a :: "'a array"   and h :: "heap" shows "execute (Array.len a) h = Some (Array.length h a, h)"
frefI: fixes P :: "'d \<Rightarrow> bool"   and R :: "('b \<times> 'd) set"   and f :: "'b \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and S :: "('e \<times> 'f) set" assumes "\<And>x y. \<lbrakk>P y; (x, y) \<in> R\<rbrakk> \<Longrightarrow> (f x, g y) \<in> S" shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"
relative_to_inc: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and U :: "'b set" assumes "P S" shows "(P relative_to U) (U \<inter> S)"
lincomb_empty: fixes a :: "'c \<Rightarrow> 'a" shows "V.module.lincomb a {} = \<zero>\<^bsub>V\<^esub>"
lunit_agreement: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.lunit a = \<l>[a]"
ord_Zp_of_a_inv: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> nonzero Zp" shows "ord_Zp a = ord_Zp (\<ominus> a)"
invar: fixes s :: "('a, 'b) HashMap.hashmap" shows "hm.invar s"