hcju/isabelleps huggingface dataset:
AE_borel_eq: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and f :: "'b \<Rightarrow> real"   and t :: "'c"   and g :: "'b \<Rightarrow> real" assumes "filtrated_prob_space M F"   and "f \<in> borel_measurable (F t)"   and "g \<in> borel_measurable (F t)"   and "AE w in M. f w = g w" shows "{w \<in> space M. f w \<noteq> g w} \<in> sets (F t) \<and> emeasure M {w \<in> space M. f w \<noteq> g w} = 0"
one_mult_mat_vec: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "1\<^sub>m n *\<^sub>v v = v"
vcomp_in_hhom: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.seq \<nu> \<mu>"   and "src\<^sub>C \<nu> = a"   and "trg\<^sub>C \<nu> = b" shows "\<guillemotleft>\<nu> \<cdot>\<^sub>C \<mu> : a \<rightarrow>\<^sub>C b\<guillemotright>"
qGoodInp_def: fixes qinp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm option" shows "qGoodInp qinp \<equiv> liftAll qGood qinp \<and> |{i. qinp i \<noteq> None}| <o |UNIV|"
dirichlet_prod_inverse: fixes f :: "nat \<Rightarrow> 'a"   and i :: "'a" assumes "f 1 * i = (1::'a)" shows "dirichlet_prod f (dirichlet_inverse f i) = (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
timpl_closure_no_Abs_eq: fixes t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and s :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and TI :: "('g set \<times> 'g set) set" assumes "t \<in> timpl_closure s TI"   and "\<forall>f\<in>funs_term t. \<not> is_Abs f" shows "t = s"
sclem: fixes n :: "nat"   and xs :: "int list" shows "length (concat (map (bin_to_bl n) xs)) = length xs * n"
Lxy_distinct: fixes xs :: "'a list"   and S :: "'a set" assumes "distinct xs" shows "distinct (Lxy xs S)"
vifintersection_vsubset_iff: fixes I :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "I \<noteq> 0" shows "(A \<subseteq>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)) = (\<forall>i\<in>elts I. A \<subseteq>\<^sub>\<circ> f i)"
elims: fixes x :: "'a ns_constraint"   and y :: "'a le_constraint" assumes "lec_of_nsc x = y" obtains p :: "linear_poly" and c :: "'a" where "x = LEQ_ns p c" and "y = Le_Constraint Leq_Rel p c"   | v :: "linear_poly" and va :: "'a" where "x = GEQ_ns v va" and "y = undefined"
powhr_le_cancel_iff: fixes x :: "real star"   and a :: "real star"   and b :: "real star" assumes "1 < x" shows "(x powhr a \<le> x powhr b) = (a \<le> b)"
plus_absorb1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "o[F](\<lambda>x. f x + g x) = o[F](g)"
Goedel_Form_Proofs_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and "proof" :: "'d set"   and prfOf :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and encPf :: "'d \<Rightarrow> 'b"   and Pf :: "'c" shows "Goedel_Form_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi fls prv bprv enc S dsj proof prfOf encPf Pf \<equiv> Repr_SelfSubst var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc S \<and> CleanRepr_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc fls dsj proof prfOf encPf Pf"
ccField_to_elem: fixes x :: "Vars.var"   and G :: "CoCalls" assumes "x \<in> ccField G" shows "\<exists>y. x--y\<in>G"
quotientP_def: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "P /// \<phi> \<equiv> S2P (Collect P // P2R \<phi>)"
esing_simps_2: fixes x' :: "'c"   and x :: "'c"   and n :: "'d" assumes "x' \<noteq> x" shows "(esing x\<cdot>n) x' = \<bottom>"
mat_det_similarity: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A \<noteq> 0" shows "mat_det (similarity A M) = mat_det M"
special_axioms_hold: fixes A :: "fmla"   and e :: "(name, nat) finfun" assumes "A \<in> special_axioms" shows "eval_fmla e A"
gf_def: shows "gf = [g, f]\<^sub>\<circ>"
start_state_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list" shows "start_state P C M = (let (D, Ts, T, mxs, mxl\<^sub>0, b) = method P C M in (None, start_heap P, [([], Null # replicate mxl\<^sub>0 undefined, C, M, 0)]))"
notin_set_nthsI: fixes x :: "'g"   and xs :: "'g list"   and I :: "nat set" assumes "x \<notin> set xs" shows "x \<notin> set (nths xs I)"
sum: shows "(\<Sum>(a, n)\<leftarrow>N. lec_of_nsc (a *R n)) = Le_Constraint Leq_Rel 0 d"
that: fixes l :: "('e, 'a) FingerTreeStruc"   and e :: "'e"   and a :: "'a"   and r :: "('e, 'a) FingerTreeStruc" assumes "splitTree p i s = (l, (e, a), r)"   and "toList s = toList l @ (e, a) # toList r"   and "\<not> p (i + annot l)"   and "p (i + annot l + a)"   and "ft_invar l"   and "ft_invar r" shows "thesis"
simps_1: fixes \<sigma> :: "'a OCL_Types.type" shows "OclSuper \<squnion> \<sigma> = OclSuper"
complement_succ_def: fixes A :: "('d, 'c) nba"   and a :: "'d" shows "complement_succ A a \<equiv> \<lambda>(f, P). {(g, st_succ A a g P) |g. g \<in> lr_succ A a f}"
nonempty_update_with_aux: fixes v :: "'d"   and k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and ps :: "('c \<times> 'd) list" shows "AList.update_with_aux v k f ps \<noteq> []"
div_mult2_numeral_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and k :: "num"   and l :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (divide a (numeral.numeral one plus k)) (numeral.numeral one plus l) = divide a (numeral.numeral one plus (k * l))"
H_2: shows "rep_Bonk_Schramm_extension y \<in> extended_distance_set"
limit_Append_path_nonelem_split': fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> limit (Append (\<Z> k (Suc u)) k) (\<P> k u)"   and "p \<notin> \<P> k u" shows "\<exists>q ts. p = q @ ts \<and> is_continuation k u q ts"
cases: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c list"   and a2 :: "'c list" assumes "ord.lexordp_eq less a1 a2" obtains ys :: "'c list" where "a1 = []" and "a2 = ys__"   | x :: "'c" and y :: "'c" and xs :: "'c list" and ys :: "'c list" where "a1 = x__ # xs__" and "a2 = y__ # ysa__" and "less x__ y__"   | x :: "'c" and y :: "'c" and xs :: "'c list" and ys :: "'c list" where "a1 = x # xs" and "a2 = y # ys" and "\<not> less x y" and "\<not> less y x" and "ord.lexordp_eq less xs ys"
monoid_mult_hom_axioms: shows "monoid_mult_hom (inv hom)"
smaller_lits_are_false_if_cl_not_valid: fixes S :: "'a eclause set"   and C :: "'a literal set"   and t :: "'a trm" assumes "\<not> validate_ground_clause (int_clset S) C" shows "smaller_lits_are_false t C S"
add_spairs_alt_spp: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "\<And>x. x \<in> set bs \<Longrightarrow> Inl (spp_of p, spp_of x) \<notin> app_pair spp_of ` set ps" shows "map (app_pair spp_of) (add_spairs ps bs p) = add_spairs_spp (map (app_pair spp_of) ps) (map spp_of bs) (spp_of p)"
emax_eq: fixes x :: "('a, 'b) IEEE.float itself" shows "emax x = 2 ^ LENGTH('a) - 1"
correctCompositionDiffLevelsA1: shows "correctCompositionDiffLevels sA1"
induct: fixes P :: "(('b \<rightarrow> 'c \<rightarrow> 'c) \<rightarrow> 'c \<rightarrow> ('b, 'd) Stream \<rightarrow> 'c) \<Rightarrow> bool" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> f z v. Fixrec.run\<cdot>(match_Stream\<cdot>v\<cdot>(\<Lambda> h s. Fixrec.succeed\<cdot>(case h\<cdot>s of Done \<Rightarrow> z | Skip\<cdot>s' \<Rightarrow> x\<cdot>f\<cdot>z\<cdot>(Stream\<cdot>h\<cdot>s') | Yield\<cdot>xa\<cdot>s' \<Rightarrow> f\<cdot>xa\<cdot>(x\<cdot>f\<cdot>z\<cdot>(Stream\<cdot>h\<cdot>s'))))))" shows "P foldrS"
branch_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (branch_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
sinh_square_eq: fixes x :: "'a" shows "(sinh x)\<^sup>2 = (cosh x)\<^sup>2 - (1::'a)"
lr_coset_assoc: fixes g :: "'a"   and h :: "'a"   and P :: "'a set" assumes "g \<in> carrier H'"   and "h \<in> carrier H'"   and "P \<subseteq> carrier H'" shows "g <#\<^bsub>H'\<^esub> (P #>\<^bsub>H'\<^esub> h) = g <#\<^bsub>H'\<^esub> P #>\<^bsub>H'\<^esub> h"
lunit_char_2: fixes a :: "'a" assumes "ide a" shows "C.tensor C.unity (C.lunit a) = C.tensor \<iota> a \<cdot> C.assoc' C.unity C.unity a"
red_single_nonzero1: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single p q f t" shows "p \<noteq> 0"
5: shows "ks \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> ms = ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms"
rbt_comp_delete_def: fixes k :: "'a"   and t :: "('a, 'c) rbt" shows "local.rbt_comp_delete k t = paint color.B (local.rbt_comp_del k t)"
game2_d_bad: shows "spmf (map_spmf snd game2_d) True \<le> real q2 / 2 ^ len"
reduc_fair_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and Ns :: "'a set llist" assumes "calculus Bot Inf entails Red_I Red_F" shows "calculus.reduc_fair Inf Red_I Red_F Ns = (inference_system.Inf_from Inf (Liminf_llist Ns - Sup_llist (lmap Red_F Ns)) \<subseteq> Sup_llist (lmap Red_I Ns))"
simps_2: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_Wbis (c1 ;; c2) = (SC_WbisT c1 \<and> SC_Wbis c2 \<or> SC_Wbis c1 \<and> SC_discr c2 \<or> SC_ZObis (c1 ;; c2) \<or> SC_WbisT (c1 ;; c2))"
simps_1: fixes uu :: "('c, 'd) hd" shows "args (Hd uu) = []"
set_return_status_def: shows "set_return_status = (\<lambda>\<sigma>. Some ((), \<sigma>\<lparr>return_status := True\<rparr>))"
gbinomial_parallel_sum: fixes a :: "'a"   and n :: "nat" shows "(\<Sum>k\<le>n. a + of_nat k gchoose k) = a + of_nat n + (1::'a) gchoose n"
in_Q0: fixes A :: "'a set" assumes "A \<in> sets M"   and "A \<inter> \<Union> (range Q) = {}" shows "emeasure M A = 0 \<and> \<integral>\<^sup>+ x. f x * indicator A x \<partial>M = 0 \<or> 0 < emeasure M A \<and> \<integral>\<^sup>+ x. f x * indicator A x \<partial>M = \<infinity>"
simple_arg_pos_def: fixes rel :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) set"   and f :: "'a \<times> nat"   and i :: "nat" shows "simple_arg_pos rel f i \<equiv> \<forall>ts. i < snd f \<longrightarrow> length ts = snd f \<longrightarrow> (Fun (fst f) ts, ts ! i) \<in> rel"
assms_1: shows "f maps\<^bsub>CatDom (F ;;: G)\<^esub> X to Y"
cvm_fold_def: fixes g :: "'d \<Rightarrow> 'f \<Rightarrow> 'd \<times> 'e"   and cvm :: "'d"   and ss :: "'f list" shows "cvm_fold g cvm ss = foldl (\<lambda>(cvm, ss) s. apsnd (\<lambda>s'. ss @ [s']) (g cvm s)) (cvm, []) ss"
c_eq: shows "column j (Gram_Schmidt_upt_k A k) = column i (Gram_Schmidt_upt_k A k)"
adv_P2_def: fixes m1 :: "'msg1"   and m2 :: "'msg2"   and D :: "'view2 \<Rightarrow> bool spmf" shows "adv_P2 m1 m2 D = \<bar>spmf (R2 m1 m2 \<bind> D) True - spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S2 m2 o2 \<bind> D)) True\<bar>"
divide_self_if: fixes a :: "'a" shows "a / a = (if a = (0::'a) then 0::'a else (1::'a))"
assms_1: shows "C\<mu> \<in> grounding_of_cls C"
m_def: shows "m = card (GAUSS.E p (int q))"
cod_char: fixes f :: "'b" shows "hhom\<^sub>D.cod f = (if hhom\<^sub>D.arr f then D.cod f else D.null)"
assms_2: fixes b :: "'b" assumes "b \<in> B" shows "\<exists>a. a \<in> A"
symm_closure_of_symm_rel: fixes Rel :: "('a \<times> 'a) set" assumes "sym Rel" shows "symcl Rel = Rel"
eSubst_nchotomy: fixes eY :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and eX :: "'c withERR"   and y :: "'var" shows "(\<exists>Y. eY = OK Y \<and> igWls MOD (asSort ys) Y) \<and> (\<exists>s X. eX = OK X \<and> igWls MOD s X) \<or> eSubst MOD ys eY y eX = ERR"
Contrs_def: shows "Contrs = {z. \<exists>A Gamma. z = (A # Gamma, {A # A # Gamma})}"
finite_imp_nat_seg_image_inj_on: fixes A :: "'c set" assumes "finite A" shows "\<exists>n f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
snd_Gauss_Jordan_in_ij_PA_eq: fixes P :: "(('b, 'd) vec, 'd) vec"   and A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'c" shows "snd (Gauss_Jordan_in_ij_PA (P, A) i j) = Gauss_Jordan_in_ij A i j"
prod_relI: fixes a :: "'d"   and a' :: "'e"   and R1 :: "('d \<times> 'e) set"   and b :: "'f"   and b' :: "'g"   and R2 :: "('f \<times> 'g) set" assumes "(a, a') \<in> R1"   and "(b, b') \<in> R2" shows "((a, b), a', b') \<in> R1 \<times>\<^sub>r R2"
exec_to_exec_merge_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>merge_guards c,s\<rangle> \<Rightarrow> t"
strand_size_map_fun_lt_2: fixes X :: "('a, 'b) Term.term list"   and f :: "'a" shows "size\<^sub>s\<^sub>t (map Send X) < size\<^sub>s\<^sub>t [send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t]"
case_3: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and f3 :: "('d \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'c"   and x3 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "(case Apply x3 of Const x \<Rightarrow> f1 x | Load x \<Rightarrow> f2 x | Apply x \<Rightarrow> f3 x) = f3 x3"
rep_eq: fixes x :: "\<o> \<Rightarrow> \<o>" shows "eval\<o> (forall\<^sub>0 x) = (\<lambda>s w. \<forall>xa. eval\<o> (x xa) s w)"
Gramian_matrix_def: fixes G :: "'b vec list"   and k :: "nat" shows "gs.Gramian_matrix G k = (let M = mat k n (\<lambda>(i, j). G ! i $ j) in M * M\<^sup>T)"
sel_2: fixes x1 :: "'h"   and x2 :: "'g" shows "snd (x1, x2) = x2"
poly_replicate_append: fixes n :: "nat"   and p :: "'b poly"   and x :: "'b" shows "poly (monom (1::'b) n * p) x = x ^ n * poly p x"
i_Exec_Comp_Stream_Acc_Output__eq_NoMsg_State_Idle_conv: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = NoMsg) = (output_fun (s t1) = NoMsg. t1 \<U> t2 [0\<dots>] \<oplus> t0. output_fun (s t2) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t2)))"
U_parOf_def: shows "U_parOf \<equiv> length \<circ> GE_parOf"
that: fixes n :: "nat"   and k :: "nat" assumes "smap f (gtrace (sdrop n r) (gtarget (stake n r) v)) = sconst k" shows "thesis"
sel_2: fixes x21 :: "'a"   and x22 :: "'b"   and x23 :: "'a"   and x24 :: "'b" shows "pac_src1 (Mult x21 x22 x23 x24) = x21"
multi_psub_self: fixes A :: "'k multiset" shows "(A \<subset># A) = False"
run_reachable': fixes x :: "'a"   and xs :: "'a stream"   and x\<^sub>0 :: "'a" assumes "run (x ## xs)"   and "E\<^sup>*\<^sup>* x\<^sub>0 x" shows "pred_stream (E\<^sup>*\<^sup>* x\<^sub>0) xs"
red_self: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "punit.red {p} p 0"
wi_hom_pred: fixes a :: "int" shows "word_pred (word_of_int a) = word_of_int (a - 1)"
cases: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" assumes "r\<^sup>+\<^sup>+ a1 a2" obtains a :: "'a" and b :: "'a" where "a1 = a__" and "a2 = b__" and "r a__ b__"   | a :: "'a" and b :: "'a" and c :: "'a" where "a1 = a" and "a2 = c" and "r\<^sup>+\<^sup>+ a b" and "r b c"
the_Arr_Arr: fixes T :: "ty"   and k :: "int"   and cs :: "vn \<Rightarrow> val option" shows "the_Arr (Some \<lparr>tag = Arr T k, values = cs\<rparr>) = (T, k, cs)"
P1: shows "Bet A B A0 \<and> Cong B A0 A B"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl' newWl state in InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
preserves_ide: fixes a :: "'a \<times> 'a" assumes "C.VV.ide a" shows "C.ide (fst a \<star>\<^sub>C snd a)"
is_Done_return_resumption: fixes x :: "'f" shows "is_Done (return_resumption x)"
add_mem_right_cancel: fixes x :: "V"   and y :: "V"   and z :: "V" shows "(x + y \<in>\<^sub>\<circ> x + z) = (y \<in>\<^sub>\<circ> z)"
shadow_root_delete_get_2: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root_ptr' :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "shadow_root_ptr \<noteq> shadow_root_ptr'"   and "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr' h = Some h'" shows "get shadow_root_ptr h' = get shadow_root_ptr h"
that: fixes ivl' :: "float interval" assumes "x \<in>\<^sub>r ivl'"   and "approx_form prec f (bs[n := Some ivl']) ss" shows "thesis"
compositionality: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and list :: "'g list" shows "map f (map g list) = map (f \<circ> g) list"
Cd_union: fixes d :: "'b"   and X :: "'a set"   and Y :: "'a set" shows "Cd d (X \<union> Y) = (if Cd d X = {} then Cd d Y else if Cd d Y = {} then Cd d X else {MaxR.maxR (Pd d) x y |x y. x \<in> Cd d X \<and> y \<in> Cd d Y})"
SA_trans_TVar: fixes \<Gamma> :: "binding list"   and i :: "nat"   and U :: "POPLmark.type"   and T :: "POPLmark.type" assumes "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB U\<rfloor>"   and "\<Gamma> \<turnstile> \<up>\<^sub>\<tau> (Suc i) 0 U <: T" shows "\<Gamma> \<turnstile> TVar i <: T"
set_sort_oalist: fixes xs :: "('a \<times> 'c) list"   and x :: "'o" assumes "distinct (map fst xs)" shows "set (sort_oalist x xs) = {kv \<in> set xs. snd kv \<noteq> (0::'c)}"
Indirect: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and intr :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides intr"   and "G \<turnstile> intr overrides old" shows "G \<turnstile> new overrides old"
subspace_0: fixes S :: "'b set" assumes "local.subspace S" shows "(0::'b) \<in> S"
one_is_normal: shows "{\<one>\<^bsub>Z n\<^esub>} \<lhd> Z n"
invertible_det: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "invertible_mat A = (det A \<noteq> (0::'a))"
seq_comp_refusals_finite: fixes xs :: "'c option list"   and P :: "'c option process"   and Q :: "'c option process"   and A :: "'d set"   and F :: "'d \<Rightarrow> 'c option set" assumes "xs \<in> Domain (seq_comp_failures P Q)"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> (xs, F x) \<in> seq_comp_failures P Q" shows "(xs, \<Union> (F ` A)) \<in> seq_comp_failures P Q"
Quotient3I: fixes Abs :: "'h \<Rightarrow> 'g"   and Rep :: "'g \<Rightarrow> 'h"   and R :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>a. Abs (Rep a) = a"   and "\<And>a. R (Rep a) (Rep a)"   and "\<And>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" shows "Quotient3 R Abs Rep"
status: fixes \<sigma> :: "'a status"   and f :: "'a"   and n :: "nat" shows "set (status \<sigma> (f, n)) \<subseteq> {0..<n}"
cong: fixes times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and one :: "'f" shows "comm_monoid_mult.prod times one \<equiv> comm_monoid_mult.prod times one"
sign_iso: fixes n :: "nat" assumes "2 \<le> n" shows "sym_group n Mod carrier (alt_group n) \<cong> sign_img"
mHom_0: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "f \<zero> = \<zero>\<^bsub>N\<^esub>"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "inf (inf a b) c = inf a (inf b c)"
Next: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and first :: "bool"   and nts :: "nat list"   and I :: "\<I>" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "first = (Monitor.progress \<sigma> P \<phi>' j = 0)"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [Monitor.progress \<sigma> P \<phi>' j - 1..<j] nts" shows "wf_mformula \<sigma> j P V n R (MNext I \<phi> first nts) (formula.Next I \<phi>')"
measure_of_def: fixes \<Omega> :: "'c set"   and A :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "measure_of \<Omega> A \<mu> = Abs_measure (\<Omega>, if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>}, \<lambda>a. if a \<in> sigma_sets \<Omega> A \<and> measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> then \<mu> a else 0)"
Nmlize_Hcomp_Hcomp: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "Arr t"   and "Arr u"   and "Arr v"   and "Src t = Trg u"   and "Src u = Trg v" shows "\<^bold>\<lfloor>(t \<^bold>\<star> u) \<^bold>\<star> v\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>u\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>v\<^bold>\<rfloor>\<^bold>\<rfloor>"