hcju/isabelleps huggingface dataset:
threshold_properties_1: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and \<G> :: "'b ltl set"   and i :: "nat"   and k :: "nat" assumes "threshold \<phi> w \<G> = Some i"   and "0 < i" shows "\<not> \<G> \<Turnstile>\<^sub>P af\<^sub>G \<phi> (w [i - 1 \<rightarrow> k])"
per_supset_rel_def: shows "SeprefUF.per_supset_rel \<equiv> {(p1, p2). Restr p1 (Domain p2) = p2 \<and> p1 - Domain p2 \<times> Domain p2 \<subseteq> Id}"
fmerge_def: fixes return :: "'f fset \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('f fset \<Rightarrow> 'e) \<Rightarrow> 'e"   and A :: "'f fset"   and f :: "'f \<Rightarrow> 'e" shows "fmerge return bind A f = ffold (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A |\<union>| B)))) (return {||}) (f |`| A)"
that: fixes N :: "'a set" assumes "\<And>x. x \<in> space M - N \<Longrightarrow> P x"   and "N \<in> null_sets M" shows "thesis"
lookup_default_update: fixes d :: "'a"   and k :: "'b"   and v :: "'a"   and m :: "('b, 'a) mapping" shows "Mapping.lookup_default d (Mapping.update k v m) k = v"
EndPhase2_HInv3: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase2 s s' t"   and "HInv3_inner s p q d" shows "HInv3_inner s' p q d"
rep_eq: fixes x :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and xa :: "('e, 'c) uexpr"   and xb :: "('f, 'c) uexpr"   and xc :: "('g, 'c) uexpr" shows "\<lbrakk>trop x xa xb xc\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b) (\<lbrakk>xc\<rbrakk>\<^sub>e b))"
safe_rpd_fv_regex: fixes fv :: "'b \<Rightarrow> 'c set"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex"   and s :: "'b regex"   and test :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d" assumes "Regex.safe_regex fv safe Past Strict r"   and "s \<in> rpd test i r" shows "fv_regex fv s = fv_regex fv r"
preserves_reflects_arr: fixes f :: "'a" shows "hom'.arr (e \<star> f \<star> src f) = hom.arr f"
dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<B>\<O>\<S>\<S>_def: fixes X :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X.boss) = eval_extract X (deref_oid\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n in_post_state (deref_assocs\<^sub>2\<B>\<O>\<S>\<S> in_post_state (select\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<B>\<O>\<S>\<S> (deref_oid\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n in_post_state))))"
components_are_iso: fixes a :: "'a" assumes "B.ide a" shows "B.iso (B.\<rr> a)"
poly_rel_plus: shows "(poly_rel ===> poly_rel ===> poly_rel) (plus_poly_i ops) (+)"
numfloor_nz: fixes t :: "MIR.num" assumes "nozerocoeff t" shows "nozerocoeff (numfloor t)"
wf_syscls_xcpt_subcls_Throwable: fixes C :: "String.literal"   and P :: "'b prog" assumes "C \<in> sys_xcpts"   and "wf_syscls P" shows "P \<turnstile> C \<preceq>\<^sup>* Throwable"
start_points_path: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x"   and "start_points x \<noteq> 0" shows "backward_terminating_path_root r x"
rtrancl_beta_AppL: fixes s :: "dB"   and s' :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'" shows "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<degree> t"
constructive_security_obsf'D: fixes \<eta> :: "nat" shows "constructive_security_obsf (middle \<eta>) (ideal \<eta>) (sim_inner \<eta>) (\<I>_middle \<eta>) (\<I>_inner \<eta>) (\<I>_common \<eta>) (absorb (\<A> \<eta>) (obsf_converter (sim_outer \<eta> |\<^sub>= 1\<^sub>C))) (advantage (absorb (\<A> \<eta>) (obsf_converter (sim_outer \<eta> |\<^sub>= 1\<^sub>C))) (obsf_resource (sim_inner \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal \<eta>)) (obsf_resource (middle \<eta>)))"
hom_return: fixes x :: "'a" shows "h (return1 x) = return2 x"
csum_absorb1': fixes r2 :: "('c \<times> 'c) set"   and r1 :: "('d \<times> 'd) set" assumes "Card_order r2"   and "r1 \<le>o r2"   and "cinfinite r1 \<or> cinfinite r2" shows "r2 +c r1 =o r2"
simps_10: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x11 :: "'a"   and x12 :: "'b" shows "x31@[x32] \<noteq> x11 \<leftarrow> x12"
simps: fixes A :: "'a mat"   and b :: "'a vec" shows "mat_append_col A b = mat_of_cols (dim_row A) (cols A @ [b])"
that_4: assumes "\<not> ed_ou X \<and> open_unresolvable X" shows "thesis"
hcomp_simps_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" shows "trg (\<nu> \<star> \<mu>) = trg \<nu>"
intro: fixes H :: "'h set"   and M :: "('i, 'h, 'j) module_scheme"   and R :: "('i, 'k) ring_scheme" assumes "subgroup H (add_monoid M)"   and "submodule_axioms H R M" shows "Module.submodule H R M"
superset_preserves_strict_redundancy: fixes C :: "'a Literal set"   and S :: "'a Literal set set"   and SS :: "'a Literal set set" assumes "strictly_redundant C S"   and "S \<subseteq> SS" shows "strictly_redundant C SS"
diff_union_swap2: fixes y :: "'b"   and M :: "'b multiset"   and x :: "'b" assumes "y \<in># M" shows "add_mset x M \<ominus> y = add_mset x (M \<ominus> y)"
VectorSpaceHom_scalar_mul: fixes a :: "'f" shows "VectorSpaceHom (\<cdot>) V (\<cdot>) (a \<cdot>\<cdot> T)"
wf_SUFFIX: fixes r :: "'b rexp" assumes "local.wf n r" shows "local.wf n (SUFFIX r)"
secure_implies_g_secure: fixes P :: "'a process"   and L :: "'a \<Rightarrow> g_level" assumes "secure P g_I L" shows "g_secure P L"
NegativeArraySize_def: shows "NegativeArraySize = STR ''java/lang/NegativeArraySizeException''"
rel_refl: fixes Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "\<And>x. Ra x x" shows "list_all2 Ra x x"
bre'_pref_br: fixes Qi :: "'c set"   and \<delta> :: "('c, 'd) ta_rule set" shows "wa_precise_refine (bre'_algo Qi \<delta>) (bre_algo Qi \<delta>) br'_\<alpha>"
det_transpose: fixes A :: "('c, 'd) sq_matrix" shows "Square_Matrix.det (Square_Matrix.transpose A) = Square_Matrix.det A"
snth_in: fixes s :: "'d stream"   and X :: "'d set"   and n :: "nat" assumes "s \<in> streams X" shows "s !! n \<in> X"
order_trans: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "order r"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
n_absorb1': fixes x :: "'a"   and y :: "'a" shows "!x + !x \<cdot> !y = !x"
invertible: fixes u :: "'a" assumes "u \<in> A" shows "IA.additive.invertible u"
wtPB_Un: fixes \<Phi>1 :: "('fsym, 'psym) lit list set"   and \<Phi>2 :: "('fsym, 'psym) lit list set" shows "wtPB (\<Phi>1 \<union> \<Phi>2) = (wtPB \<Phi>1 \<and> wtPB \<Phi>2)"
runit_char_2: fixes a :: "'a" assumes "ide a" shows "\<r>\<^sub>1[a] \<otimes>\<^sub>1 \<I>\<^sub>1 = (a \<otimes>\<^sub>1 \<iota>\<^sub>1) \<cdot> \<a>\<^sub>1[a, \<I>\<^sub>1, \<I>\<^sub>1]"
simps_7: fixes a :: "MIR.num" shows "bound0 (Eq a) = numbound0 a"
simps_2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" shows "usubstappf \<sigma> U (Geq \<theta> \<eta>) = Geqo (usubstappt \<sigma> U \<theta>) (usubstappt \<sigma> U \<eta>)"
lbounds_def: fixes as :: "atom list" shows "lbounds as = concat (map (\<lambda>x. case x of Le i [] \<Rightarrow> [] | Le i (k # ks) \<Rightarrow> if 0 < k then [(i, ks)] else [] | _ \<Rightarrow> []) as)"
tendsto_uniform_limitI: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and S :: "'d set"   and l :: "'d \<Rightarrow> 'e"   and F :: "'c filter"   and x :: "'d" assumes "uniform_limit S f l F"   and "x \<in> S" shows "((\<lambda>y. f y x) \<longlongrightarrow> l x) F"
graph_restrict_simps_2: fixes G :: "('a, 'b) graph_rec_scheme"   and R :: "'a set" shows "g_E (graph_restrict G R) = rel_restrict (g_E G) R"
i_m: shows "[i..<m] = i # [Suc i..<m]"
simps_19: fixes x4 :: "32 word"   and x3 :: "bool" shows "Intg x4 \<noteq> Bool x3"
one_m_x_times: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "(1\<^sub>r\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> x) \<cdot>\<^sub>r\<^bsub>R\<^esub> \<Sigma>\<^sub>e R NPOW x R n = 1\<^sub>r\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> x^\<^bsup>R Suc n\<^esup>"
prems: shows "enat (Suc i) < llength (lselect s w)"
decode_plan'_set_is: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and k :: "nat" shows "set (decode_plan' \<Pi> \<A> k) = {\<Pi>\<^sub>\<O> ! index (\<Pi>\<^sub>\<O>) op |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op))}"
existence_ivl_subsetI: fixes T :: "real set"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and X :: "'d set"   and Y :: "'d set"   and S :: "real set"   and g :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and t :: "real"   and t0 :: "real"   and x0 :: "'d" assumes "ll_on_open_it T f X"   and "X = Y"   and "T = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> f t x = g t x"   and "t \<in> ll_on_open.existence_ivl T f X t0 x0" shows "t \<in> ll_on_open.existence_ivl S g Y t0 x0"
try_unbox_def: fixes \<tau> :: "'c"   and x :: "'d"   and \<Sigma> :: "'c option list"   and unbox :: "'d \<Rightarrow> 'e option"   and mk_instr :: "'e \<Rightarrow> 'b" shows "try_unbox \<tau> x \<Sigma> unbox mk_instr \<equiv> case unbox x of None \<Rightarrow> Error () | Some n \<Rightarrow> Ok (mk_instr n, Some \<tau> # \<Sigma>)"
interpret_floatarith_max_Var_cong: fixes f :: "floatarith"   and xs :: "real list"   and ys :: "real list" assumes "\<And>i. i < max_Var_floatarith f \<Longrightarrow> xs ! i = ys ! i" shows "interpret_floatarith f ys = interpret_floatarith f xs"
closest_bound_def: fixes Q\<^sub>b :: "'a"   and Q :: "'a set" shows "closest_bound Q\<^sub>b Q \<equiv> \<exists>f. is_bound_f Q\<^sub>b Q f \<and> (\<forall>Q\<^sub>b'. is_bound Q\<^sub>b' Q \<and> Q\<^sub>b' \<noteq> Q\<^sub>b \<longrightarrow> [[f 0 Q\<^sub>b Q\<^sub>b']])"
int_to_bv_returntype: fixes w :: "int" shows "norm_signed (int_to_bv w) = int_to_bv w"
add_mono_thms_linordered_field_2: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i = j \<and> k < l" shows "i + k < j + l"
blinfun_apply_inject: fixes x :: "'b \<Rightarrow>\<^sub>L 'c"   and y :: "'b \<Rightarrow>\<^sub>L 'c" shows "(blinfun_apply x = blinfun_apply y) = (x = y)"
iteratei_def: fixes list_it :: "'g \<Rightarrow> (('e \<times> 'f) list \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> ('e \<times> 'f) list \<Rightarrow> ('e \<times> 'f) list) \<Rightarrow> ('e \<times> 'f) list \<Rightarrow> ('e \<times> 'f) list"   and S :: "'g" shows "poly_map_iteratei_defs.iteratei list_it S \<equiv> it_to_it (list_it S)"
uminus_le_Fin_iff: fixes x :: "'a extended"   and y :: "'a" shows "(- x \<le> Fin (- y)) = (Fin y \<le> x)"
simps_6: fixes jmps :: "jump set"   and l :: "label"   and e :: "expr"   and c :: "stmt" shows "jumpNestingOkS jmps (l\<bullet> While(e) c) = jumpNestingOkS ({Cont l} \<union> jmps) c"
run_owner_ii_ex: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "pred_owner_ii n (run_asset_i n s) (run_owner_ii n s)"
map_snd_mk_fst: fixes k :: "'b"   and l :: "'a list" shows "map snd (map (Pair k) l) = l"
measurable_extract_real: shows "extract_real \<in> borel_measurable (stock_measure REAL)"
aperiodicI_pair: fixes C1 :: "'s set"   and C2 :: "'s option set" assumes "aperiodic C1"   and "K2.aperiodic C2" shows "KN.aperiodic (C1 \<times> C2)"
simps_5: fixes c :: "poly"   and s :: "tm" shows "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
le_anti_symmetry: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Le C D"   and "C D Le A B" shows "Cong A B C D"
simps_4: fixes var :: "nat"   and \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and \<Gamma> :: "real list" shows "convert_fm var (And \<phi> \<psi>) \<Gamma> = (case (convert_fm var \<phi> \<Gamma>, convert_fm var \<psi> \<Gamma>) of (None, ba) \<Rightarrow> None | (Some a, None) \<Rightarrow> None | (Some a, Some b) \<Rightarrow> Some (AndUni a b))"
inverse_transformations_inverse_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b" assumes "inverse_transformations A B F G \<sigma> \<sigma>'" shows "vertical_composite.map A B \<sigma> \<sigma>' = F"
hyps: fixes as :: "'a list" assumes "length as < length asa__"   and "distinct as" shows "distincts (orbits_list f as)"
canonical_proj_ext_def: fixes I :: "nat \<Rightarrow> 'a set"   and n :: "nat" shows "canonical_proj_ext I n = (\<lambda>a. map (\<lambda>i. I i +> a) [0..<Suc n])"
pdevs_val_independent_pdevs: fixes xs :: "(real \<times> real) list"   and e :: "nat \<Rightarrow> real" assumes "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"   and "list_all (\<lambda>x. x \<noteq> 0) xs"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "\<exists>e'. e' \<in> UNIV \<rightarrow> {- 1..1} \<and> pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list (independent_pdevs xs))"
phis_disj_1: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list"   and n' :: "'node"   and vs' :: "'val list" assumes "Mapping.lookup (phis g) (n, v) = Some vs"   and "Mapping.lookup (phis g) (n', v) = Some vs'" shows "n = n'"
k2_2: fixes k' :: "'a" assumes "k' < k2" shows "g k' = f k'"
assms_2: shows "\<not> is_unit (h - [:c1:])"
weakOutputAlpha: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and p :: "(name \<times> name) list"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>M\<lparr>\<nu>*(p \<bullet> xvec)\<rparr>\<langle>(p \<bullet> N)\<rangle> \<prec> P'"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "xvec \<sharp>* P"   and "xvec \<sharp>* (p \<bullet> xvec)"   and "(p \<bullet> xvec) \<sharp>* M"   and "distinct xvec" shows "\<Psi> : Q \<rhd> P \<Longrightarrow>M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> p \<bullet> P'"
axioms_1: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>2 :: "'f set" assumes "finite_dimensional_vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2" shows "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1"
maxim_Int: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and B :: "'d set" assumes "wo_rel r"   and "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field r"   and "wo_rel.maxim r A \<in> B" shows "wo_rel.maxim r (A \<inter> B) = wo_rel.maxim r A"
S'_S: fixes i :: "'a" shows "S' i \<subseteq> S i"
A: shows "(ca + cb, w, c') \<in> trcl (ntr fg)"
mono: fixes x :: "'c" shows "monotone (\<sqsubseteq>) (\<le>) (\<lambda>xs. f x xs (LCons x xs))"
Rep_zero_matrix_def: fixes j :: "nat"   and i :: "nat" shows "Rep_matrix 0 j i = (0::'a)"
fun_type_param_idx: fixes f :: "'fun"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and g :: "'fun"   and S :: "('fun, 'atom) Term.term list"   and i :: "nat" assumes "\<Gamma> (Fun f T) = Fun g S"   and "i < length T" shows "\<Gamma> (T ! i) = S ! i"
LeadsTo_cancel2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w A' \<union> B"   and "F \<in> B \<longmapsto>w B'" shows "F \<in> A \<longmapsto>w A' \<union> B'"
p1_def: shows "p1 = quadratic_part_1 var a b d (Neq p)"
real_transfer_operator_charact: fixes g :: "'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> real" assumes "\<And>A. A \<in> sets M \<Longrightarrow> LINT x|M. indicat_real A x * g x = LINT x|M. indicat_real A (T x) * f x"   and "integrable M f"   and "integrable M g" shows "AE x in M. real_transfer_operator f x = g x"
right_cancel: fixes x :: "'a vec"   and y :: "'a vec"   and z :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "(y + x = z + x) = (y = z)"
integrable_fst': fixes M1 :: "'b measure"   and M2 :: "'c measure"   and f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "pair_sigma_finite M1 M2"   and "integrable (M1 \<Otimes>\<^sub>M M2) f" shows "integrable M1 (\<lambda>x. LINT y|M2. f (x, y))"
shiftr1_unfold: fixes x :: "'b word" shows "shiftr1 x = x >> 1"
satisfies_atom_set_def: fixes v :: "nat \<Rightarrow> 'a"   and as :: "'a atom set" shows "v \<Turnstile>\<^sub>a\<^sub>s as \<equiv> \<forall>a\<in>as. v \<Turnstile>\<^sub>a a"
inj_on_extended: fixes scaleB :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'b \<Rightarrow> 'e"   and C :: "'b set"   and B :: "'b set"   and W :: "'b set" assumes "linear scaleB scaleC f"   and "finite C"   and "local.independent C"   and "C = B \<union> W"   and "B \<inter> W = {}"   and "{x. f x = (0::'e)} \<subseteq> local.span B" shows "inj_on f W"
comp_null_1: fixes f :: "'a" shows "L.comp L.null f = L.null"
assms_1: shows "c = ((recf.Id m n, xs, ls) # fs, rv)"
validE: fixes X :: "msg" assumes "X \<in> sym.valid" obtains x :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "X = implem_sym (Chan x A B M)" and "M \<in> payload"
ts_inv_ok_upd_ts: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and t :: "'c"   and x :: "'a \<times> 'b \<Rightarrow>f nat"   and I :: "'c \<Rightarrow> 'd option"   and x' :: "'a \<times> 'b \<Rightarrow>f nat" assumes "ts t = \<lfloor>x\<rfloor>"   and "ts_inv_ok ts I" shows "ts_inv_ok (ts(t \<mapsto> x')) I"
simps_12: fixes C' :: "char list"   and C :: "char list"   and M :: "char list"   and es :: "'a exp list" shows "not_init C' (C\<bullet>\<^sub>sM(es)) = not_inits C' es"
of_real_in_nonpos_Ints_iff: fixes x :: "real" shows "(of_real x \<in> \<int>\<^sub>\<le>\<^sub>0) = (x \<in> \<int>\<^sub>\<le>\<^sub>0)"
body: shows "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"
row_four_block_mat_1: fixes A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'a mat"   and nc2 :: "nat"   and C :: "'a mat"   and nr2 :: "nat"   and D :: "'a mat"   and i :: "nat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2"   and "i < nr1" shows "row (four_block_mat A B C D) i = row A i @\<^sub>v row B i"
id1: shows "ffp (R (n (i - 1) - 1)) = subresultant (n (i - 1) - 1) (F 1) (F 2)"
WhileFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com" assumes "\<not> bval b s" shows "(WHILE b DO c, s) \<Rightarrow> s"
pivotandupdate_bounds_consistent: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<diamond> (pivot_and_update x\<^sub>i x\<^sub>j c s) = \<diamond> s"
add_decreasing2: fixes c :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" assumes "c \<subseteq># {#}"   and "a \<subseteq># b" shows "a + c \<subseteq># b"
plus_left_inverse: fixes x :: "'a" shows "x + inver x = zer"
daMMk: fixes k :: "nat" assumes "k < n" shows "adjoint (M k) * M k \<in> carrier_mat d d"
cs_not_not_started_stable_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace c t c'"   and "snd (cs c cid) \<noteq> NotStarted"   and "channel cid = Some (p, q)" shows "snd (cs c' cid) \<noteq> NotStarted"
semilattice_neutr_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "semilattice_neutr f z \<equiv> semilattice f \<and> comm_monoid f z"
per_col: fixes B :: "'p"   and C :: "'p"   and A :: "'p"   and D :: "'p" assumes "B \<noteq> C"   and "Per A B C"   and "Col B C D" shows "Per A B D"
flip_not_fv: fixes a :: "'a"   and x :: "'b"   and b :: "'a" assumes "a \<notin> fv x"   and "b \<notin> fv x" shows "(a \<leftrightarrow> b) \<bullet> x = x"
encR: fixes S :: "'procS" shows "SourceTerm S \<sim>\<lbrakk>\<cdot>\<rbrakk> TargetTerm (\<lbrakk>S\<rbrakk>)"
inorder_delete: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "invar t"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (delete x t) = del_list x (Tree2.inorder t)"
simps_1: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and s :: "'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.mset_lit (Pos (Eq t s)) = {#t, s#}"
S3MemUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RNext rmCh mm ires p]_(m p) \<and> $S3 rmhist p \<longrightarrow> unchanged m p"
MonotProblem_axioms: shows "MonotProblem GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf gPB"
cases: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "local.trimming g" obtains "current \<Gamma> g" and "wave \<Gamma> g" and "g \<le> f" and "\<And>x. \<lbrakk>x \<in> RF\<^sup>\<circ> (TER f); x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> KIR g x" and "\<E> (TER g) - A \<Gamma> = \<E> (TER f) - A \<Gamma>"
pow_carrier_mod: fixes G :: "('c, 'd) cyclic_group_scheme"   and g :: "'c"   and k :: "nat" assumes "cyclic_group G"   and "g \<in> carrier G" shows "g [^]\<^bsub>G\<^esub> (k mod order G) = g [^]\<^bsub>G\<^esub> k"
sublist_length: fixes l :: "'d list"   and l' :: "'d list" assumes "subseq l l'" shows "length l \<le> length l'"
2: shows "(\<lambda>t. M * exp (- (Re s - c) * t)) integrable_on {A..}"
UNIV_4: shows "UNIV = {1, 2, 3, 4}"
image_in_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "'a"   and P :: "'a"   and P' :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ReflectLAt Bet Cong Y P P' A B" shows "Tarski_neutral_dimensionless.Col Bet P P' Y"