hcju/isabelleps huggingface dataset:
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e option" shows "map_option g (map_option f v) = map_option (g \<circ> f) v"
inf_sup_aci_3: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.lattice inf less_eq less sup" shows "inf x (inf y z) = inf y (inf x z)"
hyps: fixes x :: "'a" assumes "x \<in> r \<down>m P" shows "\<exists>y\<in>#P -s r \<down>m P. (x, y) \<in> r"
cn_merge_gs_split: fixes i :: "nat"   and gs :: "recf list"   and ga :: "abc_inst list"   and gb :: "nat"   and gc :: "nat"   and p :: "nat" assumes "i < length gs"   and "rec_ci (gs ! i) = (ga, gb, gc)" shows "cn_merge_gs (map rec_ci gs) p = cn_merge_gs (map rec_ci (take i gs)) p [+] (ga [+] mv_box gb (p + i)) [+] cn_merge_gs (map rec_ci (drop (Suc i) gs)) (p + Suc i)"
r6: fixes x :: "name"   and s :: "trm"   and s' :: "trm"   and t :: "trm" assumes "x \<sharp> (s, s')"   and "s \<mapsto> s'" shows "s to x in t \<mapsto> s' to x in t"
ring_simprules_24: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
lunit_char_2: fixes f :: "'a" assumes "B.ide f" shows "B.L (B.lunit f) = (\<i>[trg\<^sub>B f] \<star>\<^sub>B f) \<cdot>\<^sub>B B.\<a>' (trg\<^sub>B f) (trg\<^sub>B f) f"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a) F \<Phi>"
assms_1: shows "nat_term_order_eq to1 to2 dg ps"
e2: shows "pdevs_val e (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst Y)) (snd (fst X))))) = pdevs_val e (scaleR_pdevs (fst (fst Y)) (snd (fst X))) + e2"
fconverse_def: fixes r :: "V" shows "r\<inverse>\<^sub>\<bullet> = set {[b, a]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>lss.rel)"
rep_eq: fixes x :: "'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" shows "blinfun_apply (blinfun_vec x) = (\<lambda>xa. \<chi>i. blinfun_apply (x i) xa)"
evaluate_iff_1: fixes env :: "v sem_env"   and st :: "'a state"   and e :: "exp0"   and r :: "'a state \<times> (v, v) result" shows "evaluate True env st e r = (r = eval env e st)"
assms_4: shows "qf2 = (\<lambda>x y. x\<^sup>2 + y\<^sup>2 + B1 * x + B2 * y + C)"
rtrancl_Un_separatorE: fixes a :: "'a"   and b :: "'a"   and P :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" assumes "(a, b) \<in> (P \<union> Q)\<^sup>*"   and "\<forall>x y. (a, x) \<in> P\<^sup>* \<longrightarrow> (x, y) \<in> Q \<longrightarrow> x = y" shows "(a, b) \<in> P\<^sup>*"
partial_order_on_converse: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "partial_order_on A (r\<inverse>) = partial_order_on A r"
inf_compl_bot_left2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (inf (uminus x) y) = bot"
locked_in_vf_def: fixes v_f :: "process \<Rightarrow> val option"   and v :: "val" shows "locked_in_vf v_f v \<equiv> \<exists>Q. quorum_for Q v v_f"
eval_neg_if_dominant_term_neg': fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "trimmed_ms_aux F"   and "is_expansion_aux F f basis"   and "fst (dominant_term_ms_aux F) < 0" shows "\<forall>\<^sub>F x in at_top. f x < 0"
surjective: fixes r :: "('c, 'd, 'e) graph_scheme" shows "r = \<lparr>nodes = nodes r, edges = edges r, \<dots> = more r\<rparr>"
Liminf_bounded: fixes C :: "'d"   and X :: "'c \<Rightarrow> 'd"   and F :: "'c filter" assumes "\<forall>\<^sub>F n in F. C \<le> X n" shows "C \<le> Liminf F X"
preserves_equivalent_objects: fixes a :: "'b"   and b :: "'b" assumes "C.equivalent_objects a b" shows "D.equivalent_objects (G.map\<^sub>0 a) (G.map\<^sub>0 b)"
lens_equivI: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "X \<subseteq>\<^sub>L Y"   and "Y \<subseteq>\<^sub>L X" shows "X \<approx>\<^sub>L Y"
id_1: shows "monom 1 1 \<circ>\<^sub>p v = v"
ind_valid_graph': fixes L :: "(nat \<times> int \<times> nat) list"   and E' :: "(nat \<times> int \<times> nat) set" assumes "subgraph \<lparr>nodes = fst ` set L \<union> (snd \<circ> snd) ` set L, edges = E'\<rparr> (GD_from_list_\<alpha>_graph L L)" shows "valid_graph \<lparr>nodes = fst ` set L \<union> (snd \<circ> snd) ` set L, edges = E'\<rparr>"
suminf_minus_initial_segment: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "summable f" shows "(\<Sum>n. f (n + k)) = (\<Sum>n. f n) - sum f {..<k}"
fac_len': fixes w :: "'b list"   and u :: "'b list"   and v :: "'b list" shows "\<^bold>|w\<^bold>| \<le> \<^bold>|u \<cdot> w \<cdot> v\<^bold>|"
filter_chain_lemma: fixes F :: "'b set set"   and T :: "'b set set"   and C :: "'b set set set" assumes "Topology.filter F T"   and "C \<in> chains {V \<in> Filters\<^bsub>T\<^esub>. F \<subseteq> V}" shows "\<Union> (C \<union> {F}) \<in> Filters\<^bsub>T\<^esub>"
qo_on_predecessor_subset_conv': fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "qo_on P A"   and "B \<subseteq> A"   and "C \<subseteq> A" shows "({x \<in> A. \<exists>y\<in>B. P x y} \<subseteq> {x \<in> A. \<exists>y\<in>C. P x y}) = (\<forall>x\<in>B. \<exists>y\<in>C. P x y)"
heap_def: fixes s :: "st" shows "heap s = globs s \<circ> Inl"
terminates_replicate_generator: fixes a :: "'a" shows "terminates ((<) 0, \<lambda>n. (a, n - 1))"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes type_wf set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "l_set_disconnected_nodes_get_tag_name_axioms set_disconnected_nodes_locs get_tag_name_locs" shows "l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs"
restriction_stable: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u\<parallel>w \<longrightarrow> restrict u f c = restrict w f c"
continuous_on_Arcsin_real: shows "continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} Arcsin"
Closed: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "(\<Psi>', S, T) \<in> Rel" shows "(p \<bullet> \<Psi>', p \<bullet> S, p \<bullet> T) \<in> Rel"
finite_surj: fixes A :: "'e set"   and B :: "'f set"   and f :: "'e \<Rightarrow> 'f" assumes "finite A"   and "B \<subseteq> f ` A" shows "finite B"
avl_delete_max_1: fixes x :: "'b tree" assumes "avl x"   and "x \<noteq> ET" shows "avl (snd (delete_max x))"
inner_diff_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inner (x - y) z = inner x z - inner y z"
term_ok'_eta_norm: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term" assumes "term_ok' \<Sigma> t" shows "term_ok' \<Sigma> (eta_norm t)"
nth_butlast: fixes n :: "nat"   and xs :: "'d list" assumes "n < length (butlast xs)" shows "butlast xs ! n = xs ! n"
divergence_reflection_iff_source_target_rel_reflects_divergence: shows "enc_reflects_divergence = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_reflects_divergence Rel (STCal Source Target))"
subdag_not_sym: fixes s :: "dag"   and t :: "dag"   and P :: "bool" assumes "subdag s t"   and "subdag t s" shows "P"
rotate13: fixes A13 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and A7 :: "fmla"   and A8 :: "fmla"   and A9 :: "fmla"   and A10 :: "fmla"   and A11 :: "fmla"   and A12 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A13 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H)))))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H)))))))))))) \<turnstile> B"
deriv_anno: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P anno Q,A"
splitted_def: fixes p :: "'a list list" shows "UP.splitted p = (size (UP.roots p) = degree p)"
2: fixes theta :: "('a \<times> 'a) set" assumes "trans theta" shows "trans (Retr theta)"
emeasure_distr: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and A :: "'d set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "A \<in> sets N" shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)"
exhaust: fixes y :: "tv" obtains x1 :: "bool" where "y = Det x1"   | x2 :: "nat" where "y = Indet x2"
split_face_edges_f21_bet: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs"   and "vs \<noteq> []"   and "between (vertices f) ram2 ram1 = []" shows "\<E> f21 = {(ram1, hd vs), (last vs, ram2), (ram2, ram1)} \<union> Edges vs"
counit_simps_4: shows "src \<psi> = trg f"
stable_no_blocking_onI2: fixes ds :: "'d set"   and X :: "'x set" assumes "\<And>h X''. blocking_on ds X h X'' \<Longrightarrow> False" shows "stable_no_blocking_on ds X"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "current \<Gamma> f = ((\<forall>x. d_OUT f x \<le> weight \<Gamma> x) \<and> (\<forall>x. d_IN f x \<le> weight \<Gamma> x) \<and> (\<forall>x. x \<notin> A \<Gamma> \<longrightarrow> d_OUT f x \<le> d_IN f x) \<and> (\<forall>x. x \<in> A \<Gamma> \<longrightarrow> d_IN f x = 0) \<and> (\<forall>x. x \<in> B \<Gamma> \<longrightarrow> d_OUT f x = 0) \<and> (\<forall>x. x \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub> \<longrightarrow> f x = 0))"
assms: shows "(lin_red F)\<^sup>*\<^sup>* p q"
simps_9: fixes f1 :: "label \<Rightarrow> 'a"   and f2 :: "label \<Rightarrow> 'a"   and f3 :: "'a"   and x1 :: "label" shows "(case Break x1 of Break x \<Rightarrow> f1 x | Cont x \<Rightarrow> f2 x | Ret \<Rightarrow> f3) = f1 x1"
rk2_step_def: fixes X0 :: "'c set"   and h :: "real" shows "rk2_step X0 h = one_step X0 h (\<lambda>X0 hl hu CX. let H = lv_ivl [min hl hu] [max hl hu] in rk2_param_spec \<bind> (\<lambda>rps. let rkp = lv_ivl [rps] [rps]; s2 = lv_ivl [0] [1] in mk_safe CX \<bind> (\<lambda>_. ASSUME (ncc CX) \<bind> (\<lambda>_. let env = concat ` listset [list_of_eucl ` X0, list_of_eucl ` CX, rkp, H, s2] in approx_slp_spec (rk2_fas' DIM('c)) (2 * DIM('c)) rk2_slp env \<bind> (\<lambda>env. case env of None \<Rightarrow> RETURN None | Some env \<Rightarrow> let res' = take DIM('c) ` env in ASSERT (env_len res' DIM('c)) \<bind> (\<lambda>_. let res = eucl_of_list ` res' in ASSUME (ncc res) \<bind> (\<lambda>_. let err' = drop DIM('c) ` take (DIM('c) * 2) ` env in ASSERT (env_len err' DIM('c)) \<bind> (\<lambda>_. let err = eucl_of_list ` err' in default_reduce_argument_spec \<bind> (\<lambda>ra. reduce_spec ra res \<bind> (\<lambda>res. ASSUME (ncc res) \<bind> (\<lambda>_. safe_set res \<bind> (\<lambda>s. if s then mk_safe res \<bind> (\<lambda>res. RETURN (Some (res, err))) else RETURN None))))))))))))"
add: fixes f :: "'l \<Rightarrow> 'm"   and x :: "'l"   and y :: "'l" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
elims: fixes x :: "'a set set"   and y :: "'a set" assumes "vertices x = y" obtains G :: "'a set set" where "x = G" and "y = \<Union> G"
ring_hom_nat_pow: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and a :: "'c"   and n :: "nat" assumes "ring R"   and "ring S"   and "h \<in> ring_hom R S"   and "a \<in> carrier R" shows "h (a [^]\<^bsub>R\<^esub> n) = h a [^]\<^bsub>S\<^esub> n"
append_store_commute: fixes i1 :: "'c"   and mo :: "'c"   and e1 :: "'d"   and e2 :: "'d"   and i2 :: "'c" assumes "i1 \<noteq> mo" shows "\<langle>Append i1 e1\<rangle> \<rhd> \<langle>Store e2 mo i2\<rangle> = \<langle>Store e2 mo i2\<rangle> \<rhd> \<langle>Append i1 e1\<rangle>"
memoization_correct_2: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and empty :: "'b"   and dp :: "'a \<Rightarrow> 'c"   and dp\<^sub>T :: "'a \<Rightarrow> ('b, 'c) state"   and x :: "'a"   and v :: "'c"   and m :: "'b" assumes "dp_consistency_empty lookup update P empty"   and "dp_consistency.consistentDP lookup P dp dp\<^sub>T"   and "run_state (dp\<^sub>T x) empty = (v, m)" shows "dp_consistency.cmem lookup dp m"
derf: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(f has_derivative f' x) (at x within {a..b})"
cases: fixes a1 :: "'a list"   and a2 :: "'a list"   and a3 :: "'a list" assumes "append2 a1 a2 a3" obtains xs :: "'a list" where "a1 = []" and "a2 = xs__" and "a3 = xs__"   | xs :: "'a list" and ys :: "'a list" and zs :: "'a list" and x :: "'a" where "a1 = x # xs" and "a2 = ys" and "a3 = x # zs" and "append2 xs ys zs"
invar_def: fixes q :: "('c, 'd) BinomialTree list" shows "invar q \<equiv> queue_invar q \<and> rank_invar q"
wf_trm_param: fixes f :: "'fun"   and T :: "('fun, 'a) Term.term list"   and t :: "('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"   and "t \<in> set T" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
thmD13_part2: fixes k :: "nat" assumes "items_le k (\<J> k 0) = Gen (paths_le k (\<P> k 0))"   and "k \<le> length Doc" shows "items_le k (\<I> k) = Gen (paths_le k (\<Q> k))"
spheremap_lemma2: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "subspace S"   and "subspace T"   and "dim S < dim T"   and "S \<subseteq> T"   and "continuous_on (sphere (0::'a) 1 \<inter> S) f"   and "f ` (sphere (0::'a) 1 \<inter> S) \<subseteq> sphere (0::'a) 1 \<inter> T" shows "\<exists>c. homotopic_with_canon (\<lambda>x. True) (sphere (0::'a) 1 \<inter> S) (sphere (0::'a) 1 \<inter> T) f (\<lambda>x. c)"
isRes_def: fixes \<sigma> :: "'tp" shows "isRes \<sigma> \<equiv> \<exists>f. wtFsym f \<and> resOf f = \<sigma>"
P2: shows "P Q TS A D \<and> P Q TS C D"
elims_2: fixes x :: "object ast_effect" assumes "wf_effect_inst x" obtains a :: "object atom formula list" and d :: "object atom formula list" where "x = Effect a d" and "Ball (set a \<union> set d) (wf_fmla_atom objT)"
intro: fixes m :: "'a"   and m' :: "'a" assumes "m \<noteq> m'" shows "mut_m' m m'"
merge_list_length: fixes as :: "'b list list"   and ls :: "'b list list" shows "length (merge_list as ls) \<le> length (concat (as @ ls))"
node_tauTE': fixes i :: "nat"   and s :: "'c"   and R :: "nat set"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS i s R, \<tau>, s') \<in> node_sos S" obtains d :: "nat" and s'a :: "'c" where "s' = NodeS i s'a__ R" and "(s, \<not>unicast d, s'a__) \<in> S" and "d \<notin> R"   | s'a :: "'c" where "s' = NodeS i s'a R" and "(s, \<tau>\<^sub>s, s'a) \<in> S"
f_arg_min_list_f: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "xs \<noteq> []" shows "f (arg_min_list f xs) = Min (f ` set xs)"
nonneg_euler_genus: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map" assumes "digraph_map G M" shows "0 \<le> pre_digraph_map.euler_genus G M"
fds_divisor_sum: shows "fds divisor_sum = fds_zeta * fds id"
dom_char: fixes f :: "'c" shows "C'.dom f = (if C'.arr f then C.dom f else C.null)"
abs_eq: fixes xa :: "'c set"   and x :: "'c set" assumes "eq_onp ord_class.filter xa xa"   and "eq_onp ord_class.filter x x" shows "(Filters.filter.Abs_filter xa \<le> Filters.filter.Abs_filter x) = (xa \<subseteq> x)"
msubst_conj: fixes P :: "'b \<Rightarrow> 'a upred"   and Q :: "'b \<Rightarrow> 'a upred"   and v :: "('b, 'a) uexpr" shows "(P x \<and> Q x)\<lbrakk>x\<rightarrow>v\<rbrakk> = (msubst P v \<and> msubst Q v)"
max_normal_simple_quotient: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G"   and "finite (carrier G)" shows "max_normal_subgroup H G = simple_group (G Mod H)"
a_subid_aux2: fixes y :: "'a"   and x :: "'a" shows "ar y \<cdot> x \<le> x"
cos_periodic_nat: fixes x :: "real"   and n :: "nat" shows "cos (x + real n * (2 * pi)) = cos x"
effect_insertion_sort: fixes a :: "'b array"   and h :: "heap"   and h' :: "heap"   and u :: "unit" assumes "effect (insertion_sort a) h h' u" shows "mset (Array.get h a) = mset (Array.get h' a) \<and> sorted (Array.get h' a)"
simps_6: fixes SK :: "nat option \<times> nat set \<times> nat set" shows "InvK (SesK SK) = SesK SK"
not_lfinite_lprefix_conv_eq: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "linfinite xs" shows "lprefix xs ys = (xs = ys)"
extf_compat_list: fixes y :: "('s, 'v) tm"   and x :: "('s, 'v) tm"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "'s"   and xs :: "('s, 'v) tm list"   and xs' :: "('s, 'v) tm list" assumes "y \<noteq> x"   and "gt y x" shows "extf f gt (xs @ y # xs') (xs @ x # xs')"
divides_antisym: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a divides\<^bsub>G\<^esub> b"   and "b divides\<^bsub>G\<^esub> a" shows "a \<sim>\<^bsub>G\<^esub> b"
eq1: fixes j :: "nat"   and k :: "nat" assumes "j < length vs"   and "k < length vs" shows "(inner_prod (vs ! k) (vs ! j) = 0) = (j \<noteq> k)"
Int_absorb1: fixes B :: "'a set"   and A :: "'a set" assumes "B \<subseteq> A" shows "A \<inter> B = B"
nb\<^sub>1: shows "\<forall>C\<in>cnf (\<Phi>\<^sub>I \<Pi>). clause_semantics \<A> C"
integrable_rv: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "integrable f M" shows "f \<in> rv M"
Newman_local: fixes r :: "('b \<times> 'b) set"   and X :: "'b set" assumes "SN_on r X"   and "WCR_on r {x. SN_on r {x}}" shows "CR_on r X"
P8: shows "\<lfloor>\<lambda>w. universal (\<lambda>x xa. (xa explains x \<^bold>\<rightarrow> x dependsOn xa) w)\<rfloor>"
abrupt_if_False: fixes x :: "abopt"   and y :: "abopt" shows "abrupt_if False x y = y"
nonInterference_def: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and c :: "com" shows "nonInterference \<Gamma> c \<equiv> \<forall>s1 s2 s1' s2'. \<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2 \<and> \<langle>c,s1\<rangle> \<rightarrow>* \<langle>Skip,s1'\<rangle> \<and> \<langle>c,s2\<rangle> \<rightarrow>* \<langle>Skip,s2'\<rangle> \<longrightarrow> \<Gamma> \<turnstile> s1' \<approx>\<^sub>L s2'"
scaleR_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR 1 x = x"
inj_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj f" shows "inj_on f A"
rtrancl3p_step_converse: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and a' :: "'a"   and bs :: "'b list"   and a'' :: "'a" assumes "r a b a'"   and "rtrancl3p r a' bs a''" shows "rtrancl3p r a (b # bs) a''"
ind: fixes j :: "name"   and i :: "name"   and A :: "fm" assumes "atom j \<sharp> (i, A)" shows "induction_axiomsp (A(i::=Zero) IMP SyntaxN.All i (SyntaxN.All j (A IMP A(i::=Var j) IMP A(i::=Eats (Var i) (Var j)))) IMP SyntaxN.All i A)"
restrict_space_def: fixes M :: "'c measure"   and \<Omega> :: "'c set" shows "restrict_space M \<Omega> = measure_of (\<Omega> \<inter> space M) (sets.restricted_space M \<Omega>) (emeasure M)"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. \<lbrakk>x = y \<Longrightarrow> P xs (y # ys); \<lbrakk>x \<noteq> y; x < y\<rbrakk> \<Longrightarrow> P xs (y # ys); \<lbrakk>x \<noteq> y; \<not> x < y\<rbrakk> \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
get_lens_plus: fixes x :: "'b \<Longrightarrow> 'd"   and y :: "'c \<Longrightarrow> 'd"   and s :: "'d" shows "get\<^bsub>x +\<^sub>L y\<^esub> s = (get\<^bsub>x\<^esub> s, get\<^bsub>y\<^esub> s)"
LDD_imp_DD: fixes r :: "('a \<times> 'a) set"   and ars :: "('b \<times> 'a \<times> 'b) set"   and \<tau> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>1 :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>2 :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>3 :: "'b \<times> ('a \<times> 'b) list"   and \<tau>1 :: "'b \<times> ('a \<times> 'b) list"   and \<tau>2 :: "'b \<times> ('a \<times> 'b) list"   and \<tau>3 :: "'b \<times> ('a \<times> 'b) list" assumes "trans r"   and "irrefl r"   and "LDD ars r (\<tau>, \<sigma>, \<sigma>1, \<sigma>2, \<sigma>3, \<tau>1, \<tau>2, \<tau>3)" shows "\<exists>\<sigma>' \<tau>'. DD ars r (\<tau>, \<sigma>, \<sigma>', \<tau>')"
lfinite_conv_llength_enat: fixes xs :: "'c llist" shows "lfinite xs = (\<exists>n. llength xs = enat n)"
is_prefix_cons: fixes x :: "'b"   and a :: "'b list"   and b :: "'b list" shows "is_prefix (x # a) b = (\<exists>c. b = x # c \<and> is_prefix a c)"
simps_2: fixes as :: "'a attree list"   and s :: "'a set \<times> 'a set" shows "attack (as \<oplus>\<^sub>\<and>\<^bsup>s\<^esup>) = s"
iarray_cong': fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<And>i. i < n \<Longrightarrow> f i = g i" shows "IArray.of_fun f n = IArray.of_fun g n"
conj: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c (Q \<inter> R),(A \<inter> B)"
map\<^sub>0_def: fixes a :: "'b" shows "FoH.map\<^sub>0 a \<equiv> src\<^sub>D (FoH.map a)"
lem: fixes k :: "nat" assumes "0 < k" shows "gauss_sum k = complex_of_real (real_of_int (int (totient n) * moebius_mu (n div gcd k n)) / real (totient (n div gcd k n)))"
hypnat_le_add_diff_inverse2: fixes n :: "nat star"   and m :: "nat star" assumes "n \<le> m" shows "m - n + n = m"
group_disjoint_sum_def: fixes G :: "('b, 'c) monoid_scheme"   and A :: "'b set"   and B :: "'b set" shows "group_disjoint_sum G A B \<equiv> Group.group G \<and> subgroup A G \<and> subgroup B G"
matchs_induct: fixes P :: "term list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "term list"   and a1 :: "'a list" assumes "P [] []"   and "\<And>t ts u us. (\<And>a. match t u = Some a \<Longrightarrow> P ts us) \<Longrightarrow> P (t # ts) (u # us)"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
sup_right_unit: fixes x :: "'a" shows "- x \<squnion> bot = - x"
ran_m_def: fixes N :: "('d, 'c) fmap" shows "ran_m N = the `# fmlookup N `# dom_m N"
A8: shows "Esigma (5 * 17) = Esigma 5 * Esigma 17"