hcju/isabelleps huggingface dataset:
DERIV_neg_dec_left: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real" assumes "(f has_real_derivative l) (at x)"   and "l < 0" shows "\<exists>d>0. \<forall>h>0. h < d \<longrightarrow> f x < f (x - h)"
intros_1: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> analz H"
fps_divide_times_eq: fixes g :: "'b fps"   and f :: "'b fps" assumes "g \<noteq> 0" shows "f * g / g = f"
relabel: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Network c s t"   and "Height_Bounded_Labeling c s t f l"   and "Network.relabel_precond c t f l u" shows "((f, Network.gap_relabel_effect c f l u), f, l) \<in> Network.gap_algo_rel c s t"
mset_map: fixes f :: "'l \<Rightarrow> 'k"   and xs :: "'l list" shows "mset (map f xs) = image_mset f (mset xs)"
fd_def: fixes x :: "'a"   and y :: "'a" shows "domain_semiringl.fd (\<cdot>) (\<lambda>x. ad (ad x)) x y = ad (ad (x \<cdot> y))"
wf_prog_def: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "wf_prog wf_md P \<equiv> wf_syscls P \<and> distinct_fst P \<and> (\<forall>c\<in>set P. wf_cdecl wf_md P c)"
inj_on_m: shows "inj_on m {i. i < n}"
nth_via_drop: fixes n :: "nat"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" assumes "drop n xs = y # ys" shows "xs ! n = y"
diff_diff_inter: fixes B :: "'b set"   and A :: "'b set"   and X :: "'b set" assumes "B \<subseteq> A"   and "B \<subseteq> X" shows "(X - (A - B)) \<inter> A = B"
lexordp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "List.lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
assms_2: shows "f $ 0 * y = (1::'a)"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "D.ide (a \<otimes>\<^sub>D \<I>\<^sub>D)"
sym_conv_converse_eq: fixes r :: "('b \<times> 'b) set" shows "sym r = (r\<inverse> = r)"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "vec_vec_Tensor (x # xs) ys = local.times x ys @ vec_vec_Tensor xs ys"
eclose_sup: fixes a :: "hf"   and b :: "hf" shows "eclose (a \<squnion> b) = eclose a \<squnion> eclose b"
renaming_preserves_satisfiability: fixes atom_ordering :: "('a \<times> 'a) set"   and S :: "'a Literal set set"   and Sel :: "'a set" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.satisfiable S" shows "propositional_atoms.satisfiable (propositional_atoms.rename_formula Sel S)"
remove_child_def: shows "Shadow_DOM.remove_child \<equiv> l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child Shadow_DOM.get_child_nodes Shadow_DOM.set_child_nodes Shadow_DOM.get_owner_document get_disconnected_nodes set_disconnected_nodes"
isomorphic_as_groups_def: fixes \<G> :: "'c set \<times> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<times> 'c"   and \<G>' :: "'d set \<times> ('d \<Rightarrow> 'd \<Rightarrow> 'd) \<times> 'd" shows "(\<G> \<cong>\<^sub>G \<G>') = (let (G, composition, unit) = \<G>; (G', composition', unit') = \<G>' in \<exists>\<eta>. group_isomorphism \<eta> G composition unit G' composition' unit')"
assms_3: shows "ks \<notin> specKeysSecrets P"
learn: shows "learn_fin psi ({0\<^sup>\<infinity>} \<union> U_single) s"
connected_component_subset: fixes S :: "'c set"   and x :: "'c" shows "connected_component_set S x \<subseteq> S"
\<eta>_map_simp: fixes y :: "'b" assumes "B.ide y" shows "transformation_by_components.map (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map \<eta>o y = G (F y)"
vec_of_list_dot_rewrite: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs = length ys" shows "vec_of_list xs \<bullet> vec_of_list ys = sum_list (map2 (*) xs ys)"
inv_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "inv (inv x) = x"
less_oFixD: fixes x :: "ordinal"   and F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "x < oFix F a"   and "mono F"   and "F x = x" shows "x < a"
finite_g: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "g e \<noteq> \<top>"
keys_lower: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "keys (lower p v) = {u \<in> keys p. u \<prec>\<^sub>t v}"
enabledE: fixes A :: "state \<times> state \<Rightarrow> bool"   and s :: "state" assumes "s \<Turnstile> Enabled A" obtains u :: "state" where "A (s, u)"
inorder_baliL: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.inorder (baliL l a r) = Tree2.inorder l @ a # Tree2.inorder r"
rem_effectless_works_5_i: fixes as :: "('c \<times> ('d, 'e) fmap) list" shows "subseq (rem_effectless_act as) as"
simps_7: fixes t :: "'c local.term" shows "Arr \<^bold>\<r>\<^bold>[t\<^bold>] = Arr t"
diff_bernpoly: fixes n :: "nat"   and x :: "real" shows "bernpoly n (x + 1) - bernpoly n x = real n * x ^ (n - 1)"
mem_s_top: fixes H :: "'a set"   and K :: "'a set"   and u :: "'a" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "u \<in> H \<diamondop> K" shows "\<exists>a\<in>H. \<exists>b\<in>K. a \<cdot> b = u"
Update_EmptySet: shows "(\<lambda>d. d) \<in> {L |L. Update L}"
induced_automorphism_half_chamber_system_gallery_map_f: fixes Cs :: "'a set list" assumes "order.greater_eq (f \<turnstile> folding_g.\<C>) (set Cs)"   and "folding_g.gallery Cs" shows "folding_g.gallery (\
 \<Turnstile> Cs)"
unstream_cong: fixes a :: "('a, 'b) Stream"   and b :: "('a, 'c) Stream" assumes "a \<approx> b" shows "unstream\<cdot>a = unstream\<cdot>b"
inject_2: fixes x21 :: "'a attree list"   and x22 :: "'a set \<times> 'a set"   and y21 :: "'a attree list"   and y22 :: "'a set \<times> 'a set" shows "(x21 \<oplus>\<^sub>\<and>\<^bsup>x22\<^esup> = y21 \<oplus>\<^sub>\<and>\<^bsup>y22\<^esup>) = (x21 = y21 \<and> x22 = y22)"
inner_dirsumI: fixes M :: "'a set"   and Ns :: "'a set list" assumes "M = (\<Sum>N\<leftarrow>Ns. N)"   and "add_independentS Ns" shows "M = (\<Oplus>N\<leftarrow>Ns. N)"
simps: fixes i :: "nat" shows "local.F i = (if i = 0 then 1 else if i = 1 then map_poly to_fract G1 else if i = 2 then map_poly to_fract G2 else let G = pseudo_mod (local.F (i - 2)) (local.F (i - 1)) in if local.F (i - 1) = 0 \<or> G = 0 then 0 else Polynomial.smult (inverse (local.b i)) G)"
norm_pres_extensionE: fixes g :: "('b \<times> real) set"   and E :: "'b set"   and p :: "'b \<Rightarrow> real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "g \<in> norm_pres_extensions E p F f" obtains H :: "'b set" and h :: "'b \<Rightarrow> real" where "g = graph H h" and "linearform H h" and "Subspace.subspace H E" and "Subspace.subspace F H" and "graph F f \<subseteq> graph H h" and "\<forall>x\<in>H. h x \<le> p x"
bij_betw_inv: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "\<exists>g. bij_betw g B A"
mk_v_def: fixes I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE"   and \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and sol :: "(real, 'd) vec" shows "mk_v I ODE \<nu> sol = (THE \<omega>. Vagree \<omega> \<nu> (- semBV I ODE) \<and> Vagree \<omega> (mk_xode I ODE sol) (semBV I ODE))"
sumtr: shows "\<forall>n. trace (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n) \<le> trace \<rho>"
cases: fixes A :: "'a set"   and B :: "'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> bool"   and R :: "'d set"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "pred_converter A B C R conv" obtains "\<forall>x\<in>results_converter (\<I>_uniform A UNIV) (\<I>_uniform UNIV R) conv. B x" and "\<forall>out\<in>outs_converter (\<I>_uniform A UNIV) (\<I>_uniform UNIV R) conv. C out"
emeasure_T': fixes a :: "(int, 'n) vec" shows "emeasure lebesgue (T' a) = emeasure lebesgue (T a)"
partition_P: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and yes :: "'a list"   and no :: "'a list" assumes "List.partition P xs = (yes, no)" shows "(\<forall>p\<in>set yes. P p) \<and> (\<forall>p\<in>set no. \<not> P p)"
differentiable_image_in_sets_lebesgue: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue"   and "DIM('a) \<le> DIM('b)"   and "f differentiable_on S" shows "f ` S \<in> sets lebesgue"
disjnt_equiv_class: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "equiv A r" shows "disjnt (r `` {a}) (r `` {b}) = ((a, b) \<notin> r)"
induced_arrowI_2: fixes \<chi>' :: "(bool, bool list) concrete_category.arr \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "D.cones_map (induced_arrow a' \<chi>') (D.mkCone e) = \<chi>'"
unique_connecting_path_unique: fixes v :: "'c"   and xs :: "'c list"   and w :: "'c" assumes "v \<leadsto>xs\<leadsto>\<^bsub>T\<^esub> w" shows "xs = v \<leadsto>\<^bsub>T\<^esub> w"
assert_bound_nolhs_index_valid: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and as :: "('a \<times> 'b atom) set"   and a :: "'a \<times> 'b atom" assumes "AssertBoundNoLhs assert_bound"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "index_valid as s" shows "index_valid (insert a as) (assert_bound a s)"
mult_right: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "(\<lambda>x. f x * h x) \<in> \<Theta>[F](\<lambda>x. g x * h x)"
IRSpec_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals" shows "IRSpec ch mm rs \<equiv> (\<forall>p. RPSpec ch mm rs p) \<and> (\<forall>l. #l \<in> #MemLoc \<longrightarrow> MSpec ch mm rs l)"
nonempty_has_size: fixes S :: "'b multiset" shows "(S \<noteq> {#}) = (0 < size S)"
f_simps_2: fixes x :: "'a"   and i :: "nat" assumes "x \<notin> W i" shows "f i x = 0"
load: fixes \<C> :: "t_context"   and n :: "nat"   and a :: "nat"   and tp_sx :: "(tp \<times> sx) option"   and t :: "t"   and off :: "nat" assumes "memory \<C> = Some n"   and "load_store_t_bounds a (option_projl tp_sx) t" shows "\<C> \<turnstile> [Load t tp_sx a off] : [T_i32] _> [t]"
ex_contid_maximal: fixes S :: "'a set"   and I :: "'a set"   and T :: "'a set set" assumes "S = {1\<^sub>r}"   and "\<zero> \<notin> S"   and "ideal R I"   and "I \<inter> S = {}"   and "T = {J. ideal R J \<and> S \<inter> J = {} \<and> I \<subseteq> J}" shows "\<exists>mx. maximal_set T mx"
adj_not_same: fixes a :: "'a" assumes "(a, a) \<in> parcs H" shows "False"
simps_3: fixes A :: "'b set"   and B :: "'b set"   and e1 :: "'b"   and t1' :: "'b list"   and e2 :: "'b"   and t2' :: "'b list" shows "merge A B (e1 # t1') (e2 # t2') = (if e1 = e2 then e1 # merge A B t1' t2' else if e1 \<in> A \<inter> B then e2 # merge A B (e1 # t1') t2' else e1 # merge A B t1' (e2 # t2'))"
prems_1: fixes i' :: "nat"   and j' :: "nat" assumes "i' < j'"   and "i' < length (x # xs)"   and "j' < length (x # xs)"   and "filter P (x # xs) ! i = (x # xs) ! i'"   and "filter P (x # xs) ! j = (x # xs) ! j'" shows "thesis"
sublist_at'_def: fixes xs :: "'b list"   and ys :: "'b list"   and i :: "nat" shows "sublist_at' xs ys i \<equiv> take (length xs) (drop i ys) = xs"
submatrix_split: fixes A :: "'b mat"   and I :: "nat set"   and J :: "nat set" shows "submatrix A I J = submatrix (submatrix A UNIV J) I UNIV"
ipurge_tr_set: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "set (ipurge_tr I D u xs) \<subseteq> set xs"
\<G>_sat: fixes \<psi> :: "'a ltl"   and j :: "nat" assumes "G \<psi> \<in> \<G>" shows "suffix (l + j) w \<Turnstile> G \<psi>"
L5_Controller: fixes y :: "nat \<Rightarrow> nat list"   and lIn :: "nat \<Rightarrow> bit"   and lOut :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat"   and a :: "bit" assumes "Controller_L y lIn lOut z"   and "lOut t = a"   and "z t = []" shows "lIn t = a"
sees_wf_mdecl: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "wf_mdecl wf_md P D (M, b, Ts, T, m)"
a5: shows "X \<subseteq> E - (stateful \<union> backflows E)"
P: shows "P = (\<lambda>x y. y \<in> getRel S_Idt G `` {x})"
nc_zero: fixes x :: "('a \<times> 'a set) set" shows "x \<inter> NC \<cdot> {} = {}"
assms_2: fixes x :: "complex" assumes "x \<in> U" shows "f x integrable_on cbox a b"
succs_mono: fixes p :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and i :: "instr"   and mxs :: "nat"   and mpc :: "nat"   and rT :: "ty"   and \<tau>' :: "ty list \<times> ty err list"   and \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" assumes "wf_prog p P"   and "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'" shows "set (succs i \<tau> pc) \<subseteq> set (succs i \<tau>' pc)"
assms_2: shows "1 \<le> k \<and> k \<le> p ^ n - 1"
int_pow_neg_int: fixes x :: "int \<times> int"   and n :: "nat" assumes "x \<in> carrier lucas_lehmer_ring" shows "[(- int n)] \<cdot>\<^bsub>lucas_lehmer_ring\<^esub> x = \<ominus>\<^bsub>lucas_lehmer_ring\<^esub> ([n] \<cdot>\<^bsub>lucas_lehmer_ring\<^esub> x)"
path_strategies_eventually_constant: fixes P :: "'a llist" assumes "\<not> lfinite P"   and "lset P \<subseteq> S"   and "valid_path P"   and "path_conforms_with_strategy p P well_ordered_strategy" shows "\<exists>n. \<forall>m\<ge>n. path_strategies P $ n = path_strategies P $ m"
sel_5: fixes x21 :: "('b, 'a) rbt"   and x22 :: "color"   and x23 :: "'b"   and x24 :: "'a"   and x25 :: "('b, 'a) rbt" shows "val (Node x21 x22 x23 x24 x25) = x24"
induct: fixes P :: "'c llist \<Rightarrow> bool"   and a0 :: "'c llist" assumes "P LNil"   and "\<And>x xs. P (LCons x xs)" shows "P a0"
op_inter_ivl_coll_scaleR2_def: fixes X :: "('c \<times> 'd) set"   and Y :: "'c set" shows "op_inter_ivl_coll_scaleR2 X Y = op_inter_fst_ivl_coll_scaleR2 X Y \<bind> (\<lambda>eivls. scaleR2_rep_coll eivls \<bind> (\<lambda>((l, u), ivls). op_ivl_of_ivl_coll ivls \<bind> (\<lambda>ivl. let R = op_inter_fst ivl Y in scaleRe_ivl_coll_spec l u (filter_empty_ivls (mk_coll R)))))"
ideD_3: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "cod a = a"
mHom_mem: fixes N :: "('e, 'b, 'f) Module_scheme"   and f :: "'a \<Rightarrow> 'e"   and m :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M" shows "f m \<in> carrier N"
strategy_attracts_via_lset: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts_via p \<sigma> v0 A W" shows "lset P \<inter> W \<noteq> {}"
SC_siso_imp_SC_ZObis: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.SC_siso tval aval indis c" shows "PL_Indis.SC_ZObis tval aval indis c"
zipWithL_strict_2: fixes f :: "'d \<rightarrow> 'c \<rightarrow> 'b"   and x :: "'d"   and xs :: "'d LList" shows "zipWithL\<cdot>f\<cdot>(LCons\<cdot>x\<cdot>xs)\<cdot>\<bottom> = \<bottom>"
assms_1: shows "\<PP>\<^sub>\<infinity> \<phi> \<G>' w"
emeasure_count_space_finite: fixes X :: "'a set"   and A :: "'a set" assumes "X \<subseteq> A"   and "finite X" shows "emeasure (count_space A) X = of_nat (card X)"
inf_sup_aci_1: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and x :: "'a::type"   and y :: "'a::type" assumes "class.lattice (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type)" shows "(inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (x::'a::type) (y::'a::type) = inf y x"
strict_trans2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "b \<succ> a"   and "c \<succeq> b" shows "c \<succ> a"
HoHV_char: fixes \<mu>\<nu>\<tau> :: "'a \<times> 'a \<times> 'a" shows "HoHV \<mu>\<nu>\<tau> = B.HoVH (DN \<mu>\<nu>\<tau>)"
simps_1: fixes uu :: "'d"   and x :: "'d" shows "the_default uu (Some x) = x"
H: fixes r :: "'a llist" assumes "r \<in> finpref A t"   and "infsuff A r \<subseteq> u" shows "R"
pseudofunctor_def: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi> :: "'c \<times> 'c \<Rightarrow> 'd" shows "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi> \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<and> weak_arrow_of_homs V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) \<and> (composite_functor (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) V\<^sub>C V\<^sub>D (\<lambda>\<mu>\<nu>. H\<^sub>C (fst \<mu>\<nu>) (snd \<mu>\<nu>)) F \<and> composite_functor (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) (subcategory.comp (product_category.comp V\<^sub>D V\<^sub>D) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>D (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>D (snd \<mu>\<nu>) \<and> src\<^sub>D (fst \<mu>\<nu>) = trg\<^sub>D (snd \<mu>\<nu>))) V\<^sub>D (weak_arrow_of_homs.FF V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) (\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<and> natural_isomorphism (subcategory.comp (product_category.comp V\<^sub>C V\<^sub>C) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>C (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>C (snd \<mu>\<nu>) \<and> src\<^sub>C (fst \<mu>\<nu>) = trg\<^sub>C (snd \<mu>\<nu>))) V\<^sub>D ((\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<circ> weak_arrow_of_homs.FF V\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D src\<^sub>D trg\<^sub>D F) (F \<circ> (\<lambda>\<mu>\<nu>. H\<^sub>C (fst \<mu>\<nu>) (snd \<mu>\<nu>))) \<Phi> \<and> pseudofunctor_axioms V\<^sub>C H\<^sub>C \<a>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D F \<Phi>"
drop: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and p :: "'d" assumes "Matching_Ternary.matches \<gamma> m action.Drop p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m action.Drop], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalDeny"
set_lvars_id: fixes s :: "state" shows "set_lvars (locals (snd s)) s = s"
DSourceIsSource: fixes C :: "CSet"   and i :: "AbstrLevelsID"   and S :: "CSet" assumes "C \<in> DSources i S" shows "C \<in> Sources i S"
redT_updTs_Some: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'h"   and xw :: "'f \<times> 'g \<Rightarrow>f nat"   and tas :: "('h, 'f, 'i) new_thread_action list" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_oks ts tas" shows "redT_updTs ts tas t = \<lfloor>xw\<rfloor>"
af_subsequence_U_GF_advice: fixes i :: "nat"   and n :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltln"   and X :: "'b ltln set"   and \<phi> :: "'b ltln" assumes "i \<le> n"   and "suffix n w \<Turnstile>\<^sub>n af \<psi> (w [i \<rightarrow> n])[X]\<^sub>\<nu>"   and "\<And>j. j < i \<Longrightarrow> suffix n w \<Turnstile>\<^sub>n af \<phi> (w [j \<rightarrow> n])[X]\<^sub>\<nu>" shows "suffix (Suc n) w \<Turnstile>\<^sub>n af (\<phi> U\<^sub>n \<psi>) (prefix (Suc n) w)[X]\<^sub>\<nu>"
th: shows "(0 \<le> real_of_int a + b) = (real_of_int (- a) + - b \<le> 0)"
fstE: fixes x :: "'d \<times> 'e"   and a :: "'d"   and b :: "'e"   and P :: "'d \<Rightarrow> bool" assumes "x = (a, b)"   and "P (fst x)" shows "P a"
D: shows "I (fst (snd (cfs ! 0))) = 0"
subst_rename: fixes y :: "name"   and t :: "lam"   and x :: "name"   and u :: "lam" assumes "y \<sharp> t" shows "([(y, x)] \<bullet> t)[y::=u] = t[x::=u]"
Inf_pres_map_dual_var: fixes f :: "'b \<Rightarrow> 'c" shows "Inf_pres f = Sup_pres (\<partial>\<^sub>F f)"
assms: shows "is_pappus2 A B C A' B' C' P Q R"
pure_fresh: fixes a :: "atom"   and x :: "'b" shows "a \<sharp> x"
peval_Cnst_2: fixes xs :: "'a list" shows "peval xs (PExpr1 (PCnst 1)) = \<one>"
sets_count_space: fixes \<Omega> :: "'c set" shows "sets (count_space \<Omega>) = Pow \<Omega>"
WHILEI_invisible_refine: fixes I' :: "'c \<Rightarrow> bool"   and s' :: "'c"   and s :: "'d"   and R :: "('d \<times> 'c) set"   and I :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres" assumes "I' s' \<Longrightarrow> (s, s') \<in> R"   and "\<And>s s'. \<lbrakk>(s, s') \<in> R; I' s'\<rbrakk> \<Longrightarrow> I s"   and "\<And>s s'. \<lbrakk>(s, s') \<in> R; I' s'; I s; b' s'\<rbrakk> \<Longrightarrow> b s"   and "\<And>s s'. \<lbrakk>(s, s') \<in> R; I' s'; I s; b s\<rbrakk> \<Longrightarrow> f s \<le> sup (\<Down> R (ASSUME (b' s') \<bind> (\<lambda>_. f' s'))) (\<Down> R (RETURN s'))" shows "WHILE\<^bsup>I\<^esup> b f s \<le> \<Down> R (WHILE\<^bsup>I'\<^esup> b' f' s')"
relpow_fun_conv: fixes a :: "'b"   and b :: "'b"   and n :: "nat"   and R :: "('b \<times> 'b) set" shows "((a, b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> R))"
2: shows "norm1 f = (\<Sum>i = 0..<n. \<bar>F i\<bar>)"
span_alt: fixes B :: "'c set" shows "vs2.span B = {\<Sum>x | f x \<noteq> (0::'a). f x *b x |f. {x. f x \<noteq> (0::'a)} \<subseteq> B \<and> finite {x. f x \<noteq> (0::'a)}}"
mask_lower_twice: fixes n :: "nat"   and m :: "nat"   and x :: "'b word" assumes "n \<le> m" shows "(x AND NOT (mask n)) AND NOT (mask m) = x AND NOT (mask m)"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "hom.in_hom f a' a"   and "hom'.iso (e \<star> f \<star> src f)" shows "hom.iso f"
span_add_eq: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> ideal S" shows "(x + y \<in> ideal S) = (y \<in> ideal S)"