hcju/isabelleps huggingface dataset:
normalize_unit_factor_eqI: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b"   and "unit_factor a = unit_factor b" shows "a = b"
restrict_to_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "restrict_to P A = (\<lambda>x y. x \<in> A \<and> y \<in> A \<and> P x y)"
IH: shows "vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
addfeqsim: fixes t :: "nat \<Rightarrow> 'b" shows "first t ## t \<approx> t"
refl: fixes x :: "'a" assumes "x \<in> A" shows "bo x x"
cod_simp: fixes f :: "'a \<times> 'a" assumes "S.VV.arr f" shows "S.VV.cod f = S.VxV.cod f"
cc: fixes T :: "'a set" assumes "T \<in> \<F>" shows "closed T \<and> connected T"
label_incr_simp: fixes n :: "label"   and i :: "nat"   and m :: "label"   and j :: "nat" assumes "n \<oplus> i = m \<oplus> i + j" shows "n = m \<oplus> j"
eq_fract_1: fixes b :: "'a"   and d :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<noteq> (0::'a)"   and "d \<noteq> (0::'a)" shows "(Fraction_Field.Fract a b = Fraction_Field.Fract c d) = (a * d = c * b)"
remap_lang2: fixes f :: "'Q \<Rightarrow> 'b"   and t :: "'L tree" assumes "inj_on f Q"   and "t \<in> ta_lang (ta_remap f TA)" shows "t \<in> ta_lang TA"
inj_on_eq_iff: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and x :: "'g"   and y :: "'g" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x = f y) = (x = y)"
assms: shows "a \<notin> {uv, rev_G uv, rev_H uw, rev_H vw}"
atom_eq_iff: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and atom :: "'a \<Rightarrow> atom"   and a :: "'a"   and b :: "'a" assumes "class.at_base permute atom" shows "(atom a = atom b) = (a = b)"
bex_imageD: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x\<in>f ` A. P x" shows "\<exists>x\<in>A. P (f x)"
Sup_upper: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and x :: "'g"   and A :: "'g set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
sync_order_committed_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and J :: "nat \<Rightarrow> ('c, 'b) justifying_execution" shows "sync_order_committed P E J = (\<forall>n. sync_order P (justifying_exec (J n)) |` committed (J n) = inv_imageP (sync_order P E) (action_translation (J n)) |` committed (J n))"
cosh_dist_at_least_1: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2" shows "1 \<le> cosh_dist a b"
permute_plus: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "perm"   and q :: "perm"   and x :: "'a" assumes "class.pt permute" shows "permute (p + q) x = permute p (permute q x)"
add_le_cancel_left: fixes c :: "'k multiset"   and a :: "'k multiset"   and b :: "'k multiset" shows "(c + a \<subseteq># c + b) = (a \<subseteq># b)"
f_def: fixes env_v_pes_err_v_s :: "v sem_env \<times> v \<times> (pat \<times> exp0) list \<times> v \<times> 'a state" shows "f env_v_pes_err_v_s = (case env_v_pes_err_v_s of (env, v, pes, err_v, s) \<Rightarrow> evaluate_match True env s v pes err_v)"
assm: shows "\<exists>t. (t, as) \<in> ts_tuple_rel (set auxlist) \<and> P t"
seqE: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "VxVxV.seq g f" obtains "seq (fst g) (fst f)" and "VxV.seq (snd g) (snd f)"
eq_iff_3: fixes var :: "Syntax.var"   and "term" :: "Syntax.term"   and vara :: "Syntax.var"   and terma :: "Syntax.term" shows "(Lam var term = Lam vara terma) = ([[atom var]]lst. term = [[atom vara]]lst. terma)"
hyp_wrong: fixes j :: "nat"   and t :: "nat" shows "\<phi> (the (s (e_butlast (local.prefixes j (Suc t))))) \<noteq> \<psi> j"
game2_def: shows "game2 = exec_gpv rf_encrypt \<A>1 Map.empty \<bind> (\<lambda>(((p0, p1), \<sigma>), s1). if valid_plain p0 \<and> valid_plain p1 then coin_spmf \<bind> (\<lambda>b. let pb = if b then p0 else p1 in rf_encrypt s1 pb \<bind> (\<lambda>(cipher, s2). exec_gpv rf_encrypt (\<A>2 cipher \<sigma>) s2 \<bind> (\<lambda>(b', s3). return_spmf (b' = b)))) else coin_spmf)"
cr_mini_alg_unique_def: shows "cr_mini_alg_unique \<equiv> \<lambda>x y. x = Rep_mini_alg_unique y"
no_vs_change_snapshot_s_vs_is_valid_bound: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list"   and vs :: "'a set"   and s :: "('a, bool) fmap" assumes "finite PROB"   and "vars_change as vs s = []"   and "no_effectless_act as"   and "sat_precond_as s as"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> problem_plan_bound (snapshot PROB (fmrestrict_set vs s))"
star_1l: fixes x :: "'b \<times> 'c" shows "less_eq_prod (times_prod' x (prod.qstar x)) (prod.qstar x)"
XY: shows "random_variable (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))"
L_pos_def: shows "L_pos = take (length A_pos) L"
Lambda: fixes r :: "dB" assumes "IT r" shows "IT (Abs r)"
parts_impl_inj: fixes M' :: "msg"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and x' :: "tag"   and A' :: "agent"   and B' :: "agent" assumes "M' \<in> payload"   and "implem (Chan x A B M) \<in> parts {implem (Chan x' A' B' M')}" shows "x = x' \<and> A = A' \<and> B = B' \<and> M = M'"
of_int_add: fixes w :: "int"   and z :: "int" shows "of_int (w + z) = of_int w + of_int z"
simps_2: fixes P :: "'e \<Rightarrow> bool"   and g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" shows "dropWhile_raw P g (False, s) = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip (False, s') | Yield a s' \<Rightarrow> Yield a (False, s'))"
exhaust_2: fixes x :: "2" shows "x = 1 \<or> x = 2"
less_fun_def: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" shows "(f \<prec> g) = (f \<lesssim> g \<and> \<not> g \<lesssim> f)"
t_idem: fixes x :: "'a" shows "tx \<cdot> tx = tx"
in_face_cycle_sets: fixes a :: "'b" assumes "a \<in> arcs H" shows "H.face_cycle_set a \<in> H.face_cycle_sets"
eval_oracle'_def: shows "eval_oracle' = (\<lambda>c_o c_a ((id, occ), s') guess. map_spmf (\<lambda>b'. case occ of None \<Rightarrow> ((), (Suc id, if b' then Some id else None), s') | Some j\<^sub>0 \<Rightarrow> ((), (Suc id, Some j\<^sub>0), s')) (eval c_o c_a s' guess))"
assms_2: shows "x \<subseteq>\<^sub>L y"
hassoc_isEmpty_def: fixes ht :: "('a \<times> 'b) list" shows "hassoc_isEmpty ht \<equiv> return (length ht = 0)"
order_less_le_subst1: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and b :: "'d"   and c :: "'d" assumes "a < f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a < f c"
inv2c5: shows "\<forall>p. inpt s p \<in> allInput s \<and> (chosen s = NotAnInput \<longrightarrow> outpt s p = NotAnInput)"
Zp_continuous_is_Zp_closed: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_continuous f" shows "f \<in> carrier (function_ring (carrier Zp) Zp)"
soundnessFAll: fixes u :: "vbl"   and A :: "formula"   and Gamma :: "formula list" assumes "u \<notin> freeVarsFL (FAll Pos A # Gamma)"   and "validS (instanceF u A # Gamma)" shows "validS (FAll Pos A # Gamma)"
empty_interior_affine_hull: fixes S :: "'a set" assumes "finite S"   and "card S \<le> DIM('a)" shows "interior (affine hull S) = {}"
build_ip_partition_no_empty_elems: fixes wi :: "'a wordinterval"   and c :: "parts_connection"   and rs :: "'a simple_rule list" assumes "wi \<in> set (build_ip_partition c rs)" shows "\<not> wordinterval_empty wi"
inject_7: fixes x91 :: "nat"   and x92 :: "'a fm"   and y91 :: "nat"   and y92 :: "'a fm" shows "(ExN x91 x92 = ExN y91 y92) = (x91 = y91 \<and> x92 = y92)"
show_r_zero: fixes G :: "('h, 'i) ring_scheme"   and x :: "'h"   and a :: "'h" assumes "abelian_group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x \<oplus>\<^bsub>G\<^esub> a = x) = (a = \<zero>\<^bsub>G\<^esub>)"
ListUpdateE_updates: fixes xs :: "('a \<times> 'b \<times> 'c) list"   and x :: "'a \<times> 'b \<times> 'c"   and l'' :: "'a \<Rightarrow> 'b" assumes "distinct (map fst xs)"   and "x \<in> set xs" shows "ListUpdateE l'' xs (fst x) = fst (snd x)"
T_rev_def: fixes xs :: "'b list" shows "T_rev xs = length xs + 1"
prems_2: fixes ys :: "'a list"   and x :: "'a" assumes "\<not> List.member ys x"   and "a # xs \<approx> x # ys" shows "thesis"
sc_assoc: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and a :: "'i"   and b :: "'i"   and m :: "'h" assumes "R module M"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "m \<in> carrier M" shows "(a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> (b \<cdot>\<^sub>s\<^bsub>M\<^esub> m)"
beta_cfun: fixes f :: "'a \<Rightarrow> 'b"   and u :: "'a" assumes "cont f" shows "(\<Lambda> x. f x)\<cdot>u = f u"
subsequence_diagonalization_lemma: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"   and r :: "nat \<Rightarrow> 'a" assumes "\<And>i r. \<exists>k. strict_mono k \<and> P i (r \<circ> k)"   and "\<And>i r k1 k2 N. \<lbrakk>P i (r \<circ> k1); \<And>j. N \<le> j \<Longrightarrow> \<exists>j'\<ge>j. k2 j = k1 j'\<rbrakk> \<Longrightarrow> P i (r \<circ> k2)" obtains k :: "nat \<Rightarrow> nat" where "strict_mono k" and "\<And>i. P i (r \<circ> k)"
merge_states_def: fixes x :: "nat"   and y :: "nat"   and t :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "merge_states x y t = (if y < x then merge_states_aux x y t else merge_states_aux y x t)"
insert_closed: fixes g :: "'d"   and gens :: "'d set" assumes "g \<in> gens" shows "\<iota> g \<in> carrier \<F>\<^bsub>gens\<^esub>"
lincomb_list_as_lincomb: fixes vs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "set vs \<subseteq> carrier_vec n" shows "n.lincomb_list c vs = n.lincomb (m.mk_coeff vs c) (set vs)"
subdag_size: fixes t :: "dag"   and s :: "dag" assumes "subdag t s" shows "size s < size t"
p: fixes as :: "('c poly \<times> nat) set"   and p :: "'c poly" assumes "monic_factorization as p" shows "p = (\<Prod>(a, i)\<in>as. a ^ Suc i)"
empty: fixes \<C> :: "t_context" shows "\<C> \<turnstile> [] : [] _> []"
vars_Variable: fixes N :: "nat" shows "vars (Var N) = {N}"
n_in_obs: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and S :: "'b set" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.valid_node sourcenode targetnode valid_edge n"   and "n \<in> S" shows "CFG.obs sourcenode targetnode valid_edge n S = {n}"
if_split_asm: fixes P :: "'d \<Rightarrow> bool"   and Q :: "bool"   and x :: "'d"   and y :: "'d" shows "P (if Q then x else y) = (\<not> (Q \<and> \<not> P x \<or> \<not> Q \<and> \<not> P y))"
compact_space_fip: fixes X :: "'c topology" shows "compact_space X = (\<forall>\<U>. (\<forall>C\<in>\<U>. closedin X C) \<and> (\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> \<Inter> \<F> \<noteq> {}) \<longrightarrow> \<Inter> \<U> \<noteq> {})"
cases: fixes a1 :: "'b LDCFG_node"   and a2 :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and a3 :: "'b LDCFG_node" assumes "CFGExit.path a1 a2 a3" obtains n :: "'b LDCFG_node" where "a1 = n__" and "a2 = []" and "a3 = n__" and "CFGExit.valid_node n__"   | n'' :: "'b LDCFG_node" and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list" and n' :: "'b LDCFG_node" and a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node" and n :: "'b LDCFG_node" where "a1 = n" and "a2 = a # as" and "a3 = n'" and "CFGExit.path n'' as n'" and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a" and "src a = n" and "trg a = n''"
1: shows "H * e * (H * d)\<^sup>\<star> * H * e \<le> H * e"
neg_eq_iff_add_eq_0: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (plus a b = zero)"
homeomorphism_UNIV_imp_open_map: fixes p :: "'a \<Rightarrow> 'b"   and p' :: "'b \<Rightarrow> 'a"   and f' :: "'a set" assumes "homeomorphism UNIV UNIV p p'"   and "open f'" shows "open (p ` f')"
ideD_1: fixes a :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.ide a" shows "S.arr a"
bind_lzero: fixes f :: "'f \<Rightarrow> 'e option" shows "None \<bind> f = None"
no_matching_Goto: fixes t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t'"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t"   and "no_matching_Goto \<gamma> p rs\<^sub>1" shows "thesis"
Pi_pmf_subset: fixes A :: "'b set"   and A' :: "'b set"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A"   and "A' \<subseteq> A" shows "Pi_pmf A' dflt p = map_pmf (\<lambda>f x. if x \<in> A' then f x else dflt) (Pi_pmf A dflt p)"
l: assumes "set (x # map snd txs) \<subseteq> var"   and "set (Var y # t # map fst txs) \<subseteq> trm"   and "set [] \<subseteq> fmla" shows "length (getFrN (x # map snd txs) (Var y # t # map fst txs) [] (Suc (length txs))) = Suc (length txs)"
assms_1: shows "A \<subseteq>\<^sub>\<circ> \<int>\<^sub>\<circ>"
dependent_finite: fixes S :: "'b set" assumes "finite S" shows "local.dependent S = (\<exists>u. (\<exists>v\<in>S. u v \<noteq> (0::'a)) \<and> (\<Sum>v\<in>S. u v *s v) = (0::'b))"
xs: shows "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) [Monitor.progress \<sigma> P \<phi> j..<Monitor.progress \<sigma> P' \<phi> j'] xs"
orthogonal_commute: fixes x :: "'c"   and y :: "'c" shows "orthogonal x y = orthogonal y x"
gso_oc_projection_span_2: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs"   and "i < length fs" shows "gram_schmidt.is_oc_projection n (gram_schmidt_fs.gso n fs i) (LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (gram_schmidt_fs.gso n fs ` {0..<i})) (fs ! i)"
inj_on_fst: shows "inj_on fst (set lc)"
card_le_1_if_pairwise_eq: fixes S :: "'a set" assumes "\<forall>x\<in>S. \<forall>y\<in>S. x = y" shows "card S \<le> 1"
intf: fixes a :: "'a"   and b :: "'a" shows "f integrable_on cbox a b"
simps_2: fixes n :: "nat"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "minit0 n (formula.Eq t1 t2) = MRel (eq_rel n t1 t2)"
simps: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_range \<sigma> = \<sigma> ` subst_domain \<sigma>"
Bloch_general: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex"   and t :: "real"   and r :: "real" assumes "f holomorphic_on S"   and "a \<in> S"   and "\<And>z. z \<in> frontier S \<Longrightarrow> t \<le> dist a z"   and "r \<le> t * cmod (deriv f a) / 12" obtains b :: "complex" where "ball b r \<subseteq> f ` S"
P_app_app: fixes t\<^sub>1 :: "'a"   and u\<^sub>1 :: "'b"   and t\<^sub>2 :: "'a"   and u\<^sub>2 :: "'b" assumes "P t\<^sub>1 u\<^sub>1"   and "P t\<^sub>2 u\<^sub>2" shows "P (app t\<^sub>1 t\<^sub>2) (app u\<^sub>1 u\<^sub>2)"
u_def: shows "u = Matrix.mat 4 1 (\<lambda>(i, j). if i = 0 then v $$ (0, 0) * w $$ (0, 0) else if i = 3 then v $$ (1, 0) * w $$ (1, 0) else if i = 1 then v $$ (0, 0) * w $$ (1, 0) else v $$ (1, 0) * w $$ (0, 0))"
le_cong_l: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> z"
lm004: fixes R :: "('c \<times> 'd) set" assumes "runiq R" shows "R = graph (Domain R) (toFunction R)"
corr: fixes L :: "(nat \<times> int \<times> nat) list" shows "<emp> kruskal_algo L <\<lambda>r. \<up> (set r \<subseteq> set L \<and> minimum_spanning_forest (GD_from_list_\<alpha>_graph L r) (GD_from_list_\<alpha>_graph L L))>\<^sub>t"
alw_InvInfoAtBetaUpright_aux: fixes s :: "('c, 'd) configuration stream" assumes "alw (holds InvTempUpright) s"   and "alw (holds InvIncomingInfoUpright) s"   and "holds InvInfoAtBetaUpright s"   and "alw next s" shows "alw (holds InvInfoAtBetaUpright) s"
block_img: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "image_mset ((`) bij_map) \<B> = \<B>'"
i_expand_i_drop_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" shows "f \<odot> k \<Up> (n * k) = f \<Up> n \<odot> k"
ereal_0_plus: fixes x :: "ereal" shows "ereal 0 + x = x"
piecewise_differentiable_neg: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f piecewise_differentiable_on S" shows "(\<lambda>x. - f x) piecewise_differentiable_on S"
inj_on_to_nat: fixes S :: "'a set" shows "inj_on to_nat S"
assms_2: fixes x :: "'a"   and n :: "nat" assumes "x \<in> S" shows "(f n has_derivative f' n x) (at x within S)"
foundation22: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> X \<triangleq> Y) = (X \<tau> = Y \<tau>)"
succ_of_E_def: fixes E :: "('a \<times> 'b) set" shows "succ_of_E E \<equiv> \<lambda>u. {v. (u, v) \<in> E}"
hom_add: fixes c :: "'d"   and x :: "'d"   and y :: "'d" shows "c * (x + y) = c * x + c * y"
FVar_lemma: fixes v :: "val"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and s2' :: "st"   and statDeclC :: "qtname"   and field :: "Decl.field"   and fn :: "vname"   and a :: "val"   and x2 :: "abopt"   and s2 :: "st"   and statC :: "qtname"   and G :: "prog"   and y :: "class"   and L :: "lenv"   and s1 :: "st" assumes "((v, f), Norm s2') = fvar statDeclC (static field) fn a (x2, s2)"   and "G\<turnstile>statC\<preceq>\<^sub>C statDeclC"   and "table_of (DeclConcepts.fields G statC) (fn, statDeclC) = Some field"   and "wf_prog G"   and "x2 = None \<longrightarrow> G,s2\<turnstile>a\<Colon>\<preceq>Class statC"   and "statDeclC \<noteq> Object"   and "class G statDeclC = Some y"   and "(x2, s2)\<Colon>\<preceq>(G, L)"   and "s1\<le>|s2"   and "inited statDeclC (globs s1)" shows "G,s2'\<turnstile>v\<Colon>\<preceq>type field \<and> s2'\<le>|f\<preceq>type field\<Colon>\<preceq>(G, L)"
is_singletonI: fixes x :: "'d" shows "is_singleton {x}"
subseq_Cons2_iff: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "subseq (x # xs) (y # ys) = (if x = y then subseq xs ys else subseq (x # xs) ys)"
vote_set_as_Union: fixes vs :: "nat \<Rightarrow> process \<Rightarrow> val option"   and Q :: "process set" shows "vote_set vs Q = (\<Union>a\<in>Q \<inter> voters vs. vote_set vs {a})"
Int: fixes S :: "'a set set"   and a :: "'a set"   and b :: "'a set" assumes "generate_topology S a"   and "generate_topology S b" shows "generate_topology S (a \<inter> b)"
msubstltpos: fixes vs :: "'b list"   and c :: "poly"   and a :: "poly"   and b :: "tm"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "(0::'b) < \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"   and "islin (Lt (CNP 0 a b))" shows "Ifm vs (x # bs) (msubstltpos c t a b) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) (Lt (CNP 0 a b))"
l_set_child_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_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" shows "l_set_child_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs \<equiv> l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs \<and> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs"
card_has_rev: fixes S :: "'b set"   and n :: "nat" assumes "finite S"   and "has n S" shows "n \<le> card S"
bij_betw_id_iff: fixes A :: "'d set"   and B :: "'d set" shows "bij_betw id A B = (A = B)"
h4: assumes "\<exists>(a', b', c')\<in>set les. a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)" shows "\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
assms: shows "Fun (Fu f) T \<in> timpl_closure t TI"
assms_2: shows "(i, Store e1 mo2 i) \<in> set (node_deliver_messages xs)"
sum_plus_one_le_prod_plus_one: fixes I :: "'a set"   and p :: "'a \<Rightarrow> 'b" assumes "\<And>i. i \<in> I \<Longrightarrow> (0::'b) \<le> p i" shows "sum p I + (1::'b) \<le> (\<Prod>i\<in>I. p i + (1::'b))"
test_bit_def': fixes a :: "'a word"   and x :: "nat" shows "a !! x = bit (uint a) x"
set_awalk_verts_cas: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "cas u p v" shows "set (awalk_verts u p) = {u} \<union> set (map (tail G) p) \<union> set (map (head G) p)"
IT_implies_termi: fixes t :: "dB" assumes "IT t" shows "termip (\<rightarrow>\<^sub>\<beta>) t"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "compatible_ordering A less_eq less"   and "attractive A less_eq" shows "attractive_ordering A less_eq less"
that: fixes element_ptr :: "(_) element_ptr" assumes "node_ptr = cast element_ptr" shows "thesis"
assms: fixes x :: "'a" assumes "c \<le> norm x" shows "P x"
key_sets_insert: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "key_sets X (insert Y H) = key_sets_msg X Y \<union> key_sets X H"
hyps_2: shows "P'' \<longmapsto> \<tau> \<prec> P'''"
E_nonneg: fixes X :: "real pmf" assumes "\<forall>x\<in>set_pmf X. 0 \<le> x" shows "0 \<le> E X"
prime_not_dvd_fact: fixes k :: "nat"   and n :: "nat" assumes "k < n"   and "prime n" shows "\<not> n dvd fact k"
MVT2: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "a < b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (f has_real_derivative f' x) (at x)" shows "\<exists>z>a. z < b \<and> f b - f a = (b - a) * f' z"
scale_minus_right: fixes a :: "'a"   and x :: "'c" shows "a *b - x = - (a *b x)"
primes_pi_def: shows "\<pi> = prime_sum_upto (\<lambda>p. 1)"