hcju/isabelleps huggingface dataset:
calculus_axioms: shows "calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> empty_ord.Red_F_\<G>"
simps_4: fixes I :: "interp"   and \<alpha> :: "game"   and \<beta> :: "game" shows "game_sem I (\<alpha> \<union>\<union> \<beta>) = (\<lambda>X. game_sem I \<alpha> X \<union> game_sem I \<beta> X)"
qAbs_alphaAbs_iff: fixes xs :: "'d"   and x :: "'e"   and X :: "('b, 'c, 'd, 'e, 'f) qTerm"   and A' :: "('b, 'c, 'd, 'e, 'f) qAbs" shows "(qAbs xs x X $= A') = (\<exists>x' y X'. A' = qAbs xs x' X' \<and> y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
rec_sg_def: shows "rec_sg = Cn 1 rec_minus [constn 1, Cn 1 rec_minus [constn 1, recf.id 1 0]]"
getChoices_sub_edges_fst: fixes g :: "gState\<^sub>I"   and p :: "pState"   and es :: "edge list" shows "fst ` set (getChoices g p es) \<subseteq> set es"
same1: shows "(prod_list (map ((!) (map_sgas sign)) (a # xa)) = 1) = (poly (qs ! a) x = 0 \<and> prod_list (map ((!) (map_sgas sign)) xa) = 1)"
Const\<^sub>0_zero: shows "Const\<^sub>0 (0::'b) = 0"
order_iff_strict: fixes a :: "nat list"   and b :: "nat list" shows "(a \<le>\<^sub>v b) = (a <\<^sub>v b \<or> a = b)"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
eq_iff: fixes a :: "'d"   and b :: "'d" shows "(a = b) = (a \<le> b \<and> b \<le> a)"
union_eqvt: fixes p :: "perm"   and A :: "'a set"   and B :: "'a set" shows "p \<bullet> (A \<union> B) = p \<bullet> A \<union> p \<bullet> B"
ide_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o cardSuc (cmax |UNIV| natLeq)" shows "ide (mkIde A)"
fds_deriv_fds_ln: fixes l :: "'a"   and f :: "'a fds" shows "fds_deriv (fds_ln l f) = fds_deriv f / f"
sg_condition: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<forall>a b. a \<in> H \<and> b \<in> H \<longrightarrow> a \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> b \<in> H" shows "G \<guillemotright> H "
path_conforms_with_strategy_irrelevant_updates: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy p P \<sigma>"   and "\<And>v. v \<in> lset P \<Longrightarrow> \<sigma> v = \<sigma>' v" shows "path_conforms_with_strategy p P \<sigma>'"
nstep: shows "(\<sigma>1, coll1) \<in> csmall_nstep P \<sigma> n \<and> (\<sigma>', coll2) \<in> csmall P \<sigma>1 \<and> combine coll1 coll2 = coll"
S2_live: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> WF(MClkFwd memCh crCh cst p)_c p \<longrightarrow> (S2 rmhist p \<leadsto> S3 rmhist p)"
G_def: shows "G \<equiv> \<lambda>x. of_real (ln x) * sum_upto (\<lambda>xa. F (x / real xa)) x"
subtopology_topspace: fixes U :: "'c topology" shows "subtopology U (topspace U) = U"
f_at_Suc: shows "\<forall>x y. f (Suc y) x = h y (f y x) x"
Units_inv_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L"
negligible_real_ivlI: fixes b :: "real"   and a :: "real" assumes "b \<le> a" shows "negligible {a..b}"
algebraically_independent_over_rat_def: fixes n :: "nat"   and a :: "nat \<Rightarrow> 'a" shows "algebraically_independent_over_rat n a = (\<forall>p. vars p \<subseteq> {..<n} \<and> (\<forall>m. MPoly_Type.coeff p m \<in> \<rat>) \<and> insertion a p = (0::'a) \<longrightarrow> p = 0)"
lipschitz_incr: fixes s :: "real"   and h :: "real" assumes "s \<in> {t0..t0 + e1'}"   and "h \<in> {0..t0 + e1' - s}" shows "B'-lipschitz_on (cball (solution s) \<bar>r - b\<bar>) (euler_increment f h s)"
leadsTo_induct_pre: fixes F :: "'c program"   and za :: "'c set"   and zb :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "F \<in> za \<longmapsto> zb"   and "P zb"   and "\<And>A B. \<lbrakk>F \<in> A ensures B; F \<in> B \<longmapsto> zb; P B\<rbrakk> \<Longrightarrow> P A"   and "\<And>S. \<forall>A\<in>S. F \<in> A \<longmapsto> zb \<and> P A \<Longrightarrow> P (\<Union> S)" shows "P za"
G_Inf_def: fixes M :: "'a literal multiset set" shows "G_Inf M = {uu_. \<exists>CAs DA AAs As E. uu_ = Infer (CAs @ [DA]) E \<and> gr.ord_resolve M CAs DA AAs As E}"
bvars\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('f, 'e) stateful_strand_step list"   and B :: "('f, 'e) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t (A @ B) = bvars\<^sub>s\<^sub>s\<^sub>t A \<union> bvars\<^sub>s\<^sub>s\<^sub>t B"
IH_2: fixes inputs :: "real vec list"   and j :: "nat" assumes "valid_net' m2"   and "map dim_vec inputs = input_sizes m2"   and "j < output_size' m2" shows "evaluate_net m2 inputs $ j = (\<Sum>is | is \<lhd> input_sizes m2. (\<Prod>k<length inputs. inputs ! k $ (is ! k)) * lookup (tensors_from_net m2 $ j) is)"
Gi: shows "(G_impl, G) \<in> igbg_impl_rel_ext Re node_rel"
closed_closure_eq: fixes a :: "'a set" assumes "a closed" shows "closure a = a"
differentiable_bound_general: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and \<phi> :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> 'a"   and \<phi>' :: "real \<Rightarrow> real" assumes "a < b"   and "continuous_on {a..b} f"   and "continuous_on {a..b} \<phi>"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (\<phi> has_vector_derivative \<phi>' x) (at x)"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> norm (f' x) \<le> \<phi>' x" shows "norm (f b - f a) \<le> \<phi> b - \<phi> a"
length_Keys_to_list: fixes ps :: "(('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "length (pp_pm.extended_ord.punit.Keys_to_list ps) = card (Keys (set ps))"
insertion_one: fixes f :: "nat \<Rightarrow> real" shows "insertion f (Const 1) = 1"
clearjunk_delete: fixes x :: "'d"   and al :: "('d \<times> 'e) list" shows "clearjunk (delete x al) = delete x (clearjunk al)"
compile_load: fixes step1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and step2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and final1 :: "'d \<Rightarrow> bool"   and final2 :: "'e \<Rightarrow> bool"   and load1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and load2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and order :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and match :: "'h \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and compile :: "'f \<Rightarrow> 'g option"   and p1 :: "'f"   and p2 :: "'g"   and s2 :: "'e" assumes "compiler step1 step2 final1 final2 load1 load2 order match compile"   and "compile p1 = Some p2"   and "load2 p2 s2" shows "\<exists>s1 i. load1 p1 s1 \<and> match i s1 s2"
that: fixes ys :: "'a list"   and x' :: "'a"   and zs :: "'a list" assumes "xs = ys @ x' # zs"   and "P x'"   and "\<forall>z\<in>set zs. \<not> P z" shows "thesis"
terminates_impl_no_inf_chain: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<nexists>f. f 0 = ([c], [], s) \<and> (\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow>\<^sup>+ f (Suc i))"
S'_3: shows "S' \<turnstile> \<^bold>\<bottom>"
image_subsetI: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
distrPUTL2: fixes x :: "'c"   and P :: "'c \<Rightarrow> 'd decision option"   and PL :: "('c \<Rightarrow> 'd decision option) list"   and PUT :: "'c \<Rightarrow> 'd decision option" assumes "x \<in> dom P"   and "list2policy PL x = P x"   and "disjDom PL"   and "PUT x = P x" shows "PUTList PUT x PL"
pdevs_apply_nlex_pdevs: fixes x :: "(real \<times> real) pdevs"   and n :: "nat" shows "pdevs_apply (nlex_pdevs x) n = (if Counterclockwise_2D_Arbitrary.lex 0 (pdevs_apply x n) then - pdevs_apply x n else pdevs_apply x n)"
1: shows "one_chain_line_integral F basis (\<Union> (f ` one_chain1)) = one_chain_line_integral F basis one_chain1"
arg_less_sf: fixes x :: "nat" assumes "1 < x" shows "x < sf x"
simI: fixes Q :: "ccs"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "\<And>\<alpha> Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>[Rel] Q"
pderiv_eq_0_iff: fixes p :: "'a poly" shows "(pderiv p = 0) = (degree p = 0)"
reconstruct_is_original_poly: fixes p :: "complex poly" shows "reconstruct_poly (lead_coeff p) (complex_roots_complex p) = p"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<CC>\<lparr>Cod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
q_def: shows "q = p div [:- c, 1::'a:]"
measurable_sets_borel: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'e measure"   and A :: "'e set" assumes "f \<in> borel \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "f -` A \<in> sets borel"
const_le_unat: fixes b :: "nat"   and a :: "'a word" assumes "b < 2 ^ LENGTH('a)"   and "word_of_nat b \<le> a" shows "b \<le> unat a"
brD: fixes c :: "'a"   and a :: "'c"   and \<alpha> :: "'a \<Rightarrow> 'c"   and I :: "'a \<Rightarrow> bool" assumes "(c, a) \<in> br \<alpha> I" shows "a = \<alpha> c \<and> I c"
freshFreeTransition: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> \<alpha> \<prec> P'"   and "c \<sharp> P"   and "c \<sharp> \<alpha>" shows "c \<sharp> P'"
t': shows "Ct STR ''Pure.all'' ((T \<rightarrow> propT) \<rightarrow> propT) $ Abs T t' = t"
pull_def: fixes x :: "wall"   and y :: "wall" shows "pull x y \<equiv> pull_posneg x y \<or> pull_negpos x y"
o_eq_dest_lhs: fixes a :: "'c \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'c"   and c :: "'a \<Rightarrow> 'b"   and v :: "'a" assumes "a \<circ> b = c" shows "a (b v) = c v"
prob_dom_def: fixes prob :: "(('b, 'c) fmap \<times> ('b, 'd) fmap) set" shows "prob_dom prob \<equiv> \<Union>(s1, s2)\<in>prob. action_dom s1 s2"
plus_vec_def: fixes v\<^sub>1 :: "'a Matrix.vec"   and v\<^sub>2 :: "'a Matrix.vec" shows "v\<^sub>1 + v\<^sub>2 \<equiv> Matrix.vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $ i + v\<^sub>2 $ i)"
inj_on_the_inv_into: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "inj_on f A" shows "inj_on (the_inv_into A f) (f ` A)"
publicKey_image_eq: fixes x :: "agent"   and AA :: "agent set" shows "(pubK x \<in> pubK ` AA) = (x \<in> AA)"
SN_on_induct': fixes R :: "('b \<times> 'b) set"   and s :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "SN_on R {s}"   and "\<And>t. (\<And>u. (t, u) \<in> R \<Longrightarrow> P u) \<Longrightarrow> P t" shows "P s"
simple_product_run: fixes \<delta>\<^sub>1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'b"   and \<delta>\<^sub>2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>1 :: "'b"   and q\<^sub>2 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "run \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w i = (run \<delta>\<^sub>1 q\<^sub>1 w i, run \<delta>\<^sub>2 q\<^sub>2 w i)"
block_complement_elem_iff: fixes ps :: "'a set"   and bl :: "'a set" assumes "ps \<subseteq> \<V>" shows "(ps \<subseteq> bl\<^sup>c) = (\<forall>x\<in>ps. x \<notin> bl)"
simps_7: fixes \<sigma> :: "'a basic_type" shows "UnlimitedNatural \<squnion> \<sigma> = (if UnlimitedNatural \<le> \<sigma> then \<sigma> else if \<sigma> \<le> UnlimitedNatural then UnlimitedNatural else OclAny)"
eucl_ivals_4: fixes a :: "'c" shows "{a..} \<in> sets borel"
det3_def': fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" shows "det3 p q r = fst p * snd q + snd p * fst r + fst q * snd r - snd q * fst r - snd p * fst q - fst p * snd r"
vsubstEnv_def: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" shows "vsubstEnv ys y1 y2 \<equiv> substEnv ys (Var ys y1) y2"
OFCLASS_bezout_ring_imp_class_bezout_ring: assumes "OFCLASS('b, bezout_ring_class)" shows "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus"
decompose: fixes k :: "nat"   and d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat"   and xs :: "'b list" assumes "k \<in> set (trace d i xs)" shows "\<exists>pref mids suf. xs = pref @ concat mids @ suf \<and> deltas d pref i = k \<and> (\<forall>n\<in>set (butlast (trace d i pref)). n \<noteq> k) \<and> (\<forall>mid\<in>set mids. deltas d mid k = k \<and> (\<forall>n\<in>set (butlast (trace d k mid)). n \<noteq> k)) \<and> (\<forall>n\<in>set (butlast (trace d k suf)). n \<noteq> k)"
simps_1: fixes t :: "varType"   and val :: "integer" shows "variable_inv (Var t val) = (varType_inv t \<and> val \<in> {min_var_value..max_var_value})"
in_keys_iff: fixes x :: "'b"   and s :: "'b \<Rightarrow>\<^sub>0 'c" shows "(x \<in> keys s) = (lookup s x \<noteq> (0::'c))"
less_add_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less a (plus a one)"
assms_1: shows "nonneg_cexpr V (\<Gamma>(x := t1)) e"
simps_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "CMP_BRANCH GT x y z = z"
rev_predicate1D: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "P x"   and "P \<le> Q" shows "Q x"
in_obs_intra_valid_1: fixes n' :: "'node"   and n :: "'node"   and S :: "'node set" assumes "n' \<in> obs_intra n S" shows "valid_node n"
UNIV_tau_i_is_empty: fixes \<tau> :: "'b set \<Rightarrow> 'b set" assumes "finite UNIV"   and "mono \<tau>"   and "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subset> (\<tau> ^^ i) UNIV" shows "(\<tau> ^^ card UNIV) UNIV = {}"
poly_lift_pCons: fixes a :: "'a"   and p :: "'a poly" shows "poly_lift (pCons a p) = pCons [:a:] (poly_lift p)"
plus_base_def: fixes A :: "'b tensor"   and B :: "'b tensor" shows "plus_base A B = tensor_from_vec (dims A) (vec_plus (vec A) (vec B))"
map_is_rbt: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and t :: "('c, 'e) rbt" shows "ord.is_rbt less (RBT_Impl.map f t) = ord.is_rbt less t"
flow_solves_ode: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> X" shows "(flow t0 x0 solves_ode f) (existence_ivl t0 x0) X"
abs_ge_zero: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq zero (abs a)"
fls_X_power_nth: fixes n :: "nat"   and k :: "int" shows "fls_nth (fls_X ^ n) k = (if k = int n then 1::'a else (0::'a))"
uniqImpliesNotLastMemButlast: fixes l :: "'b list" assumes "uniq l" shows "last l \<notin> set (butlast l)"
reduced_GB_subset_monic_dgrad_p_set: shows "reduced_GB F \<subseteq> monic ` F"
eq: shows "Y = LCons x ` ltl ` (Y \<inter> {xs. \<not> lnull xs})"
path_is_rtrancl: fixes E :: "('a \<times> 'a) set"   and u :: "'a"   and l :: "'a list"   and v :: "'a" assumes "path E u l v" shows "(u, v) \<in> E\<^sup>*"
MGF_asm: fixes A :: "'a set"   and t :: "'c" assumes "\<forall>pn\<in>U. P A {mgf (f pn)}" shows "P A {mgf t}"
arr_dom: fixes f :: "'a" assumes "R.arr f" shows "R.arr (R.dom f)"
div_div_eq_right: 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 c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b"   and "dvd.dvd times b a" shows "divide a (divide b c) = times (divide a b) c"
tensor_mat_id: fixes dims :: "nat list"   and vars :: "nat set" shows "tensor_mat dims vars (1\<^sub>m (partial_state.d1 dims vars)) (1\<^sub>m (partial_state.d2 dims vars)) = 1\<^sub>m (state_sig.d dims)"
sminus_zero: fixes v :: "'t" shows "v \<ominus> (0::'a) = v"
rep_eq: shows "zhmsetmset 1 = {#0#}\<^sub>z"
final_threadE: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" assumes "final_thread s t" obtains x :: "'x" where "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "final x" and "wset s t = None"
prv_imp_eqvEL: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "prv (Syntax_with_Connectives.eqv cnj imp \<phi>1 \<phi>2)" shows "prv (imp \<phi>1 \<phi>2)"
hrcis_mult: fixes r1 :: "real star"   and a :: "real star"   and r2 :: "real star"   and b :: "real star" shows "hrcis r1 a * hrcis r2 b = hrcis (r1 * r2) (a + b)"
IS: fixes xs :: "fin_string"   and k :: "nat"   and x :: "char" assumes "P xs k" shows "P (Abs_fin_string (x # Rep_fin_string xs)) (Suc k)"
in_hhom_def: fixes \<mu> :: "'d"   and a :: "'d"   and b :: "'d" shows "D.in_hhom \<mu> a b \<equiv> D.arr \<mu> \<and> src\<^sub>D \<mu> = a \<and> trg\<^sub>D \<mu> = b"
Schottky_lemma2: fixes x :: "real" assumes "0 \<le> x" obtains n :: "nat" where "0 < n" and "\<bar>x - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1 / 2"
case_prod_transfer: fixes A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'f \<Rightarrow> bool" shows "((A ===> B ===> C) ===> rel_prod A B ===> C) case_prod case_prod"
mult_smult_left: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "Polynomial.smult a p * q = Polynomial.smult a (p * q)"
simps_7: fixes t :: "('a, 'b) trm" shows "SIGT (Differential t) = SIGT t"
f_def: shows "f = (\<lambda>n t. except t (- (X \<union> {x. elem_index x < n})))"
impE: fixes cid :: "'bid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'BB) \<Rightarrow> nat \<Rightarrow> bool" assumes "pb.eval cid t t' n (\<gamma> \<longrightarrow>\<^sup>b \<gamma>')" shows "pb.eval cid t t' n \<gamma> \<longrightarrow> pb.eval cid t t' n \<gamma>'"
has_vderiv_on_union: fixes f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and s :: "real set"   and t :: "real set" assumes "(f has_vderiv_on g) (s \<union> closure s \<inter> closure t)"   and "(f has_vderiv_on g) (t \<union> closure s \<inter> closure t)" shows "(f has_vderiv_on g) (s \<union> t)"
singular_subdivision_power_diff: fixes n :: "nat"   and p :: "nat"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "(singular_subdivision p ^^ n) (a - b) = (singular_subdivision p ^^ n) a - (singular_subdivision p ^^ n) b"
rconsNlist_inv: fixes n :: "nat"   and t :: "('a, 'b) FingerTreeStruc"   and xs :: "('a, 'b) Node list" assumes "FingerTreeStruc.is_leveln_ftree n t"   and "FingerTreeStruc.is_measured_ftree t"   and "\<forall>x\<in>set xs. FingerTreeStruc.is_leveln_node n x \<and> FingerTreeStruc.is_measured_node x" shows "FingerTreeStruc.is_leveln_ftree n (FingerTreeStruc.rconsNlist t xs) \<and> FingerTreeStruc.is_measured_ftree (FingerTreeStruc.rconsNlist t xs)"
set_mode_locs_impl: shows "set_mode_locs = set_mode_locs"
not_numeral_less_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (numeral.numeral one plus n) one"
mean_divisor_count_asymptotics'': shows "(\<lambda>m. real (card {(n, d). n \<in> {0<..m} \<and> d dvd n}) / real (card {0<..m})) \<sim>[sequentially] (\<lambda>x. ln (real x))"
snapshot_pair: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" shows "snapshot PROB s = {(p, e). (p, e) \<in> PROB \<and> agree p s \<and> agree e s}"
fully_faithful_functor_axioms: shows "fully_faithful_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L"
shift2_invD_1: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "shift2_inv qs" shows "valid_decomp X qs"
inverse_power_tendsto_zero: shows "(\<lambda>x. inverse (of_nat x ^ Suc 0)) \<longlonglongrightarrow> (0::'b)"
r_cancel_one: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier R"   and "a \<in> carrier R" shows "(a \<oplus> x = x) = (a = \<zero>)"
lookup_default_update_neq: fixes k :: "'b"   and k' :: "'b"   and d :: "'c"   and v :: "'c"   and m :: "('b, 'c) mapping" assumes "k \<noteq> k'" shows "lookup_default d (update k v m) k' = lookup_default d m k'"
cases: fixes x :: "('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" obtains bs :: "('t \<Rightarrow>\<^sub>0 'b) list" and ss :: "'t list" and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" where "x = (bs, ss, ps)"
comm_mult: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "a \<in> R"   and "b \<in> R" shows "multiplication a b = multiplication b a"
rewrite_MultiportPorts_def: fixes m :: "'c common_primitive match_expr" shows "rewrite_MultiportPorts m \<equiv> normalize_match (replace_primitive_matchexpr (is_MultiportPorts, multiportports_sel) rewrite_MultiportPorts_one m)"
bind_refine: fixes M :: "'e nres"   and R' :: "('e \<times> 'f) set"   and M' :: "'f nres"   and f :: "'e \<Rightarrow> 'g nres"   and R :: "('g \<times> 'h) set"   and f' :: "'f \<Rightarrow> 'h nres" assumes "M \<le> \<Down> R' M'"   and "\<And>x x'. (x, x') \<in> R' \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "M \<bind> f \<le> \<Down> R (M' \<bind> f')"
member: shows "bl2 \<in># remove1_mset bl \<B>"
assms_4: shows "h \<circ> f = g \<circ> h"
Nil_lenlex_iff1: fixes ns :: "'a list"   and r :: "('a \<times> 'a) set" shows "(([], ns) \<in> lenlex r) = (ns \<noteq> [])"
exhaust: fixes y :: "A" obtains "y = \<bottom>"   | "y = A"