hcju/isabelleps huggingface dataset:
cmp_simps_3: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "S.trg (S.cmp\<^sub>U\<^sub>P (f, g)) = S.trg (S.UP f)"
cproper_interval_natural_def: shows "cproper_interval = proper_interval"
poly_induct2: fixes p :: "nat \<Rightarrow> 'a"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "p \<in> carrier P"   and "\<And>p. \<lbrakk>p \<in> carrier P; deg R p = 0\<rbrakk> \<Longrightarrow> Q p"   and "\<And>p. \<lbrakk>0 < deg R p; p \<in> carrier P; Q (Cring_Poly.truncate R p)\<rbrakk> \<Longrightarrow> Q p" shows "Q p"
real_of_pos_eq_transfer: fixes x :: "real\<^sup>+"   and y :: "real\<^sup>+" shows "(x = y) = (real_of_pos x = real_of_pos y)"
cobounded1: fixes a :: "'c"   and b :: "'c" shows "a \<sqinter> b \<le> a"
fls_conv_base_factor_to_fps_shift_subdegree: fixes f :: "'b fls" shows "f = fls_shift (- fls_subdegree f) (fls_regpart_as_fls (fls_base_factor f))"
proj_specific_secrets_anti_mono: fixes l :: "'b"   and t :: "('fun, 'var) Term.term"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and Sec' :: "('fun, 'var) Term.term set" assumes "proj_specific l t \<A> Sec"   and "Sec' \<subseteq> Sec" shows "proj_specific l t \<A> Sec'"
all_n_lists_def: fixes P :: "'a list \<Rightarrow> bool"   and n :: "nat" shows "all_n_lists P n = (\<forall>xs\<in>set (List.n_lists n enum). P xs)"
select_sets1: fixes M :: "'c measure measure"   and N :: "'c measure" assumes "sets M = sets (subprob_algebra N)" shows "sets M = sets (subprob_algebra (select_sets M))"
induct: fixes P :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat \<Rightarrow> real"   and a1 :: "nat" assumes "\<And>t. P t 0"   and "\<And>t. P t 1"   and "\<And>n t. \<lbrakk>2 \<le> n; P t (nat \<lfloor>real n / 2\<rfloor>); P t (nat \<lceil>real n / 2\<rceil>)\<rbrakk> \<Longrightarrow> P t n" shows "P a0 a1"
adv_P1_def: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'i spmf"   and S1 :: "'g \<Rightarrow> 'j \<Rightarrow> 'i spmf"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'k spmf"   and S2 :: "'h \<Rightarrow> 'l \<Rightarrow> 'k spmf"   and funct :: "'g \<Rightarrow> 'h \<Rightarrow> ('j \<times> 'l) spmf"   and m1 :: "'g"   and m2 :: "'h"   and D :: "'i \<Rightarrow> bool spmf" assumes "sim_det_def R1 S1 R2 S2 funct" shows "sim_det_def.adv_P1 R1 S1 funct m1 m2 D \<equiv> \<bar>spmf (R1 m1 m2 \<bind> D) True - spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S1 m1 o1 \<bind> D)) True\<bar>"
Max_atm_of_set_mset_commute: fixes C :: "'b literal multiset" assumes "C \<noteq> {#}" shows "Max (atm_of ` set_mset C) = atm_of (Max_mset C)"
ldistinct_lmap: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "ldistinct (lmap f xs) = (ldistinct xs \<and> inj_on f (lset xs))"
1: shows "\<guillemotleft>\<r>[S.cod \<mu>] \<cdot>\<^sub>S R \<mu> : R (S.dom \<mu>) \<Rightarrow>\<^sub>S S.cod \<mu>\<guillemotright>"
square_free_imp_separable: fixes f :: "'a poly" assumes "square_free f" shows "separable f"
sgnx_times: fixes F :: "real filter"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "F \<noteq> bot"   and "f sgnx_able F"   and "g sgnx_able F" shows "sgnx (\<lambda>x. f x * g x) F = sgnx f F * sgnx g F"
nn_integral_measurable_subprob_algebra2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and M :: "'c measure"   and N :: "'d measure"   and L :: "'c \<Rightarrow> 'd measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"   and "L \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "(\<lambda>x. integral\<^sup>N (L x) (f x)) \<in> borel_measurable M"
cSup_upper: fixes x :: "'d multiset"   and X :: "'d multiset set" assumes "x \<in> X"   and "subset_mset.bdd_above X" shows "x \<subseteq># Sup X"
Par_def: fixes x2 :: "ccs"   and x1 :: "ccs" shows "x2 \<parallel> x1 \<equiv> Abs_ccs (Par_Rep (Rep_ccs x2) (Rep_ccs x1))"
bi_total_alt_def: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "bi_total A = (left_total A \<and> right_total A)"
preserves_reflects_arr: fixes f :: "'a" shows "A.arr (A.map f) = A.arr f"
supp_Inr: fixes x :: "'b" shows "supp (Inr x) = supp x"
singleton_matrix_le: fixes j :: "nat"   and i :: "nat"   and a :: "'b"   and b :: "'b" shows "(singleton_matrix j i a \<le> singleton_matrix j i b) = (a \<le> b)"
type_less_eq_x_Collection: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<le> Collection \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = Set \<rho>__" and "\<rho>__ \<le> \<sigma>"   | \<rho> :: "'a OCL_Types.type" where "\<tau> = OrderedSet \<rho>'__" and "\<rho>'__ \<le> \<sigma>"   | \<rho> :: "'a OCL_Types.type" where "\<tau> = Bag \<rho>''__" and "\<rho>''__ \<le> \<sigma>"   | \<rho> :: "'a OCL_Types.type" where "\<tau> = Sequence \<rho>'''__" and "\<rho>'''__ \<le> \<sigma>"   | \<rho> :: "'a OCL_Types.type" where "\<tau> = Collection \<rho>" and "\<rho> \<le> \<sigma>"
smallest_accepting_rank_properties_6: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "mojmir_def.smallest_accepting_rank \<Sigma> \<delta> q\<^sub>0 w F = Some i" shows "i < semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0"
rel_fun_map_fun1: fixes h :: "'b \<Rightarrow> 'a"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "rel_fun (BNF_Def.Grp UNIV h)\<inverse>\<inverse> A f g" shows "rel_fun (=) A (map_fun h id f) g"
assoc: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Type_Semigroups.semigroup f" shows "f (f a b) c = f a (f b c)"
assms: shows "u \<turnstile>\<^bsub>v,n\<^esub> And M1 M2"
lsorted_ldropn: fixes xs :: "'a llist"   and n :: "nat" assumes "lsorted xs" shows "lsorted (ldropn n xs)"
strict_sorted_equal: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_sorted xs"   and "strict_sorted ys"   and "set ys = set xs" shows "ys = xs"
rev_sub_add: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "h \<in> carrier P" shows "rev_compose R g (f \<oplus>\<^bsub>P\<^esub> h) = rev_compose R g f \<oplus>\<^bsub>P\<^esub> rev_compose R g h"
conv_abscissa_perzeta': fixes q :: "real" assumes "q \<notin> \<int>" shows "conv_abscissa (fds_perzeta q) \<le> 0"
iso_trisplit_def: shows "iso_trisplit = (\<lambda>(((s11, s12), s13), (s21, s22), s23). (((s11, s21), s12, s22), s13, s23), \<lambda>(((s11, s21), s12, s22), s13, s23). (((s11, s12), s13), (s21, s22), s23))"
card_of_ordLeq: fixes A :: "'a set"   and B :: "'b set" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = ordLeq3 (card_of A) (card_of B)"
sams_sym: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "SAMS A B C D E F" shows "SAMS D E F A B C"
assms_7: shows "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"
le_mod_add_eq_imp_add_mod_le: fixes a :: "nat"   and b :: "nat"   and k :: "nat"   and m :: "nat" assumes "a \<le> b"   and "(a + k) mod m = b mod m" shows "a + k mod m \<le> b"
sappend_snil_id_right: fixes xs :: "[:'r:]" shows "xs :@ [::] = xs"
basis_Rm: shows "is_basis (set_of_vector cart_basis_Rm)"
assms_1: shows "chain_reparam_chain' one_chain1 one_chain2"
R: fixes x :: "'a"   and e :: "real" assumes "x \<in> S"   and "0 < e" shows "\<exists>d U. 0 < d \<and> d \<le> e \<and> S \<inter> ball x d \<subseteq> U \<and> U \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue U < e * Sigma_Algebra.measure lebesgue (ball x d)"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>\<delta>. gauge {a..b} \<delta> \<and> (\<forall>D. fine \<delta> (a, b) D \<longrightarrow> \<bar>rsum D f - k\<bar> < e)"
altc_sample_state_def: fixes altc_sample :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and p :: "'c"   and f :: "'d \<Rightarrow> ('a, 'b) stateT" shows "altc_sample_state altc_sample p f = StateT (\<lambda>s. altc_sample p (\<lambda>x. run_state (f x) s))"
set_mset_def: fixes M :: "'b multiset" shows "set_mset M = {x. 0 < count M x}"
d2: shows "d2 = ((e2 + q - x) mod q * b + (q - r)) mod q"
intros: fixes xs :: "'d list"   and gs :: "'d list"   and gs' :: "'d list" assumes "xs \<noteq> []" shows "cancel1 (gs @ xs @ xs @ gs') (gs @ xs @ gs')"
lookup_HSem_other: fixes y :: "'var"   and h :: "('var \<times> 'exp) list"   and \<rho> :: "'var \<Rightarrow> 'value" assumes "y \<notin> domA h" shows "(\<lbrace> h \<rbrace>\<rho>) y = \<rho> y"
intros_1: fixes At :: "atom"   and T :: "atom set"   and H :: "chmsg set" assumes "At \<in> T" shows "At \<in> extr T H"
ind_in_set_def: fixes A :: "'a set"   and x :: "'a" shows "ind_in_set A x = card {i \<in> A. i < x}"
simps: fixes fs :: "face list"   and n :: "nat"   and f :: "face list list"   and h :: "nat list" shows "vertices (Graph fs n f h) = [0..<n]"
lattice_of_swap: fixes fs :: "'a vec list"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and gs :: "'a vec list" assumes "set fs \<subseteq> carrier_vec n"   and "i < length fs"   and "j < length fs"   and "i \<noteq> j"   and "gs = fs[i := fs ! j, j := fs ! i]" shows "vec_module.lattice_of n gs = vec_module.lattice_of n fs"
is_local_def: fixes V :: "'a set"   and \<phi> :: "('a set \<times> 'b) set \<Rightarrow> ('a set \<times> 'b) set" shows "is_local V \<phi> \<equiv> local_ring_morphism \<phi> stfx.carrier_stalk stfx.add_stalk stfx.mult_stalk (stfx.zero_stalk V) (stfx.one_stalk V) stx.carrier_stalk stx.add_stalk stx.mult_stalk (stx.zero_stalk (identity S \<^sup>\<inverse> S V)) (stx.one_stalk (identity S \<^sup>\<inverse> S V))"
assms_2: shows "P \<turnstile> \<langle>es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
conv_add: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y \<rhd> (1::'a) = (x \<rhd> (1::'a)) \<squnion> (y \<rhd> (1::'a))"
arr_Rel_converse_Rel_eq_iff: fixes \<alpha> :: "V"   and F :: "V"   and G :: "V" assumes "arr_Rel \<alpha> F"   and "arr_Rel \<alpha> G" shows "(F\<inverse>\<^sub>R\<^sub>e\<^sub>l = G\<inverse>\<^sub>R\<^sub>e\<^sub>l) = (F = G)"
mset_subset_add_mset: fixes x :: "'b"   and N :: "'b multiset"   and M :: "'b multiset" shows "(add_mset x N \<subset># add_mset x M) = (N \<subset># M)"
foldl_mult_start: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'a"   and as :: "'b list"   and x :: "'a" shows "foldl (\<lambda>r x. r * f x) v as * x = foldl (\<lambda>r x. r * f x) (v * x) as"
numeral_less_eint_iff: fixes m :: "num"   and n :: "int" shows "(eint (numeral m) < eint n) = (numeral m < n)"
abs_exp_cancel: fixes x :: "real" shows "\<bar>exp x\<bar> = exp x"
f_expand_one: fixes k :: "nat"   and a :: "'d message_af" assumes "0 < k" shows "[a] \<odot> k = a # NoMsg\<^bsup>k - Suc 0\<^esup>"
spanning_forest_spanning: fixes f :: "'a"   and g :: "'a" assumes "spanning_forest f g" shows "spanning (- - g) f"
leg0_is_map: shows "is_left_adjoint r\<^sub>0"
comp_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y ; z = x ; (y ; z)"
preprm_singleton_involution: fixes s :: "'b \<times> 'b" shows "preprm_compose [s] [s] =p preprm_id"
simps_18: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f3 :: "'c \<Rightarrow> 'c \<Rightarrow> instr list \<Rightarrow> 'b"   and f4 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and x21 :: "bool"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat" shows "(case Read\<^sub>s\<^sub>b x21 x22 x23 x24 of Write\<^sub>s\<^sub>b x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Read\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f2 x xa xb xc | Prog\<^sub>s\<^sub>b x xa xb \<Rightarrow> f3 x xa xb | Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> f4 x xa xb xc) = f2 x21 x22 x23 x24"
deadlock1_imp_\<tau>s_deadlock2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "r1.deadlock s1" shows "\<exists>s2'. \<tau>mRed2 s2 s2' \<and> r2.deadlock s2' \<and> s1 \<approx>m s2'"
preserves_comp: fixes g :: "'c"   and f :: "'c" assumes "Cop.seq g f" shows "F (g \<cdot>\<^sup>o\<^sup>p f) = F g \<cdot>\<^sub>S F f"
preserves_seq: fixes g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.seq g f" shows "S.seq (S.map g) (S.map f)"
matchCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "[a\<frown>b]P \<longmapsto> Rs" obtains P :: "pi" and ba :: "name" where "a = ba \<and> b = ba \<and> Pb__ = P" and "P \<longmapsto> Rs"
keys_dom_lookup: fixes m :: "('d, 'e) mapping" shows "Mapping.keys m = dom (Mapping.lookup m)"
assms_1: shows "\<forall>i\<le>n. \<forall>j\<le>n. M i j \<noteq> \<infinity> \<longrightarrow> get_const (M i j) \<in> \<int>"
set_incl_pred: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool" shows "(A \<le> B) = (\<forall>a. A a \<longrightarrow> B a)"
expansion_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'a set"   and c :: "'a" assumes "expansion_on A f"   and "B \<subseteq> A"   and "C \<subseteq> A"   and "c \<in> f B"   and "c \<in> f C" shows "c \<in> f (B \<union> C)"
ys: shows "ys \<in> {xs. (distinct \<circ> map fst) xs}"
assm: shows "\<forall>x xa. xa \<noteq> x \<longrightarrow> (\<nexists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> ((0 < \<parallel>ext v\<parallel> \<and> len v ts xa = ext v \<and> restrict v (res ts) xa = lan v \<and> |lan v| = 1) \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1) \<and> True)) \<and> True))"
collapse: fixes prod :: "'d \<times> 'e" shows "(fst prod, snd prod) = prod"
transfer: shows "(\<lambda>f g. Abs_nd_fun (Rep_nd_fun f \<circ>\<^sub>K Rep_nd_fun g)) = (\<cdot>)"
pderiv_i_def: fixes ops :: "'b arith_ops_record"   and xs :: "'b list" shows "pderiv_i ops xs = pderiv_main_i ops (arith_ops_record.one ops) (tl xs)"
red_union: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.red (F \<union> G) p q = (punit.red F p q \<or> punit.red G p q)"
igSwapBinpIPresIGWlsBinpSTR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSwapBinpIPresIGWlsBinpSTR MOD \<equiv> \<forall>zs z1 z2 delta binp. igWlsBinp MOD delta (igSwapBinp MOD zs z1 z2 binp) = igWlsBinp MOD delta binp"
min_dnf_rep_abs: fixes \<phi> :: "'b ltln" shows "min_dnf (rep_ltln\<^sub>P (abs_ltln\<^sub>P \<phi>)) = min_dnf \<phi>"
power_eq_iff_eq_base: 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 n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "0 < n"   and "less_eq zero a"   and "less_eq zero b" shows "(power.power one times a n = power.power one times b n) = (a = b)"
prod_list_m: fixes fs :: "int poly list" shows "p.prod_list_m fs = p.Mp (prod_list fs)"
CondFalse: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c\<^sub>2 :: "('a, 'b, 'c) com"   and c\<^sub>1 :: "('a, 'b, 'c) com" assumes "s \<notin> b"   and "\<Gamma>\<turnstile>c\<^sub>2 \<down> Normal s" shows "\<Gamma>\<turnstile>Cond b c\<^sub>1 c\<^sub>2 \<down> Normal s"
generate_empty: shows "generate H {} = {\<one>\<^bsub>H\<^esub>}"
f16: shows "user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) = user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))"
Continuous_Disjunctous: fixes H :: "'d upred \<Rightarrow> 'd upred" assumes "Continuous H" shows "Disjunctuous H"
exp_prod2: fixes a :: "nat"   and x :: "nat" assumes "1 < a"   and "2 ^ x = a * 2" shows "even a"
P: shows "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp tpD tpFD"
tickFree_def: fixes s :: "'a event list" shows "tickFree s = (tick \<notin> set s)"
iarray_exhaust2: fixes xs :: "'b iarray"   and ys :: "'b iarray" shows "(xs = ys) = (IArray.list_of xs = IArray.list_of ys)"
None: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d option" shows "ord_option ord None x"
mono: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "continuous F" shows "mono F"
simps: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "('a, 'c, 'e) Vardi.var"   and a2 :: "('b, 'd, 'e) Vardi.var" shows "rel_var R S a1 a2 = (\<exists>x. a1 = map_var fst fst x \<and> a2 = map_var snd snd x \<and> set1_var x \<subseteq> {(x, y). R x y} \<and> set2_var x \<subseteq> {(x, y). S x y})"
nhd_def: fixes T :: "'b set set"   and x :: "'b" shows "nhds\<^bsub>T\<^esub> x = {U. U \<subseteq> carrier\<^bsub>T\<^esub> \<and> (\<exists>m. m open\<^bsub>T\<^esub> \<and> x \<in> m \<and> m \<subseteq> U)}"
Camestres: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A Q B"   and "A E C" shows "B E C"
density_context_empty: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "density_context V V' \<Gamma> \<delta>" shows "density_context {} (V \<union> V') \<Gamma> (\<lambda>_. 1)"
necessary_validity_def: shows "necessary_validity \<equiv> \<lambda>\<phi>. \<forall>v. [\<phi> in v]"
map_simp_ide: fixes a :: "'c local.term set" assumes "ide a" shows "\<nu>.map a = mkarr (DOM a\<^bold>\<down>)"
simps_2: fixes carrier :: "'f set"   and mult :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and one :: "'f"   and more :: "'g" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
encode_operator_effects_is_cnf_i: fixes t :: "nat"   and \<Pi> :: "('b, 'c) strips_problem_scheme"   and op :: "'b strips_operator" shows "is_cnf \<^bold>\<And>map (\<lambda>v. \<^bold>\<not> (Atom (Operator t (index (\<Pi>\<^sub>\<O>) op))) \<^bold>\<or> Atom (State (Suc t) (index (\<Pi>\<^sub>\<V>) v))) (add_effects_of op)"
Cauchy_theorem_aux: fixes s :: "complex set"   and pts :: "complex set"   and f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and h :: "complex \<Rightarrow> real" assumes "open s"   and "connected (s - pts)"   and "finite pts"   and "pts \<subseteq> s"   and "f holomorphic_on s - pts"   and "valid_path g"   and "pathfinish g = pathstart g"   and "path_image g \<subseteq> s - pts"   and "\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0"   and "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts))" shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
induct: fixes x :: "('b com option \<times> 'b) list"   and P :: "('b com option \<times> 'b) list \<Rightarrow> bool" assumes "x \<in> cptn"   and "\<And>Pa s. P [(Pa, s)]"   and "\<And>Pa t xs s. \<lbrakk>(Pa, t) # xs \<in> cptn; P ((Pa, t) # xs)\<rbrakk> \<Longrightarrow> P ((Pa, s) # (Pa, t) # xs)"   and "\<And>Pa s Q t xs. \<lbrakk>(Pa, s) -c\<rightarrow> (Q, t); (Q, t) # xs \<in> cptn; P ((Q, t) # xs)\<rbrakk> \<Longrightarrow> P ((Pa, s) # (Q, t) # xs)" shows "P x"
nat_pow_Suc: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> Suc n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x"
ABC': shows "\<forall>t\<in>set A. \<forall>s\<in>set B. (\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> Unifier \<delta> t s) \<longrightarrow> Q t \<and> Q s \<or> R t \<and> R s"
assms_2: shows "\<forall>\<^sub>F x in at x. f x = g x"
preprm_compose_push: fixes \<pi> :: "('b \<times> 'b) list"   and a :: "'b"   and b :: "'b" shows "preprm_compose \<pi> (preprm_unit a b) =p preprm_compose (preprm_unit (preprm_apply \<pi> a) (preprm_apply \<pi> b)) \<pi>"
front_tickFree_charn: fixes s :: "'c event list" shows "front_tickFree s = (s = [] \<or> (\<exists>a t. s = t @ [a] \<and> tickFree t))"
wf_def: fixes r :: "('c \<times> 'c) set" shows "wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
fct_gen_subalgebra_cong: fixes M :: "'b measure"   and P :: "'b measure"   and N :: "'c measure"   and Q :: "'c measure"   and X :: "'b \<Rightarrow> 'c" assumes "space M = space P"   and "sets N = sets Q" shows "fct_gen_subalgebra M N X = fct_gen_subalgebra P Q X"
eval_elim_cases_8: fixes G :: "prog"   and s :: "st"   and unop :: "unop"   and e :: "expr"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (UnOp unop e)\<succ>\<rightarrow> (v, s')" obtains v :: "val" where "va__ = In1 (eval_unop unop v)" and "G\<turnstile>Norm s \<midarrow>e-\<succ>v\<rightarrow> s'"
left_neutral: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and z :: "'g"   and a :: "'g" assumes "monoid f z" shows "f z a = a"
env_restr_below_itself: fixes S :: "'c set"   and m :: "'c \<Rightarrow> 'd" shows "m f|` S \<sqsubseteq> m"
subset_diff: fixes B :: "'b set"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "B \<subseteq> A"   and "finite A" shows "prod g A = prod g (A - B) * prod g B"
cases: fixes x :: "'a list \<times> (nat + nat set) list" obtains w :: "'a list" and I :: "(nat + nat set) list" where "x = (w, I)"
InvariantUniqAfterBackjump: fixes M :: "(Literal \<times> bool) list"   and p :: "(Literal \<times> bool) list"   and bClause :: "Literal list"   and bLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantUniq M"   and "isPrefix p M"   and "isUnitClause bClause bLiteral (elements p)"   and "M' = p @ [(bLiteral, False)]" shows "InvariantUniq M'"
tmin_def: shows "tmin = (LEAST t. iterate t step cfg = ([], rv))"
test_bit_size: fixes w :: "'b word"   and n :: "nat" assumes "w !! n" shows "n < size w"
Pi_def: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
assms_5: shows "Ana_invar_subst (ik\<^sub>s\<^sub>t \<A> \<union> assignment_rhs\<^sub>s\<^sub>t \<A>)"
assms_3: shows "\<forall>x. (\<forall>y. y \<in> P \<longrightarrow> y \<le> x) \<and> x \<in> P \<longrightarrow> R x"
code: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v0 (if deadend v0 then LNil else if v0 \<in> VV p then greedy_conforming_path p \<sigma> \<sigma>' (\<sigma> v0) else greedy_conforming_path p \<sigma> \<sigma>' (\<sigma>' v0))"
conversep_le_swap: fixes r :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and s :: "'k \<Rightarrow> 'j \<Rightarrow> bool" shows "(r \<le> s\<inverse>\<inverse>) = (r\<inverse>\<inverse> \<le> s)"