hcju/isabelleps huggingface dataset:
gt_hd_def: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and \<xi> :: "('a, 'b) hd"   and \<zeta> :: "('a, 'b) hd" assumes "Lambda_Free_Term.ground_heads gt_sym arity_sym arity_var ground_heads_var" shows "ground_heads.gt_hd gt_sym ground_heads_var \<xi> \<zeta> = (\<forall>g\<in>ground_heads.ground_heads ground_heads_var \<xi>. \<forall>f\<in>ground_heads.ground_heads ground_heads_var \<zeta>. gt_sym g f)"
intros: fixes c :: "('test, 'atom) com" assumes "\<And>s c' s'. \<lbrakk>mustT c s; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> s \<approx> s' \<and> discr0 c'"   and "\<And>s s'. \<lbrakk>mustT c s; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> s \<approx> s'" shows "discr0 c"
config'_embed: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and A :: "('b \<Rightarrow> 'c) \<times> ('b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c)"   and s0 :: "'b \<times> 'c"   and qs :: "'d list" shows "On_Off.config'_rand step (On_Off.embed A) (return_pmf s0) qs = return_pmf (On_Off.config' step A s0 qs)"
run_return_nondet: fixes x :: "'c" shows "run_nondet (return_nondet x) = Monad_Overloading.return {#x#}"
Un_def: fixes A :: "'x set"   and B :: "'x set" shows "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
sums_zeta: fixes s :: "complex" assumes "1 < Re s" shows "(\<lambda>n. of_nat (Suc n) powr - s) sums zeta s"
HAEvents_def: shows "HAEvents = fst \<circ> snd \<circ> snd \<circ> Rep_hierauto"
thmD5: fixes I :: "item set"   and k :: "nat"   and u :: "nat"   and T :: "(symbol \<times> character list) set" assumes "I \<subseteq> Gen (\<P> k u)"   and "k \<le> length Doc"   and "T \<subseteq> \<Z> k u" shows "\<pi> k T I \<subseteq> Gen (\<P> k u)"
add_left_commute: fixes V :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "x + (y + z) = y + (x + z)"
cupcake_state_preserve: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and res :: "'b state \<times> (v, v) result" assumes "Big_Step_Unclocked_Single.evaluate env s e res"   and "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "s = fst res"
T'_def: shows "T' \<equiv> {x \<in> U. setdist {x} T \<le> setdist {x} S}"
mrbst_inter_altdef: shows "mrbst_inter = mrbst_inter_diff True"
fpower_distr: fixes f :: "'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "fstar f \<circ> g = (\<Sqinter>i. fpower f i \<circ> g)"
nat_of_hatom_def: shows "nat_of_hatom \<equiv> SOME f. bij f"
finite_imp_less_Inf: fixes X :: "'a set"   and x :: "'a"   and a :: "'a" assumes "finite X"   and "x \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> a < x" shows "a < Inf X"
is_lub_unique: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "S <<| x"   and "S <<| y" shows "x = y"
iT_Minus_cut_eq: fixes k :: "nat"   and t :: "nat"   and I :: "nat set" assumes "k \<le> t" shows "k \<ominus> I \<down>\<le> t = k \<ominus> I"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>zs A ys xs gt. \<lbrakk>zs \<in> lists A; ys \<in> lists A; xs \<in> lists A; \<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x; ext gt zs ys; ext gt ys xs\<rbrakk> \<Longrightarrow> ext gt zs xs" shows "ext_trans_axioms ext"
strong_reduction_simulations_impl_bisimulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "strong_reduction_simulation Rel Cal"   and "strong_reduction_simulation (Rel\<inverse>) Cal" shows "strong_reduction_simulation Rel Cal \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Cal Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Cal P' \<and> (P', Q') \<in> Rel))"
lt_of_nat_term_order_alt: fixes to :: "'b nat_term_order" shows "lt_of_nat_term_order to = (\<lambda>u v. ko.lt (key_order_of_nat_term_order_inv to) v u)"
assms_3: shows "g \<in> borel_measurable (lebesgue_on S)"
keys_update: fixes k :: "'d"   and v :: "'e"   and m :: "('d, 'e) mapping" shows "Mapping.keys (Mapping.update k v m) = insert k (Mapping.keys m)"
zip_Cons_Cons: fixes x :: "'e"   and xs :: "'e list"   and y :: "'f"   and ys :: "'f list" shows "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
disj_def: fixes \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.disj \<gamma> \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<or> \<gamma>' t n"
name_fresh_abs: fixes b :: "name"   and a :: "name"   and x :: "'b" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
is_Hcomp_def: fixes "term" :: "'a local.term" shows "is_Hcomp term = (\<exists>x31 x32. term = x31 \<^bold>\<star> x32)"
join_bottom_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
lhd_ltake: fixes n :: "enat"   and xs :: "'c llist" assumes "n \<noteq> 0" shows "lhd (ltake n xs) = lhd xs"
pell_approximation_lemma: fixes s :: "nat"   and x :: "real" assumes "0 < s" shows "\<exists>u t. 0 < u \<and> coprime (int u) t \<and> 1 / real s \<in> {\<bar>real_of_int t - real u * x\<bar><..1 / real u}"
diff_minus_eq_add: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.group_add minus plus zero uminus" shows "minus a (uminus b) = plus a b"
suffix_Cons: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "suffix xs (y # ys) = (xs = y # ys \<or> suffix xs ys)"
cut_ge_Min_all: fixes t :: "'b"   and I :: "'b set" assumes "t \<le> iMin I" shows "I \<down>\<ge> t = I"
intI: fixes A :: "'b \<Rightarrow> bool" assumes "\<And>w. A w" shows "\<turnstile> A"
bdia_def: fixes R :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'd"   and y :: "'f" shows " \<langle>R| f y = \<Squnion>{f x |x. (x, y) \<in> R}"
map_simp_ide: fixes a :: "'a \<times> 'a \<times> 'a" assumes "VVV.ide a" shows "\<alpha>.map a = \<a>[fst a, fst (snd a), snd (snd a)]"
CR3_RED_def: fixes t :: "trm"   and \<tau> :: "ty" shows "CR3_RED t \<tau> \<equiv> \<forall>t'. t \<mapsto> t' \<longrightarrow> t' \<in> RED \<tau>"
Sup: fixes Sup :: "'e set \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and M :: "'e set"   and f :: "'e \<Rightarrow> 'e" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq M"   and "\<forall>x\<in>M. x \<in> ccpo.iterates Sup less_eq f" shows "Sup M \<in> ccpo.iterates Sup less_eq f"
comp_mkArr: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and B :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and F :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr \<Rightarrow> (('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and C :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and G :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr \<Rightarrow> (('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.arr (S.mkArr A B F)"   and "S.arr (S.mkArr B C G)" shows "S.comp (S.mkArr B C G) (S.mkArr A B F) = S.mkArr A C (G \<circ> F)"
front_tickFree_dw_closed: fixes s :: "'a event list"   and t :: "'a event list" assumes "front_tickFree (s @ t)" shows "front_tickFree s"
vec_of_list_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "vec_of_list (map f xs) = map_vec f (vec_of_list xs)"
denote_while_n_sum_partial_density: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat"   and n :: "nat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m d"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat d d" shows "partial_density_operator (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n)"
equiv_lsbis2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" assumes "coalg B1 B2 s1 s2" shows "equiv B2 (lsbis2 B1 B2 s1 s2)"
A4_unique: fixes q :: "'a"   and a :: "'a"   and x :: "'a"   and b :: "'a"   and c :: "'a"   and x' :: "'a" assumes "q \<noteq> a"   and "B q a x"   and "a x \<equiv> b c"   and "B q a x'"   and "a x' \<equiv> b c" shows "x = x'"
exp_of_real_cnj: fixes x :: "real" shows "cnj (exp (\<i> * complex_of_real x)) = exp (- (\<i> * complex_of_real x))"
distrib_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
that_1: shows "\<I>1 \<le> \<I>1'"
rep_eq: fixes x :: "('b, 'c) mapping_rbt" shows "RBT_Mapping2.lookup x = rbt_comp_lookup ccomp (mapping_rbt.impl_of x)"
chosenAllocationAlg_def: fixes N :: "'c set"   and \<Omega> :: "'d list"   and b :: "'c \<times> 'd set \<Rightarrow> 'e"   and r :: "integer" shows "chosenAllocationAlg N \<Omega> b r \<equiv> randomEl (takeAll (\<lambda>x. x \<in> (argmax \<circ> sum) b (set (allAllocationsAlg N \<Omega>))) (allAllocationsAlg N \<Omega>)) r"
rbt_less_prop: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and t :: "('a, 'b) rbt" shows "ord.rbt_less less k t = (\<forall>x\<in>set (keys t). less x k)"
igSwapIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "igSwapIGOp MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
mult_scalar_monomial: fixes t :: "'a"   and c :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.mult_scalar (monomial c t) p = local.punit.monom_mult c t p"
Dom_in_Obj: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "local.Dom f \<in> Collect B.obj"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "\<F>\<^sub>SC.arr (\<F>\<^sub>SC.inclusion_of_generators f)"
strict_monoidal_functor_axioms: shows "strict_monoidal_functor (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D local.map"
fold_Nil: fixes f :: "'l \<Rightarrow> 'k \<Rightarrow> 'k" shows "fold f [] = id"
markedAndMemberImpliesIsMarkedElement: fixes m :: "'b \<times> bool"   and M :: "('b \<times> bool) list" assumes "marked m"   and "m \<in> set M" shows "element m \<in> set (markedElements M)"
all_is_set: fixes p :: "('c, 'd) fm"   and ps :: "('c, 'd) fm list"   and xs :: "nat set" assumes "all_is p ps xs" shows "{p} \<union> set (omit xs ps) = {p} \<union> set ps"
surjective: fixes r :: "('a, 'b) tagged_packet_scheme" shows "(r::('a, 'b) tagged_packet_scheme) = \<lparr>p_iiface = p_iiface r, p_oiface = p_oiface r, p_src = p_src r, p_dst = p_dst r, p_proto = p_proto r, p_sport = p_sport r, p_dport = p_dport r, p_tcp_flags = p_tcp_flags r, p_payload = p_payload r, p_tag_ctstate = p_tag_ctstate r, \<dots> = tagged_packet.more r\<rparr>"
rows: fixes i :: "nat" assumes "i < dim_row B" shows "row A i = row B i"
residue_simple: assumes "open s"   and "0 \<in> s"   and "f holomorphic_on s" shows "residue (\<lambda>w. f w / (w - 0)) 0 = f 0"
cblinfun_eq_on_canonical_basis: fixes f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and g :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<And>u. u \<in> set canonical_basis \<Longrightarrow> f *\<^sub>V u = g *\<^sub>V u" shows "f = g"
const_times_polyn1: fixes p :: "'a"   and c :: "'a" assumes "Idomain S"   and "p \<in> carrier R"   and "c \<in> carrier S"   and "c \<noteq> \<zero>\<^bsub>S\<^esub>" shows "deg R S X (c \<cdot>\<^sub>r p) = deg R S X p"
cp0: fixes X :: "'e state \<times> 'e state \<Rightarrow> Set('d)"   and Y :: "'e state \<times> 'e state \<Rightarrow> 'd"   and \<tau> :: "'e state \<times> 'e state" shows "X->including\<^sub>S\<^sub>e\<^sub>t(Y) \<tau> = \<lambda>_. X \<tau>->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. Y \<tau>) \<tau>"
comp_ide_arr: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "seq b f" shows "local.comp b f = f"
intros_3: fixes A :: "ty" shows "is_refT (A\<lfloor>\<rceil>)"
uniq_opp: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>2 :: "'c point"   and p\<^sub>1 :: "'c point" assumes "ell_field R"   and "ell_field.on_curve R a b p\<^sub>2"   and "ell_field.add R a p\<^sub>1 p\<^sub>2 = Infinity" shows "p\<^sub>2 = ell_field.opp R p\<^sub>1"
add_poly_p_empty_r: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset" shows "add_poly_p\<^sup>*\<^sup>* (p, q, r) (p, {#}, q + r)"
assms_3: shows "D.equivalent_objects (map\<^sub>0 a) (map\<^sub>0 b)"
antisym_conv: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.order less_eq less"   and "less_eq y x" shows "less_eq x y = (x = y)"
locally_iff: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" shows "locally P S = (\<forall>T x. open T \<and> x \<in> S \<inter> T \<longrightarrow> (\<exists>U. open U \<and> (\<exists>V. P V \<and> x \<in> S \<inter> U \<and> S \<inter> U \<subseteq> V \<and> V \<subseteq> S \<inter> T)))"
rotater_rev': fixes n :: "nat" shows "rotater n (rev (rev ys)) = rev (rotate n (rev ys))"
strip_list_comb: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c"   and ys :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp (term.list_comb app f ys) = (fst (term.strip_comb unapp f), snd (term.strip_comb unapp f) @ ys)"
goal_fulfilled_M: assumes "c_offending_flows m \<lparr>nodes = V, edges = E\<rparr> \<noteq> {}" shows "all_security_requirements_fulfilled M (delete_edges (generate_valid_topology_SOME M \<lparr>nodes = V, edges = E\<rparr>) (SOME F. F \<in> c_offending_flows m \<lparr>nodes = V, edges = E\<rparr>))"
GB_alt_3_finite: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite G" shows "is_Groebner_basis G = (\<forall>f\<in>pmdl G. f \<noteq> 0 \<longrightarrow> (\<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f))"
ArrayInit_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (CLEAR x[], s) \<Rightarrow> t = (t = s(x := \<lambda>_. 0))"
map_transform_restrA: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and S :: "Vars.var set"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "map_transform t ae (restrictA S \<Gamma>) = restrictA S (map_transform t ae \<Gamma>)"
all_acquired_dropWhile_in: fixes x :: "nat"   and P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" assumes "x \<in> all_acquired (dropWhile P sb)" shows "x \<in> all_acquired sb"
n_galois2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(n x \<sqsubseteq> t y \<oplus> n z) = (n x \<cdot> n y \<sqsubseteq> n z)"
neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "\<forall>x\<in>A. g x = z" shows "abel_semigroup_set.F f z g A = z"
prv_imp_refl: fixes \<phi> :: "fmla" assumes "\<phi> \<in> UNIV" shows "{} \<turnstile> \<phi> IMP \<phi>"
rtrancl_path_simps_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "rtrancl_path R x [] y = (y = x)"
baM: fixes i :: "nat"   and s :: "'s" shows "wp a (M i) s \<le> c"
closed_segment_as_ball: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b = affine hull {a, b} \<inter> cball ((a + b) /\<^sub>R 2) (norm (b - a) / 2)"
image_eq_imp_comp: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'c \<Rightarrow> 'a"   and B :: "'c set"   and h :: "'a \<Rightarrow> 'd" assumes "f ` A = g ` B" shows "(h \<circ> f) ` A = (h \<circ> g) ` B"
CI_def: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and J :: "nat set" assumes "Ionescu_Tulcea P M" shows "Ionescu_Tulcea.CI P M J = distr (Ionescu_Tulcea.C P M 0 (Ionescu_Tulcea.up_to J) (\<lambda>x. undefined)) (Pi\<^sub>M J M) (\<lambda>f. restrict f J)"
elims: fixes x :: "'fmla"   and xa :: "('trm \<times> 'var) list"   and y :: "'fmla" assumes "rawpsubst x xa = y" obtains \<phi> :: "'fmla" where "x = \<phi>''__" and "xa = []" and "y = \<phi>''__"   | \<phi> :: "'fmla" and t1 :: "'trm" and x1 :: "'var" and txs :: "('trm \<times> 'var) list" where "x = \<phi>" and "xa = (t1, x1) # txs" and "y = rawpsubst (subst \<phi> t1 x1) txs"
fin_string_enum_ex_def: shows "fin_string_enum_ex = (\<lambda>f. list_ex f vals)"
hequiv_rel: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set" assumes "Hintikka A H" shows "equiv (names H \<inter> A) (hequiv_rel A H)"
vectorspace_def: fixes V :: "'c set" shows "vectorspace V \<equiv> ((V \<noteq> {} \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> x + y \<in> V) \<and> (\<forall>x a. x \<in> V \<longrightarrow> a \<cdot> x \<in> V)) \<and> (\<forall>x y z. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> z \<in> V \<longrightarrow> x + y + z = x + (y + z)) \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> x + y = y + x) \<and> (\<forall>x. x \<in> V \<longrightarrow> x - x = (0::'c))) \<and> ((\<forall>x. x \<in> V \<longrightarrow> (0::'c) + x = x) \<and> (\<forall>x y a. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y) \<and> (\<forall>x a b. x \<in> V \<longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x)) \<and> ((\<forall>x a b. x \<in> V \<longrightarrow> (a * b) \<cdot> x = a \<cdot> b \<cdot> x) \<and> (\<forall>x. x \<in> V \<longrightarrow> 1 \<cdot> x = x)) \<and> (\<forall>x. x \<in> V \<longrightarrow> - x = - 1 \<cdot> x) \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> x - y = x + - y)"
inner_simps_6: fixes x :: "'a"   and r :: "real"   and y :: "'a" shows "x \<bullet> r *\<^sub>R y = r * (x \<bullet> y)"
cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "O[F](f) = O[F](g)"
BNTr3: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and W :: "'a set" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W \<in> WWa D f a" shows "minimum_elem (Iod D W) W a"
a_i_neg: fixes m :: "int" assumes "m < 0" shows "\<infinity> * ant m = - \<infinity>"
ideal_zero: 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 set"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "comm_ring R addition multiplication zero unit"   and "ideal A R add mult zero unit" shows "zero \<in> A"
measurable_count_space_eq2: fixes A :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "finite A" shows "(f \<in> M \<rightarrow>\<^sub>M count_space A) = (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
enabled_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and s :: "state" shows "s \<Turnstile> Enabled A \<equiv> \<exists>u. A (s, u)"
pfa_copy_def: shows "pfa_copy \<equiv> arl_copy"
mono_h: shows "mono_on h {1..} \<or> mono_on (\<lambda>x. - h x) {1..}"
floorlog_def: fixes b :: "nat"   and a :: "nat" shows "floorlog b a = (if 0 < a \<and> 1 < b then nat \<lfloor>log (real b) (real a)\<rfloor> + 1 else 0)"
step: fixes conv :: "('a, 'b, 'c, 'd) converter"   and a :: "'a" assumes "X conv"   and "a \<in> outs_\<I> \<I>" shows "plossless_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). X conv' \<or> plossless_converter \<I> \<I>' conv')"
fixp_above_def: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f" shows "bourbaki_witt_fixpoint.fixp_above lub leq f a = (if a \<in> Field leq then lub (bourbaki_witt_fixpoint.iterates_above lub leq f a) else a)"
fix_const: fixes c :: "'e" shows "(\<mu> x. c) = c"
intros: fixes r :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'h"   and b :: "'i" assumes "r a b" shows "r\<inverse>\<inverse> b a"
Abs: fixes env :: "nat \<Rightarrow> LambdaType.type"   and T :: "LambdaType.type"   and t :: "dB"   and U :: "LambdaType.type" assumes "env\<langle>0:T\<rangle> \<turnstile> t : U" shows "env \<turnstile> dB.Abs t : T \<Rightarrow> U"
fract_poly_smult_eqE: fixes p :: "'a poly"   and c :: "'a fract"   and q :: "'a poly" assumes "map_poly to_fract p = smult c (map_poly to_fract q)" obtains a :: "'a" and b :: "'a" where "c = to_fract b / to_fract a" and "smult a p = smult b q" and "coprime a b" and "normalize a = a"
A: shows "X \<subseteq> {x. x \<le> (1::'a) \<and> x * \<top> \<sqinter> x ^ o = x}"
abs_eq: shows "bot = Abs_lifted_pair (\<lambda>f. (bot, Filters.filter.Abs_filter UNIV))"
fds_const_converges: fixes c :: "'a"   and s :: "'a" shows "fds_converges (fds_const c) s"
mono_inv_bval': fixes S1 :: "'a st option"   and S2 :: "'a st option"   and b :: "bexp"   and bv :: "bool" assumes "S1 \<le> S2" shows "inv_bval' b bv S1 \<le> inv_bval' b bv S2"
vec_lambda_unique: fixes f :: "('d, 'c) vec"   and g :: "'c \<Rightarrow> 'd" shows "(\<forall>i. f $ i = g i) = (vec_lambda g = f)"
sel_rews_3: shows "appF\<cdot>ValTT = \<bottom>"
extended_distance_symmetric: fixes x :: "'c Bonk_Schramm_extension_unfolded"   and y :: "'c Bonk_Schramm_extension_unfolded" shows "extended_distance x y = extended_distance y x"
prv_subst: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "prv \<phi>" shows "prv (subst \<phi> t x)"
better_OrR1_substc: fixes d :: "coname"   and a :: "coname"   and M :: "trm"   and z :: "name"   and P :: "trm" assumes "d \<sharp> [a].M" shows "OrR1 <a>.M d{d:=z.P} = fresh_fun (\<lambda>a'. Cut <a'>.OrR1 <a>.M a' z.P)"