hcju/isabelleps huggingface dataset:
read_buffer_typed: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" assumes "\<Gamma> = Map.empty"   and "\<S> = ({}, {})"   and "P = {}" shows "\<exists>\<Gamma>' \<S>' P'. has_type \<Gamma> \<S> P read_buffer \<Gamma>' \<S>' P'"
subjects_reduction2: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and es' :: "char list exp list"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>es,(h, l, sh),b\<rangle> [\<rightarrow>] \<langle>es',(h', l', sh'),b'\<rangle>"   and "P,E \<turnstile> (h, l, sh) \<surd>"   and "iconfs sh es"   and "P,E,h,sh \<turnstile> es [:] Ts" shows "\<exists>Ts'. P,E,h',sh' \<turnstile> es' [:] Ts' \<and> P \<turnstile> Ts' [\<le>] Ts"
nn_integral_stream_space: fixes M :: "'b measure"   and f :: "'b stream \<Rightarrow> ennreal" assumes "prob_space M"   and "f \<in> borel_measurable (stream_space M)" shows "integral\<^sup>N (stream_space M) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ X. f (x ## X) \<partial>stream_space M \<partial>M"
openproc_axioms: shows "openproc paodv opaodv id"
ternary_to_bool_bool_to_ternary: fixes X :: "bool" shows "ternary_to_bool (bool_to_ternary X) = Some X"
congD: fixes a :: "'a"   and b :: "'a"   and n :: "'a" assumes "[a = b] (mod n)" shows "a mod n = b mod n"
wf_sub_rel_of_1: shows "sub_rel_of g subs"
Limsup_compose_continuous_mono: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "continuous_on UNIV f"   and "mono f"   and "F \<noteq> bot" shows "Limsup F (\<lambda>n. f (g n)) = f (Limsup F g)"
negligible_imp_measurable: fixes S :: "'c set" assumes "negligible S" shows "S \<in> lmeasurable"
act_rews_3: fixes v :: "state \<Rightarrow> 'a"   and s :: "state"   and t :: "state" shows "(s, t) \<Turnstile> unchanged v \<equiv> v t = v s"
gauss_sum_int_conv_gauss_sum: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "int" assumes "dcharacter n \<chi>" shows "dcharacter.gauss_sum_int n \<chi> k = dcharacter.gauss_sum n \<chi> (nat (k mod int n))"
subset_fSuc_strict: fixes f :: "nat \<Rightarrow> 'b set"   and u :: "nat"   and v :: "nat" assumes "\<And>u. f u \<subseteq> f (Suc u)"   and "u < v" shows "f u \<subseteq> f v"
distinct_selsort: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "distinct (linorder_list0.selsort le xs) = distinct xs"
keys_eq_empty_iff: fixes c :: "'d \<Rightarrow>\<^sub>0 'e" shows "(keys c = {}) = (c = 0)"
simps: fixes x :: "indices list" shows "right_app x = indices.Right # x"
distrib: fixes x :: "int"   and y :: "int" shows "L (x mod 2 ^ LENGTH('b)) (y mod 2 ^ LENGTH('b)) mod 2 ^ LENGTH('b) = L x y mod 2 ^ LENGTH('b)"
reduced_labelling_3: fixes n :: "nat"   and x :: "nat \<Rightarrow> nat" shows "reduced n x = n \<or> x (reduced n x) \<noteq> 0"
frontier_ic_union_ci: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a < b"   and "b < c" shows "b \<in> frontier ({a..<b} \<union> {b<..c})"
intros_20: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> t"   and "\<not> isAbr t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
has_integral_restrict: fixes S :: "'b set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "'c" assumes "S \<subseteq> T" shows "((\<lambda>x. if x \<in> S then f x else (0::'c)) has_integral i) T = (f has_integral i) S"
DBM_reset_sound_empty: fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and c :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "'c \<Rightarrow> 'd" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "v c \<le> n"   and "DBM_reset M n (v c) d M'"   and "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> M'" shows "\<not> u \<turnstile>\<^bsub>v,n\<^esub> M"
iteration_sub_denest: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<infinity> \<le> x\<^sup>\<infinity> \<cdot> (y \<cdot> x\<^sup>\<infinity>)\<^sup>\<infinity>"
execute: fixes p :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and initial :: "'a \<Rightarrow> bool"   and a :: "'b" assumes "p \<in> transition_system_initial.nodes execute enabled initial"   and "enabled a p" shows "execute a p \<in> transition_system_initial.nodes execute enabled initial"
contour_integrable_reversepath_eq: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g" shows "(f contour_integrable_on reversepath g) = (f contour_integrable_on g)"
get_root_node_si_locs_def: shows "get_root_node_si_locs = get_ancestors_si_locs"
less_funI: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<exists>k. f k < g k \<and> (\<forall>k'<k. f k' = g k')" shows "less_fun f g"
path_component_of: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "path_component_of X x y = (\<exists>T. path_connectedin X T \<and> x \<in> T \<and> y \<in> T)"
P2E: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> \<P>\<^sup>2" obtains r :: "recf" where "recfn 2 r" and "\<forall>x y. eval r [x, y] = f x y"
inf_ereal_def: fixes x :: "ereal"   and y :: "ereal" shows "inf x y = min x y"
a_comm': fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_ad x ; rel_ad y = rel_ad y ; rel_ad x"
f_shrink_last_snoc: fixes k :: "nat"   and ys :: "'b list"   and xs :: "'b list" assumes "0 < k"   and "length ys = k"   and "length xs mod k = 0" shows "(xs @ ys) \<div>\<^sub>l k = xs \<div>\<^sub>l k @ [last ys]"
induced_function_integral_restr_nonergodic_1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "integrable (restrict_space M A) (induced_function A f)"
rec_ifz_def: shows "rec_ifz = PR (recf.Id 2 0) (recf.Id 4 3)"
singleton_set_holds: fixes M :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and m :: "'a" assumes "\<forall>x\<in>M. \<forall>y\<in>M. f x = f y"   and "m \<in> M" shows "f m = the_elem (f ` M)"
set_pmf_mix_eq: fixes a :: "real"   and p :: "'a pmf" assumes "a \<in> {0..1}" shows "mix_pmf a p p = p"
prems_1: shows "G1 \<^bold>\<rightarrow> G2 \<in> set (subformulae F)"
intros_8: fixes f :: "'d"   and F :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'd) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and g :: "'f set"   and P :: "'f set"   and c :: "('f, 'e, 'd) com"   and Q :: "'f set"   and A :: "'f set" assumes "f \<in> F"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(g \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g c Q,A"
Un_empty_left: fixes B :: "'c set" shows "brouwer.comp_inf.covector B"
IH1: assumes "\<exists>pn. p1 \<in> subterms (\<Gamma> pn)"   and "((\<sigma>, p1), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i" shows "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
fref_to_Down_curry: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c nres"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'f nres"   and P :: "'d \<times> 'e \<Rightarrow> bool"   and A :: "(('a \<times> 'b) \<times> 'd \<times> 'e) set"   and B :: "('c \<times> 'f) set"   and x' :: "'d"   and y' :: "'e"   and x :: "'a"   and y :: "'b" assumes "(uncurry f, uncurry g) \<in> [P]\<^sub>f A \<rightarrow> \<langle>B\<rangle>nres_rel"   and "P (x', y')"   and "((x, y), x', y') \<in> A" shows "f x y \<le> \<Down> B (g x' y')"
app_length_lt_exists: fixes xsa :: "'b list"   and zsa :: "'b list"   and xs :: "'b list"   and ys :: "'b list" assumes "xsa @ zsa = xs @ ys"   and "length xsa \<le> length xs" shows "xsa @ drop (length xsa) xs = xs"
to_Gromov_completion_def: fixes x :: "'c" shows "to_Gromov_completion x = abs_Gromov_completion (\<lambda>n. x)"
assms_2: shows "cl = Func_native i (t1s _> t2s) tfs es"
assms_1: shows "D \<in> inferred_clause_sets R S N"
path_from_to_first': fixes v :: "'a"   and xs :: "'a list"   and x :: "'a"   and xs' :: "'a list"   and w :: "'a" assumes "v \<leadsto>(xs @ x # xs')\<leadsto> w" shows "v \<notin> set xs'"
setmonotone_mk_regular2: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "setmonotone (mk_regular2 P F)"
assms_2: fixes kv :: "'c"   and x :: "'e" shows "f kv x \<le> f' kv x"
env_app_add: fixes e1 :: "'b environment"   and e2 :: "'b environment"   and x :: "char list"   and X :: "'b" assumes "ok e1"   and "env_dom e1 \<inter> env_dom e2 = {}"   and "ok e2"   and "x \<notin> env_dom e1"   and "x \<notin> env_dom e2" shows "e1+e2\<lparr>x:X\<rparr> = e1\<lparr>x:X\<rparr>+e2"
noetherian_domain_def: fixes R :: "('c, 'd) ring_scheme" shows "noetherian_domain R \<equiv> noetherian_ring R \<and> domain R"
PDP_nz: fixes x :: "'a"   and d :: "'a \<Rightarrow>\<^sub>L 'a" assumes "(x, d) \<in> PDP" shows "f x \<bullet> n \<noteq> 0"
field_differentiable_at_sin: fixes z :: "'a" shows "sin field_differentiable at z"
borel_measurable_lebesgue_on_preimage_borel: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue)"
inorder_ins: fixes t :: "'a tree"   and n :: "nat"   and d :: "nat"   and x :: "'a"   and t' :: "'a tree" assumes "Sorted_Less.sorted (inorder t)" shows "(ins n d x t = Same \<longrightarrow> ins_list x (inorder t) = inorder t) \<and> (ins n d x t = Bal t' \<longrightarrow> ins_list x (inorder t) = inorder t') \<and> (ins n d x t = Unbal t' \<longrightarrow> ins_list x (inorder t) = inorder t')"
stc_not_Infinitesimal: fixes x :: "complex star" assumes "stc x \<noteq> 0" shows "x \<notin> Infinitesimal"
similar_mat_witD2_2: fixes A :: "'c mat"   and n :: "nat"   and m :: "nat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "Q * P = 1\<^sub>m n"
intra_proc_additional_edge: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>a''. valid_edge a'' \<and> sourcenode a'' = targetnode a \<and> targetnode a'' = sourcenode a' \<and> kind a'' = (\<lambda>cf. False)\<^sub>\<surd>"
simps: fixes v1 :: "('c \<times> 'a) list"   and uu :: "('b \<times> ('b, 'c, 'a) namespace) list"   and n :: "'c" shows "cupcake_nsLookup (Bind v1 uu) n = map_of v1 n"
fps_one_nth_Cons: fixes x :: "'d"   and xs :: "'d list" shows "1 $ (x # xs) = (0::'c)"
fgA: shows "(\<lambda>x. f x \<otimes> g x) \<in> A \<rightarrow> carrier G"
bounded_iff: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "(sup b c \<le> a) = (b \<le> a \<and> c \<le> a)"
mono_subset_elem: fixes f :: "'a set \<Rightarrow> 'b set"   and A :: "'a set"   and B :: "'a set"   and x :: "'b" assumes "mono f"   and "A \<subseteq> B"   and "x \<in> f A" shows "x \<in> f B"
the1_equality: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<exists>!x. P x"   and "P a" shows "(THE x. P x) = a"
eq: shows "(SIGMA j:UNIV. {j}) = (UNIV - J) \<times> X \<union> Sigma J F"
resets_mono: fixes xs :: "('c \<Rightarrow> real) stream"   and i :: "nat"   and j :: "nat"   and x :: "'c"   and c :: "'c" assumes "PTA_Reachability.pairwise eq_elapsed xs"   and "pred_stream (\<lambda>u. u \<in> V) xs"   and "stream_trans xs"   and "\<forall>k\<ge>i. k \<le> j \<longrightarrow> zero x ([xs !! k]\<^sub>\<R>)"   and "x \<in> \<X>"   and "c \<in> \<X>"   and "i \<le> j" shows "(xs !! j) c \<le> (xs !! i) c"
csols_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open T f X" shows "ll_on_open.csols T f X t0 x0 = {(x, t1). {t0--t1} \<subseteq> T \<and> x t0 = x0 \<and> (x solves_ode f) {t0--t1} X}"
carrier_submanifold: shows "sub.carrier = S \<inter> src.carrier"
MD0: fixes i :: "nat" assumes "i < n - 1"   and "Suc i \<noteq> k" shows "mat_delete (make_mat n k B) 0 k $$ (i, 0) = (0::'a)"
wf_ciD2_ci_app: fixes ci :: "('a, 'b) check_instr"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list" assumes "ci_app ci ins P h stk loc C M pc frs" shows "check_instr' ins P h stk loc C M pc frs"
prod_ord_rawD: fixes xs :: "('a \<times> 'd) list \<times> 'o"   and ys :: "('a \<times> 'e) list \<times> 'o"   and P :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and k :: "'a" assumes "oalist_inv xs"   and "oalist_inv ys"   and "prod_ord_raw P xs ys"   and "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)" shows "P k (lookup_raw xs k) (lookup_raw ys k)"
insert_union_subset_iff: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "(add_mset a A \<subset># B) = (a \<in># B \<and> A \<subset># B - {#a#})"
finite_measure_mono: fixes M :: "'a measure"   and A :: "'a set"   and B :: "'a set" assumes "finite_measure M"   and "A \<subseteq> B"   and "B \<in> sets M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M B"
line_integral_sum_basis_1: fixes basis :: "'a set"   and F :: "'a \<Rightarrow> 'a"   and \<gamma> :: "real \<Rightarrow> 'a" assumes "finite basis"   and "\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>" shows "line_integral F basis \<gamma> = (\<Sum>b\<in>basis. line_integral F {b} \<gamma>)"
enat_eSuc_iff: fixes y :: "int"   and x :: "eint" shows "(eint y = x + 1) = (\<exists>n. y = n + 1 \<and> eint n = x)"
row_transpose: fixes j :: "nat"   and A :: "'a mat" assumes "j < dim_col A" shows "row A\<^sup>t j = col A j"
norm_power_ineq: fixes x :: "'b"   and n :: "nat" shows "norm (x ^ n) \<le> norm x ^ n"
centre_in_ball: fixes x :: "'b"   and e :: "real" shows "(x \<in> ball x e) = (0 < e)"
dp_heap_axioms_def: fixes Q :: "heap \<Rightarrow> bool"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and lookup_st :: "'c \<Rightarrow> (heap, 'd option) state"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap"   and update_st :: "'c \<Rightarrow> 'd \<Rightarrow> (heap, unit) state" shows "dp_heap_axioms Q lookup lookup_st update update_st \<equiv> rel_fun (=) (heap_mem_defs.rel_state Q (=)) lookup_st lookup \<and> rel_fun (=) (rel_fun (=) (heap_mem_defs.rel_state Q (=))) update_st update"
simps_2: fixes b :: "bit"   and bs :: "bit list" shows "norm_signed (b # bs) = (case b of \<zero> \<Rightarrow> if norm_unsigned bs = [] then [] else b # norm_unsigned bs | \<one> \<Rightarrow> b # rem_initial b bs)"
square_free_factorization_smult_prod_listI: fixes p :: "'a poly"   and c :: "'a"   and bs1 :: "('a poly \<times> nat) list"   and b :: "'a"   and bs :: "'a poly list"   and i :: "nat"   and bs2 :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs1 @ (smult b (prod_list bs), i) # bs2)"   and "\<And>b. b \<in> set bs \<Longrightarrow> 0 < degree b" shows "square_free_factorization p (c * b ^ Suc i, bs1 @ map (\<lambda>b. (b, i)) bs @ bs2)"
le_sup_iff: fixes z :: "'b"   and x :: "'b"   and y :: "'b" shows "(z \<le> x \<sqinter> y) = (z \<le> x \<and> z \<le> y)"
simps_1: shows "typeof Unit = \<lfloor>Void\<rfloor>"
same_together: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> H" shows "\<exists>t\<in>{j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} \<rightarrow> A. \<exists>g\<in>{j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} \<rightarrow> f ` {j. j \<le> n}. surj_to g {j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} (f ` {j. j \<le> n}) \<and> \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n = \<Sigma>\<^sub>e M (\<lambda>k. t k \<cdot>\<^sub>s g k) (card (f ` {j. j \<le> n}) - Suc 0)"
numeral_code_1: shows "Numeral1 = (1::'b)"
offs_pred_zero_cons_less: fixes ns :: "nat list"   and ub :: "nat"   and x :: "'c"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" assumes "offs_pred ns ub (x # xs) index key mi ma"   and "index key x (length ns) mi ma = i"   and "i < length ns"   and "0 < i"   and "offs_set_prev ns (x # xs) index key mi ma i = {}" shows "offs_next ns ub (x # xs) index key mi ma 0 < offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma 0"
bij_betw_comp_iff2: fixes f' :: "'a \<Rightarrow> 'b"   and A' :: "'a set"   and A'' :: "'b set"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "bij_betw f' A' A''"   and "f ` A \<subseteq> A'" shows "bij_betw f A A' = bij_betw (f' \<circ> f) A A''"
degree_of_int_poly: fixes p :: "int poly" shows "Polynomial.degree (of_int_poly p) = Polynomial.degree p"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "F g (insert x A) = g x \<^bold>* F g A"
join_meet_absorb: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> x \<sqinter> y = x"
padic_set_memI: fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "\<And>m. f m \<in> carrier (residue_ring (p ^ m))"   and "\<And>m n. m < n \<Longrightarrow> residue (p ^ m) (f n) = f m" shows "f \<in> padic_set p"
ipurge_tr_aux_single_dom: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list" shows "ipurge_tr_aux I D {u} xs = ipurge_tr I D u xs"
meet_le: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> x \<sqinter> y"
CondI: fixes G :: "prog"   and s :: "state"   and e :: "expr"   and b :: "val"   and s1 :: "state"   and e1 :: "expr"   and e2 :: "expr"   and v :: "val"   and s2 :: "state" assumes "G\<turnstile>s \<midarrow>e-\<succ>b\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<rightarrow> s2" shows "G\<turnstile>s \<midarrow>e ? e1 : e2-\<succ>(if normal s1 then v else undefined)\<rightarrow> s2"
gSwapBinpPresGWlsBinp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSwapBinpPresGWlsBinp MOD \<equiv> \<forall>zs z1 z2 delta binp' binp. wlsBinp delta binp' \<and> gWlsBinp MOD delta binp \<longrightarrow> gWlsBinp MOD delta (gSwapBinp MOD zs z1 z2 binp' binp)"
boundOutputDistinct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'" shows "distinct (bn \<alpha>)"
until_def: fixes \<gamma>' :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma>' \<UU>\<^sub>b \<gamma> \<equiv> \<lambda>t n. \<exists>n''\<ge>n. \<gamma> t n'' \<and> (\<forall>n'\<ge>n. n' < n'' \<longrightarrow> \<gamma>' t n')"
r_def: shows "r = (\<lambda>x. \<psi> x - \<theta> x)"
arcos_fixed: fixes H :: "'a set"   and a :: "'a"   and h :: "'a" assumes "M +> H"   and "a \<in> carrier M"   and "h \<in> H" shows "a \<uplus>\<^bsub>M\<^esub> H = (h \<plusminus> a) \<uplus>\<^bsub>M\<^esub> H"
suc_measurable: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat"   and c :: "bool" assumes "f \<in> borel_measurable (F (Suc n))" shows "(\<lambda>w. f (c ## w)) \<in> borel_measurable (F n)"
L_While_X: fixes X :: "char list set"   and b :: "bexp"   and c :: "com" shows "X \<subseteq> L (WHILE b DO c) X"
lasso_run_rel_def_internal: fixes R :: "('c \<times> 'd) set" shows "lasso_run_rel R \<equiv> br run_of_lasso (\<lambda>_. True) O (nat_rel \<rightarrow> R)"
subsumes_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset" shows "substitution_ops.subsumes subst_atm C D = (\<exists>\<sigma>. substitution_ops.subst_cls subst_atm C \<sigma> \<subseteq># D)"
narrow_RefT: fixes G :: "prog"   and R :: "ref_ty"   and T :: "ty" assumes "G\<turnstile>RefT R\<succ>T" shows "\<exists>t. T = RefT t"
flowD_finite: fixes g :: "'v option \<times> 'v option \<Rightarrow> ennreal"   and e :: "'v option \<times> 'v option" assumes "flow \<Psi> g" shows "g e \<noteq> \<top>"
dghm_ArrMap_is_arr': fixes a :: "V"   and b :: "V"   and f :: "V"   and A :: "V"   and B :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"   and "A = \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"   and "B = \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : A \<mapsto>\<^bsub>\<BB>\<^esub> B"
polymul_properties_3: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
fremove_def: fixes x :: "'a"   and A :: "'a fset" shows "fremove x A = A |-| {|x|}"
SIGMA_CSUM: fixes I :: "'a set"   and As :: "'a \<Rightarrow> 'b set" shows "|Sigma I As| = (CSUM i:|I|. |As i|)"
intros_2: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "x \<sharp> (\<Gamma>, s, t)"   and "(x, T\<^sub>1) # \<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2" shows "\<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1\<rightarrow>T\<^sub>2"
Quotient_uint32: shows "Quotient (=) Abs_uint32 Rep_uint32 cr_uint32"
is_dghm_axioms: shows "\<HH>' : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
right_diff_distrib: fixes x :: "(real, 3) Finite_Cartesian_Product.vec"   and y :: "(real, 3) Finite_Cartesian_Product.vec"   and z :: "(real, 3) Finite_Cartesian_Product.vec" shows "cross3 x (y - z) = cross3 x y - cross3 x z"
dghm_const_components_1: fixes \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" shows "dghm_const \<CC> \<DD> a f\<lparr>ObjMap\<rparr> = vconst_on (\<CC>\<lparr>Obj\<rparr>) a"
drop_bit_drop_bit: fixes m :: "nat"   and n :: "nat"   and a :: "'a" shows "drop_bit m (drop_bit n a) = drop_bit (m + n) a"
A: shows "ipurge_tr_rev_aux I D U xs = x # ws \<longrightarrow> (\<exists>ys zs. xs = ys @ x # zs \<and> ipurge_tr_rev_aux I D (sources_aux I D U (x # zs)) ys = [] \<and> (\<exists>v\<in>sources_aux I D U zs. (D x, v) \<in> I))"
basic_trans_rules_32: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" assumes "a = b"   and "b \<in> A" shows "a \<in> A"
simps: fixes n :: "nat" shows "shows_nat n = (if n < 10 then shows_string (digit2string n) else shows_nat (n div 10) \<circ> shows_string (digit2string (n mod 10)))"
prefix_append_iff: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"
distinct_25: fixes x41 :: "'a formula"   and x42 :: "'a formula"   and x51 :: "'a formula"   and x52 :: "'a formula" shows "x41 \<^bold>\<and> x42 \<noteq> x51 \<^bold>\<or> x52"
inv_ide: fixes a :: "'a" assumes "Cop.ide a" shows "Cop.inv a = a"
defNode_code: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and v :: "'val" shows "uninst.defNode p g v \<equiv> hd (filter (\<lambda>n. v \<in> uninst.allDefs p g n) (\<alpha>n g))"