hcju/isabelleps huggingface dataset:
normalize_block: fixes init :: "'d \<Rightarrow> 'd"   and bdy :: "('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "Language.normalize (block init bdy return c) = block init (Language.normalize bdy) return (\<lambda>s t. Language.normalize (c s t))"
substNext_code_correct: fixes g :: "'g" assumes "uninst_code.ssa.redundant uses phis g" shows "uninst.substNext (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g)) g = uninst_code.substNext_code phis g (uninst_code.chooseNext' uses phis g)"
remove_preserves_type_wf: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.remove child \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h'"
suminf_mult: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "summable f" shows "(\<Sum>n. c * f n) = c * suminf f"
wtPB_def: fixes \<Phi> :: "('a, 'b) lit list set" shows "local.wtPB \<Phi> \<equiv> Ball \<Phi> local.wtC"
simps_15: fixes v :: "formula"   and va :: "\<I>"   and vb :: "formula" shows "get_and_list (Until v va vb) = [Until v va vb]"
term_ok_eta_expand: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "typ_ok \<Theta> \<tau>" shows "term_ok \<Theta> (Abs \<tau> (f $ Bv 0))"
has_integral_neg_iff: fixes f :: "'a \<Rightarrow> 'b"   and k :: "'b"   and S :: "'a set" shows "((\<lambda>x. - f x) has_integral k) S = (f has_integral - k) S"
distrib_lincomb: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and rs :: "'a list"   and ms :: "'b list" assumes "RModuleHom R smult M smult' T"   and "set rs \<subseteq> R"   and "set ms \<subseteq> M" shows "T (scalar_mult.lincomb smult rs ms) = scalar_mult.lincomb smult' rs (map T ms)"
apartment_gallery_least_length: fixes X :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "X \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D" shows "SimplicialComplex.maxsimpchain X (C # (ARG_MIN length Cs. SimplicialComplex.maxsimpchain X (C # Cs @ [D])) @ [D])"
that: fixes S :: "'a set"   and N :: "'a set"   and N' :: "'a set" assumes "A = S \<union> N"   and "N \<subseteq> N'"   and "N' \<in> null_sets M"   and "S \<in> sets M" shows "thesis"
simps_3: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and t' :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.rank_invar (t # t' # bq) = (BinomialHeapStruc.rank t < BinomialHeapStruc.rank t' \<and> BinomialHeapStruc.rank_invar (t' # bq))"
arr_char: fixes f :: "'a \<times> 'a" shows "H.VV.arr f = (V.arr (fst f) \<and> V.arr (snd f) \<and> local.dom (fst f) = cod (snd f))"
top_on_sup: fixes o1 :: "'a st option"   and X :: "char list set"   and o2 :: "'a st option" assumes "top_on\<^sub>o o1 X"   and "top_on\<^sub>o o2 X" shows "top_on\<^sub>o (o1 \<squnion> o2) X"
lift_RealIntVal2_def: fixes f :: "int \<Rightarrow> int \<Rightarrow> int"   and g :: "real \<Rightarrow> real \<Rightarrow> real" shows "lift_RealIntVal2 f g \<equiv> map_int_pair (\<lambda>a b. IntVal (f a b)) (map_real_pair (\<lambda>a b. RealVal (g a b)) id)"
fresh_map_of: fixes x :: "'b"   and \<Gamma> :: "('b \<times> 'c) list"   and a :: "atom" assumes "x \<in> domA \<Gamma>"   and "a \<sharp> \<Gamma>" shows "a \<sharp> the (map_of \<Gamma> x)"
assms: shows "v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w"
cms\<^sub>2'_def: shows "cms\<^sub>2' = cms\<^sub>2[k := (c\<^sub>2', snd (cms\<^sub>1' ! k))]"
simps_1: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "General.filter_min_aux rel [] ys = ys"
mod_geq: fixes m :: "nat"   and n :: "nat" assumes "\<not> m < n" shows "m mod n = (m - n) mod n"
prefixBeforeLastMarkedIsPrefixBeforeLastLevel: fixes M :: "('b \<times> bool) list" assumes "markedElements M \<noteq> []" shows "prefixBeforeLastMarked M = prefixToLevel (currentLevel M - 1) M"
keys_eq_supp: fixes s :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys s = supp_fun (lookup s)"
involved_def: fixes kinvolved :: "'b \<Rightarrow> 'a set"   and oa :: "'b option" shows "Kernel.involved kinvolved oa \<equiv> case oa of None \<Rightarrow> {} | Some a \<Rightarrow> kinvolved a"
strict_mono_o: fixes r :: "'a \<Rightarrow> 'b"   and s :: "'c \<Rightarrow> 'a" assumes "strict_mono r"   and "strict_mono s" shows "strict_mono (r \<circ> s)"
cab: fixes n :: "nat" shows "c n \<in> {a..b}"
atLeastLessThan_empty_iff2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "({} = subset_mset.atLeastLessThan a b) = (\<not> a \<subset># b)"
terminates_wfE: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" assumes "terminates g" obtains R :: "('a \<times> 'a) set" where "wf R" and "\<And>s. fst g s \<Longrightarrow> (snd (snd g s), s) \<in> R"
Hoare_dgr_def: fixes P :: "I \<Rightarrow> 'a"   and D :: "I \<times> I \<Rightarrow> 'a \<Rightarrow> 'a"   and Q :: "I \<Rightarrow> 'a" shows "Hoare_dgr P D Q \<equiv> \<exists>X. (\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (i, u) j) \<and> P = \<Squnion>range X \<and> Q = \<Squnion>range X \<sqinter> - grd (step D)"
le_slength_plus: fixes xs :: "[:'a:]"   and n :: "Integer" assumes "xs \<noteq> \<bottom>"   and "n \<noteq> \<bottom>" shows "le\<cdot>n\<cdot>(slength\<cdot>xs + n) = TT"
fbd_bot_pres: shows "bot_pres bd\<^sub>\<F>"
domains_char: fixes f :: "'c" shows "C.domains f = (if C.arr f then {C.dom f} else {})"
borel_measurable_lebesgue_integral: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e"   and N :: "'d measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M restrict_space (measure_pmf B) (set_pmf B))" shows "(\<lambda>x. B.expectation (f x)) \<in> borel_measurable N"
max_input_Minus: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "max_input (Minus a1 a2) = max (max_input a1) (max_input a2)"
floor_real_def: fixes x :: "real" shows "\<lfloor>x\<rfloor> = (THE z. real_of_int z \<le> x \<and> x < real_of_int (z + 1))"
eventually_mem_existence_ivl: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and b :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "ll_on_open_it T f X"   and "b \<in> ll_on_open.existence_ivl T f X t0 x0" shows "\<forall>\<^sub>F x in at x0. b \<in> ll_on_open.existence_ivl T f X t0 x"
c1: shows "U\<^sup>\<dagger> \<in> carrier_mat n n"
wset_thread_okI: fixes ts :: "'d \<Rightarrow> ('e \<times> 'f \<Rightarrow>f nat) option"   and ws :: "'d \<Rightarrow> 'g wait_set_status option" assumes "\<And>t. ts t = None \<Longrightarrow> ws t = None" shows "wset_thread_ok ws ts"
subsets_2: shows "{m. coeff p' m \<noteq> (0::'a)} \<subseteq> S"
cpow_def: fixes r :: "('b \<times> 'b) set" shows "cpow r = |Pow (Field r)|"
refl: fixes x :: "'a" assumes "x \<in> D" shows "cmp x x = EQUAL"
plossless_gpv_absorb: fixes \<I>' :: "('i, 'j) \<I>"   and gpv :: "('i, 'j) distinguisher"   and \<I> :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter" assumes "plossless_gpv \<I>' gpv"   and "plossless_converter \<I>' \<I> conv"   and "\<I>' \<turnstile>g gpv \<surd>"   and "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>" shows "plossless_gpv \<I> (absorb gpv conv)"
cmp_simps'_2: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "src\<^sub>D (\<Phi>\<^sub>F (\<mu>, \<nu>)) = F.map\<^sub>0 (src\<^sub>C \<nu>)"
known_ptrs_defs: fixes h :: "(_) heap" shows "DocumentClass.known_ptrs h = (\<forall>ptr\<in>fset (object_ptr_kinds h). DocumentClass.known_ptr ptr)"
app_red_cong2: fixes e2 :: "exp"   and e2' :: "exp"   and e1 :: "exp" assumes "e2 \<longrightarrow>* e2'" shows "EApp e1 e2 \<longrightarrow>* EApp e1 e2'"
iso_def: fixes f :: "'a" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
ord_eq_le_eq_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a = b"   and "b \<le> c"   and "c = d" shows "a \<le> d"
OKI_def: fixes inp :: "'a \<Rightarrow> 'b option" shows "OKI inp = lift OK inp"
option_while'_None: fixes ra :: "'a"   and s :: "'s" assumes "(Some ra, None) \<in> option_while' (\<lambda>a. C a s) (\<lambda>a. B a s)" shows "(Some (ra, s), None) \<in> whileLoop_results C (\<lambda>a. gets_the (B a))"
summandFresh: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "P \<in> summands Q"   and "x \<sharp> Q" shows "x \<sharp> P"
ENC_Or: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" assumes "wf_formula n (FOr \<phi> \<psi>)" shows "lang n (ENC n (FOV (FOr \<phi> \<psi>))) - {[]} \<subseteq> lang n (ENC n (FOV \<phi>)) \<inter> lang n (ENC n (FOV \<psi>)) - {[]}"
impl_times: fixes a :: "'b N"   and b :: "'b N" shows "(a l\<rightarrow> b) * a = (b l\<rightarrow> a) * b"
PDataValue_def: shows "PDataValue = fst \<circ> Rep_pdata"
closed_hyperplane: fixes a :: "'b"   and b :: "real" shows "closed {x. a \<bullet> x = b}"
assms_4: shows "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"
A: shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A S"
assms_3: fixes ys :: "bool list" assumes "xs = ys @ [False]" shows "P"
WHILET_nofail_imp_rwof_nofail: fixes m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres"   and s :: "'a" assumes "nofail (m0 \<bind> WHILE\<^sub>T cond step)"   and "rwof m0 cond step s"   and "cond s" shows "nofail (step s)"
ign_ign_res: fixes P :: "'b assn" shows "P\<down>\<down> = P\<down>"
circline_set_I: fixes H :: "circline"   and z :: "complex_homo" assumes "on_circline H z" shows "z \<in> circline_set H"
r_nf_t_1: fixes y :: "nat"   and i :: "nat"   and x :: "nat" assumes "eval r_result [pdec1 y, pdec12 y, pdec22 y] \<down>= 0 \<or> pdec2 y \<noteq> prod_encode (i, x)" shows "eval r_nf_t [y, i, x] \<down>= 1"
min_int_poly_normalize: fixes x :: "'b" shows "normalize (min_int_poly x) = min_int_poly x"
cases: fixes a1 :: "nat"   and a2 :: "Regions.intv" assumes "Regions.valid_intv a1 a2" obtains d :: "nat" and c :: "nat" where "a1 = c__" and "a2 = Regions.intv.Const d__" and "0 \<le> d__" and "d__ \<le> c__"   | d :: "nat" and c :: "nat" where "a1 = ca__" and "a2 = Regions.intv.Intv d" and "0 \<le> d" and "d < ca__"   | c :: "nat" where "a1 = c" and "a2 = Regions.intv.Greater c"
assms: shows "B \<rightarrow>\<beta>\<^sup>* B'"
i_shrink_i_take_mult: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k" shows "f \<Down> (n * k) \<div> k = f \<div> k \<Down> n"
eq_inf_reflp: fixes w :: "'item stream" shows "w =\<^sub>I w"
tuple_simps_3: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.commutative_square C f g h k" shows "partial_magma.cod C (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k) = elementary_category_with_pullbacks.pbdom C prj0 f g"
collapse: fixes prod :: "'e \<times> 'f" shows "(fst prod, snd prod) = prod"
index_minus_mat_1: fixes i :: "nat"   and B :: "'a mat"   and j :: "nat"   and A :: "'a mat" assumes "i < dim_row B"   and "j < dim_col B" shows "(A - B) $$ (i, j) = A $$ (i, j) - B $$ (i, j)"
diff_inf_eq_sup: fixes 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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "minus a (inf b c) = plus a (sup (uminus b) (uminus c))"
RSpanI_lincomb_arb_len_coeffs: fixes rs :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M" shows "rs \<bullet>\<cdot> ms \<in> RSpan ms"
OclOr_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "X or Y \<equiv> not (not X and not Y)"
deleteAround'_def: fixes g :: "graph"   and v :: "nat"   and ps :: "(nat \<times> nat) list" shows "deleteAround' g v ps \<equiv> let fs = facesAt g v; vs = \<lambda>f. let n1 = f \<bullet> v; n2 = f \<bullet> n1 in if |vertices f| = 4 then [n1, n2] else [n1]; ws = concat (map vs fs) in removeKeyList ws ps"
bis_def: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set" shows "bis Retr \<equiv> \<Union> {theta. bisim Retr theta}"
Derives1_append_prefix: fixes v :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and w :: "symbol list"   and u :: "symbol list" assumes "Derives1 v i r w"   and "is_sentence u" shows "Derives1 (u @ v) (i + length u) r (u @ w)"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (f \<star> g \<star> h)"
parallel_HSem_ind: fixes P :: "(Vars.var \<Rightarrow> Value) \<Rightarrow> (Vars.var \<Rightarrow> Value) \<Rightarrow> bool"   and \<rho>\<^sub>1 :: "Vars.var \<Rightarrow> Value"   and \<Gamma>\<^sub>1 :: "(Vars.var \<times> exp) list"   and \<rho>\<^sub>2 :: "Vars.var \<Rightarrow> Value"   and \<Gamma>\<^sub>2 :: "(Vars.var \<times> exp) list" assumes "adm (\<lambda>\<rho>'. P (fst \<rho>') (snd \<rho>'))"   and "P \<bottom> \<bottom>"   and "\<And>y z. P y z \<Longrightarrow> P (\<rho>\<^sub>1 ++\<^bsub>domA \<Gamma>\<^sub>1\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>1 \<^bold>\<rbrakk>\<^bsub>y\<^esub>) (\<rho>\<^sub>2 ++\<^bsub>domA \<Gamma>\<^sub>2\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>2 \<^bold>\<rbrakk>\<^bsub>z\<^esub>)" shows "P (\<lbrace>\<Gamma>\<^sub>1\<rbrace>\<rho>\<^sub>1) (\<lbrace>\<Gamma>\<^sub>2\<rbrace>\<rho>\<^sub>2)"
sorted_less_set_eq: fixes xs :: "'a list" assumes "Sorted_Less.sorted xs" shows "xs = sorted_list_of_set (set xs)"
bounded_clinear_finite_dim: fixes f :: "'b \<Rightarrow> 'c" assumes "clinear f" shows "bounded_clinear f"
simps_1: fixes a :: "'b"   and w :: "nat \<Rightarrow> 'b" shows "(a ## w) 0 = a"
sup_idem: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) x = x"
fbox_loopI: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a \<Rightarrow> 'a set" assumes "P \<le> I"   and "I \<le> Q"   and "I \<le> |F] I" shows "P \<le> |LOOP F INV I ] Q"
typing_sndE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and P :: "'a trm"   and \<sigma> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> Snd P : \<sigma>" shows "\<exists>\<tau>. \<Gamma> \<turnstile> P : TPair \<tau> \<sigma>"
cong_mult_uneq': fixes d :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and e :: "'a" assumes "coprime d a"   and "[b \<noteq> c] (mod a)"   and "[d = e] (mod a)" shows "[b * d \<noteq> c * e] (mod a)"
is_acc_run_def: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" shows "b_graph.is_acc_run (degeneralize_ext m) r \<equiv> graph_defs.is_run (degeneralize_ext m) r \<and> b_graph.is_acc (degeneralize_ext m) r"
sign_simps_18: fixes aa :: "'d"   and ba :: "'d"   and ca :: "'d" shows "aa * (ba + ca) = aa * ba + aa * ca"
simps_1: shows "Tree2.set_tree \<langle>\<rangle> = {}"
assms_1: shows "xvec \<sharp>* \<Psi>"
assms_2: shows "\<not> bdd_below {u n / real n |n. N0 < n}"
"0_4": shows "list_all stateful_strand_step.is_Send (unlabel (transaction_send T))"
card_equiv_rel_eq_Bell: fixes A :: "'b set" assumes "finite A" shows "card {R. equiv A R} = Bell (card A)"
simps_2: fixes basepoint :: "'b Point"   and direction :: "'b Vector"   and more :: "'c" shows "direction \<lparr>basepoint = basepoint, direction = direction, \<dots> = more\<rparr> = direction"
uset_cong_aux: fixes U :: "(Ferrack.num \<times> int) list"   and x :: "real"   and bs :: "real list" assumes "\<forall>(t, n)\<in>set U. numbound0 t \<and> 0 < n" shows "(\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs U)) = (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (set U \<times> set U)"
simps_3: fixes NTDom :: "('g, 'i, 'j, 'h, 'k, 'l) Functor"   and NTCod :: "('g, 'i, 'j, 'h, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'h"   and more :: "'m" shows "NatTransMap \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NatTransMap"
5: shows "mat (row_length (A1 \<circ> A2)) (length (A1 \<circ> A2)) (A1 \<circ> A2)"
poly_mult_integral: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "polynomial K p1"   and "polynomial K p2"   and "poly_mult p1 p2 = []" shows "p1 = [] \<or> p2 = []"
induct: fixes P :: "'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b up\<^sub>d \<Rightarrow> bool"   and a0 :: "'b tree234"   and a1 :: "'b"   and a2 :: "'b tree234"   and a3 :: "'b"   and a4 :: "'b tree234"   and a5 :: "'b"   and a6 :: "'b up\<^sub>d" assumes "\<And>t1 a t2 b t3 c t4. P t1 a t2 b t3 c (T\<^sub>d t4)"   and "\<And>t1 a t2 b t3 c t4 d t5. P t1 a t2 b \<langle>t3, c, t4\<rangle> d (Up\<^sub>d t5)"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6. P t1 a t2 b \<langle>t3, c, t4, d, t5\<rangle> e (Up\<^sub>d t6)"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6 f t7. P t1 a t2 b \<langle>t3, c, t4, d, t5, e, t6\<rangle> f (Up\<^sub>d t7)"   and "\<And>a b c d f v. P a b c d \<langle>\<rangle> f (Up\<^sub>d v)" shows "P a0 a1 a2 a3 a4 a5 a6"
simps: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a1 :: "'b"   and a2 :: "'a list"   and a3 :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.path sourcenode targetnode valid_edge a1 a2 a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> CFG.valid_node sourcenode targetnode valid_edge n) \<or> (\<exists>n'' as n' a n. a1 = n \<and> a2 = a # as \<and> a3 = n' \<and> CFG.path sourcenode targetnode valid_edge n'' as n' \<and> valid_edge a \<and> sourcenode a = n \<and> targetnode a = n''))"
Union_least: fixes A :: "'e set set"   and C :: "'e set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C" shows "\<Union> A \<subseteq> C"
scaleR_add_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "real"   and x :: "'d"   and y :: "'d" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (plus x y) = plus (scaleR a x) (scaleR a y)"
scaleR_Pair: fixes r :: "real"   and a :: "'b"   and b :: "'c" shows "r *\<^sub>R (a, b) = (r *\<^sub>R a, r *\<^sub>R b)"
rev_ordsetmap_smap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and a :: "'b"   and b :: "'b" assumes "ComplexLikePoset less_eq less P"   and "a \<in> P"   and "b \<in> P"   and "order.greater_eq (ordering.poset_simplex_map less_eq less P b) (ordering.poset_simplex_map less_eq less P a)" shows "less_eq a b"
assms: shows "\<not> list_ex (is_LabelN l) A"
inter_line_line_comm: fixes C :: "Points"   and D :: "Points"   and A :: "Points"   and B :: "Points" assumes "C \<noteq> D" shows "Pascal_Property.inter (line A B) (line C D) = Pascal_Property.inter (line A B) (line D C)"
h2: shows "well_def_signs (length (qs1 @ qs2)) (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))"
height_split_min: fixes t :: "'c tree23"   and x :: "'c"   and t' :: "'c upD" assumes "split_min t = (x, t')"   and "0 < height t"   and "complete t" shows "hD t' = height t"
cost_nonneg: fixes R :: "'a set"   and i :: "nat" shows "0 \<le> cost R i"
recvmsg_simps_4: fixes P :: "'d \<Rightarrow> bool"   and ip :: "nat" shows "recvmsg P (\<not>unicast ip) = True"
aux4: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "applied_rule_rev Cp x (a # p) = \<lfloor>a\<rfloor>"   and "a \<notin> set p" shows "applied_rule_rev Cp x p = \<bottom>"
nxt_diverg: fixes b :: "nat" assumes "search b \<up>" shows "nxt b \<up>"
C3: shows "card {s. \<exists>x. s = senior x n \<and> \<not> sink (token_run s n)} \<le> card ((\<lambda>q. the (oldest_token q n)) ` ({token_run x n |x n. True} - Collect sink))"
fst_add: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "fst (x + y) = fst x + fst y"
eq_iff: fixes a :: "nat"   and b :: "nat" shows "(a = b) = (a dvd b \<and> b dvd a)"
assms_3: shows "s \<subseteq> r\<^sup>*"
IntD1: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
a_set_child_nodes_def: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes ptr children = invoke a_set_child_nodes_tups ptr children"
subgroup_Int: fixes I :: "'f set"   and G :: "('f, 'g) monoid_scheme"   and J :: "'f set" assumes "subgroup I G"   and "subgroup J G" shows "subgroup (I \<inter> J) G"
eq_refl: fixes x :: "'b Word.word"   and y :: "'b Word.word" assumes "x = y" shows "x \<le>s y"
min_gallery_simps_3: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set" shows "min_gallery (x # xs @ [y]) = (x \<noteq> y \<and> is_arg_min length (\<lambda>zs. gallery (x # zs @ [y])) xs)"
util_def: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "x \<succeq>[relation] y = (u y \<le> u x)"
wf2: shows "mat (row_length A2) (length A2) A2"
phiUses_def: fixes g :: "'a"   and n :: "'node" shows "i.phiUses g n \<equiv> \<Union>n'\<in>set (i.successors g n). \<Union>v'\<in>i.phiDefs g n'. snd ` Set.filter (\<lambda>(n'', v). n'' = n) (set (zip (i.predecessors g n') (the (phis2 g (n', v')))))"
ideD_3: fixes a :: "'d" assumes "D.ide a" shows "D.cod a = a"