hcju/isabelleps huggingface dataset:
implies_e1: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>"
simps_12: fixes v :: "nat" shows "pratt2_max [v]\<^sub>2 \<top>\<^sub>2 = \<top>\<^sub>2"
sup_ge2: fixes y :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd" shows "y \<preceq> x \<oplus> y"
simps_3: fixes x3 :: "nat"   and y3 :: "nat" shows "(Greater x3 = Greater y3) = (x3 = y3)"
wlsAbs_cases: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and phiAbs :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool" assumes "wlsAbs (xs, s) A"   and "\<And>x X. \<lbrakk>isInBar (xs, s); wls s X\<rbrakk> \<Longrightarrow> phiAbs (xs, s) (Abs xs x X)" shows "phiAbs (xs, s) A"
count_prime_factorization: fixes x :: "'a"   and p :: "'a" shows "count (prime_factorization x) p = (if prime p then multiplicity p x else 0)"
simps_5: fixes \<sigma> :: "nat \<Rightarrow> val"   and v :: "expr"   and w :: "expr" shows "expr_sem \<sigma> <v ,  w> = expr_sem \<sigma> v \<bind> (\<lambda>x. expr_sem \<sigma> w \<bind> (\<lambda>y. return_val <|x, y|>))"
sigma_algebra_disjoint_iff: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "sigma_algebra \<Omega> M = (algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M))"
cod_dom: fixes f :: "'a" assumes "arr f" shows "cod (local.dom f) = local.dom f"
Lr_def: fixes F'' :: "'a filter"   and g'' :: "'a \<Rightarrow> real" shows "O[F''](g'') = {f. \<exists>c>0. \<forall>\<^sub>F x in F''. norm (f x) \<le> c * norm (g'' x)}"
lfilter_fixp_parallel_induct: fixes P :: "('c llist \<Rightarrow> 'c llist) \<Rightarrow> ('d llist \<Rightarrow> 'd llist) \<Rightarrow> bool"   and Pb :: "'c \<Rightarrow> bool"   and Pa :: "'d \<Rightarrow> bool" assumes "ccpo.admissible (prod_lub llist.lub_fun llist.lub_fun) (rel_prod llist.le_fun llist.le_fun) (\<lambda>x. P (fst x) (snd x))"   and "P (\<lambda>_. lSup {}) (\<lambda>_. lSup {})"   and "\<And>f g. P f g \<Longrightarrow> P (\<lambda>xs. case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if Pb x then LCons x (f xs') else f xs') (\<lambda>xs. case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if Pa x then LCons x (g xs') else g xs')" shows "P (lfilter Pb) (lfilter Pa)"
simps_4: fixes \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and v :: "'b list"   and i :: "nat"   and \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "sat \<sigma> v i (Or \<phi> \<psi>) = (sat \<sigma> v i \<phi> \<or> sat \<sigma> v i \<psi>)"
futures_def: fixes P :: "'e process"   and xs :: "'e list" shows "futures P xs \<equiv> {(ys, Y). (xs @ ys, Y) \<in> failures P}"
wi_hom_add: fixes a :: "int"   and b :: "int" shows "word_of_int a + word_of_int b = word_of_int (a + b)"
that: fixes ys :: "'a list" assumes "t = name $$ ys"   and "matchs xs ys = Some env" shows "thesis"
covector_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "covector x" shows "covector (x * y)"
ODord_le_def: fixes X :: "'c Order set"   and Y :: "'c Order set" shows "X \<sqsubseteq> Y = (X = Y \<or> X \<sqsubset> Y)"
l8_12: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X C D A B"
path_defs_5: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" shows "path_component S x y \<equiv> \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
msetext_dersh_irrefl_from_trans: fixes xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>z\<in>set xs. \<forall>y\<in>set xs. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> msetext_dersh gt xs xs"
sign_simps_4: fixes ae :: "'c"   and be :: "'c"   and ce :: "'c" shows "ae * be * ce = ae * (be * ce)"
open_halfspace_Re_lt: fixes b :: "real" shows "open {z. Re z < b}"
weakly_preserves_hcomp: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "C.isomorphic (F f \<star>\<^sub>C F g) (F (f \<star>\<^sub>B g))"
prj_simps_1: shows "C.seq f_src.prj\<^sub>1 \<p>\<^sub>1[src.leg0 \<cdot> f_src.prj\<^sub>0, src.leg1]"
max_wp_SetPC: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> supp (P s) \<Longrightarrow> maximal (wp (p a))"   and "\<And>s. sum (P s) (supp (P s)) = 1" shows "maximal (wp (SetPC p P))"
no_var_interference: fixes c :: "command" shows "wr_com c = {}"
T_insort_length: fixes x :: "'c"   and xs :: "'c list" shows "T_insort x xs \<le> length xs + 1"
list_wset_relD: fixes a :: "'a list"   and b :: "'c set"   and R :: "('a \<times> 'c) set" assumes "(a, b) \<in> \<langle>R\<rangle>list_wset_rel" shows "(set a, b) \<in> \<langle>R\<rangle>set_rel"
trss_add_context: fixes s :: "'f list"   and c :: "'f list multiset"   and w :: "('g, 'h) label list"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and ce :: "'f list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)"   and "mon_w fg w \<inter> mon_c fg ce = {}" shows "((s, c + ce), w, s', c' + ce) \<in> trcl (trss fg)"
ide_in_hom_1: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "\<guillemotleft>f : S.src f \<rightarrow>\<^sub>S S.trg f\<guillemotright>"
b_subset_N: fixes k :: "nat"   and j :: "nat"   and i :: "nat" shows "b k (j, i) \<subseteq> N"
zero_hom_0_def: fixes hom :: "'b \<Rightarrow> 'c" shows "zero_hom_0 hom \<equiv> zero_hom hom \<and> zero_hom_0_axioms hom"
igSwapAllIPresIGWlsAllSTR_imp_igSwapAllIPresIGWlsAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igSwapAllIPresIGWlsAllSTR MOD" shows "FixSyn.igSwapAllIPresIGWlsAll Delta MOD"
elts_graph_aux_def: shows "elts_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>elts_aux. x1 = Zero \<and> x2 = {}) \<or> (\<exists>elts_aux k. x1 = Var k \<and> x2 = {}) \<or> (\<exists>elts_aux t u. x1 = Eats t u \<and> x2 = insert u (((\<lambda>uu_. uu_) \<circ> elts_aux) t) \<and> p t (((\<lambda>uu_. uu_) \<circ> elts_aux) t)))"
summable_geometric_iff: fixes c :: "'a" shows "summable ((^) c) = (norm c < 1)"
mcont_fun_lub_apply: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "mcont luba orda (fun_lub lubb) (fun_ord ordb) f = (\<forall>x. mcont luba orda lubb ordb (\<lambda>y. f y x))"
nprv_impE2: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla"   and \<chi> :: "'fmla" assumes "nprv F (imp \<phi> \<psi>)"   and "nprv F \<phi>"   and "\<chi> = \<psi> \<or> \<chi> \<in> F"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "\<chi> \<in> fmla" shows "nprv F \<chi>"
w: shows "w (g, length ts) + sum_list (map weight (scf_list (scf (g, length ts)) ts)) \<le> weight (Fun f [])"
filter_sup_left_dist_inf: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b set"   and G :: "'b set"   and H :: "'b set" assumes "class.distrib_lattice inf less_eq less sup"   and "ord.filter less_eq F"   and "ord.filter less_eq G"   and "ord.filter less_eq H" shows "semilattice_inf.filter_sup inf less_eq F (G \<inter> H) = semilattice_inf.filter_sup inf less_eq F G \<inter> semilattice_inf.filter_sup inf less_eq F H"
simps_1: fixes f :: "'fsym" shows "TE_wtFsym (Oldf f) = wtFsym f"
N_half_def: shows "N_half = (2::'a) ^ (length rs - 3)"
Limsup_mono: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and F :: "'c filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Limsup F f \<le> Limsup F g"
homeomorphic_map_eq: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "homeomorphic_map X Y g"
assms_2: shows "{x \<in> space N. P x} \<in> sets N"
perm_aux_fold: fixes pi :: "('e \<times> 'e) list"   and x :: "'d" shows "perm_aux pi x = pi \<bullet> x"
sym: fixes x :: "'a"   and y :: "'a" assumes "R x y"   and "x \<in> S"   and "y \<in> S" shows "R y x"
indep_var_eq: fixes S :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and T :: "'b measure"   and Y :: "'a \<Rightarrow> 'b" shows "indep_var S X T Y = ((random_variable S X \<and> random_variable T Y) \<and> indep_set (sigma_sets (space M) {X -` A \<inter> space M |A. A \<in> sets S}) (sigma_sets (space M) {Y -` A \<inter> space M |A. A \<in> sets T}))"
existence_ivl_zero: fixes x0 :: "'a" assumes "x0 \<in> X" shows "0 \<in> rev.existence_ivl0 x0"
cyclic_on_obtain_list_succ: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "cyclic_on f S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = x" obtains xs :: "'b list" where "f = list_succ xs" and "set xs = S" and "distinct xs"
is_natural_2: fixes f :: "'b" assumes "arr f" shows "\<rr> (cod f) \<cdot> R f = \<rr> f"
f_def_1: shows "bij_betw f P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('a \<times> ('b, 'c) stateful_strand_step) list" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
iT_Plus_neg_Max_singleton: fixes I :: "nat set" assumes "finite I"   and "I \<noteq> {}" shows "I \<oplus>- Max I = {0}"
can_occur_Recv: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and i :: "nat"   and p :: "'d"   and q :: "'d"   and s :: "'e"   and s' :: "'e"   and m :: "'f"   and c :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "distributed_system.can_occur channel trans send recv (Recv i p q s s' m) c" shows "states c p = s \<and> channel i = Some (q, p) \<and> (\<exists>xs. msgs c i = Msg m # xs) \<and> recv i p q s s' m"
LL_def: shows "LL = real_of_int (q * (\<Prod>j = 1..n. of_int (a j))) * (\<Sum>j. real_of_int (b (j + n + 1)) / real_of_int (a (j + n + 1)))"
simps_2: fixes v :: "int" shows "benv_in_d (DI v) = {}"
abs_eq: fixes x :: "('a, 'b, 'c, 'd) Tree set['a]" shows "Conj\<^sub>\<alpha> (map_bset abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tConj x)"
plus_adds_ppI_1: fixes t :: "'a"   and v :: "'t"   and s :: "'a" assumes "t adds\<^sub>p v"   and "s adds\<^sub>p v \<ominus> t" shows "s + t adds\<^sub>p v"
terminal_arrow_from_functor_axioms: shows "terminal_arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (G x) x (\<epsilon> x)"
prv_eqv_prv_rev: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv \<phi>"   and "bprv (eqv \<chi> \<phi>)" shows "bprv \<chi>"
distinct_29: fixes x3 :: "nat"   and x71 :: "nat"   and x72 :: "msg" shows "Nonce x3 \<noteq> Crypt x71 x72"
comm_drop_exp: fixes m :: "nat"   and u :: "'b list"   and r :: "'b list" assumes "m \<noteq> 0"   and "u \<cdot> r \<^sup>@ m = r \<^sup>@ m \<cdot> u" shows "r \<cdot> u = u \<cdot> r"
closed_closure_eq: fixes a :: "'a llist set" assumes "a closed" shows "closure a = a"
Gimp: fixes s :: "'c" assumes "G' s" shows "G (\<phi>' s)"
isoI: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.inverse_arrows f g" shows "Maps.iso f"
obt_4: shows "\<forall>x\<in>S. 0 \<le> u x"
matpow_mono: fixes B :: "(('a, 'n) Finite_Cartesian_Product.vec, 'n) Finite_Cartesian_Product.vec"   and n :: "nat"   and i :: "'n"   and j :: "'n" assumes "\<And>i j. A $h i $h j \<le> B $h i $h j" shows "matpow A n $h i $h j \<le> matpow B n $h i $h j"
stabilizer_def: fixes m :: "'a set" shows "conjG.stabilizer m = {g \<in> carrier G. conjugation_action (p ^ a) g m = m}"
funion_upper1: fixes A :: "'x fset"   and B :: "'x fset" shows "A |\<subseteq>| A |\<union>| B"
set1_FGco_def: fixes x :: "('a, 'c, 'd, ('c, 'e, 'f, 'g, 'h, 'i, 'j) G, 'f, 'k, 'h, 'l, 'm, 'n) F" shows "set1_FGco x = set1_F x"
l_remove_shadow_root_get_tag_name_def: fixes get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_remove_shadow_root_get_tag_name get_tag_name_locs remove_shadow_root_locs \<equiv> \<forall>w element_ptr shadow_root_ptr h h' r ptr. w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_tag_name_locs ptr \<longrightarrow> r h h'"
strip_annotate: fixes f :: "nat \<Rightarrow> 'a"   and c :: "com" shows "strip (annotate f c) = c"
memory_write_def: fixes asi :: "word8"   and addr :: "word32"   and byte_mask :: "word4"   and data_w32 :: "word32"   and state :: "'b sparc_state" shows "memory_write asi addr byte_mask data_w32 state \<equiv> let result = memory_write_asi asi addr byte_mask data_w32 state in case result of None \<Rightarrow> None | Some s1 \<Rightarrow> Some (store_barrier_pending_mod False s1)"
FG_def: shows "FG = supp f \<union> supp g"
interior_image_subset: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "inj f"   and "\<And>x. isCont f x" shows "interior (f ` S) \<subseteq> f ` interior S"
l_delete_shadow_root_get_tag_name_get_tag_name_locs: shows "l_delete_shadow_root_get_tag_name get_tag_name_locs"
servK_authentic_bis: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "B \<noteq> Tgs"   and "evs \<in> kerbV" shows "\<exists>A ST. Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace>, ST\<rbrace> \<in> set evs"
path_connected_path_image: fixes p :: "real \<Rightarrow> 'b" assumes "path p" shows "path_connected (path_image p)"
lexord_partial_trans: fixes xs :: "'a list"   and r :: "('a \<times> 'a) set"   and ys :: "'a list"   and zs :: "'a list" assumes "\<And>x y z. \<lbrakk>x \<in> set xs; (x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r"   and "(xs, ys) \<in> lexord r"   and "(ys, zs) \<in> lexord r" shows "(xs, zs) \<in> lexord r"
simps_3: fixes t :: "('c, 'd) SkewBinomialTree"   and t' :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" shows "rank_skew_invar (t # t' # bq) = (rank t \<le> rank t' \<and> rank_invar (t' # bq))"
and_eq_not_not_or: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "and a b = not (or (not a) (not b))"
dw_closed_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "relation.dw_closed r A = (\<forall>a\<in>A. \<forall>b. r a b \<longrightarrow> b \<in> A)"
prj_chine_assoc_3: shows "Prj\<^sub>0\<^sub>0 \<cdot> chine_assoc = Prj\<^sub>0"
restrict_to_halfspace_def: fixes sctn :: "'b sctn"   and X :: "'b set" shows "restrict_to_halfspace sctn X = CHECK (\<lambda>_. ()) (normal sctn \<in> set Basis_list \<or> - normal sctn \<in> set Basis_list) \<bind> (\<lambda>_. let y = pstn sctn; b = normal sctn in ivl_rep X \<bind> (\<lambda>(i, s). let i' = if b \<le> (0::'b) then i + min (y - i \<bullet> b) 0 *\<^sub>R b else i; s' = if (0::'b) \<le> b then s + min (y - s \<bullet> b) 0 *\<^sub>R b else s in RETURN ({i'..s'} :::\<^sub>i \<langle>i_rnv\<rangle>\<^sub>ii_ivl)))"
filter: shows "filter_mset ((\<subseteq>) {x, y}) (filter_mset ((\<in>) x) A) = {#a \<in># A. x \<in> a \<and> {x, y} \<subseteq> a#}"
foldl_rule: fixes I :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and l0 :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" assumes "I \<sigma>0 [] l0"   and "\<And>l1 l2 x \<sigma>. \<lbrakk>l0 = l1 @ x # l2; I \<sigma> l1 (x # l2)\<rbrakk> \<Longrightarrow> I (f \<sigma> x) (l1 @ [x]) l2" shows "I (foldl f \<sigma>0 l0) l0 []"
cases: fixes a :: "'b set" assumes "fsigma a" obtains F :: "nat \<Rightarrow> 'b set" where "a = \<Union> (range F)" and "\<And>n. closed (F n)"
prod_ord_pairD: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "comparator.oalist_inv_raw comp ys"   and "comparator.prod_ord_pair comp P xs ys"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "P k (comparator.lookup_pair comp xs k) (comparator.lookup_pair comp ys k)"
assms_2: assumes "ip \<in> vD rt" shows "P rt ip"
param_Array: fixes R :: "('a \<times> 'b) set" shows "(Array, Array) \<in> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>array_rel"
Q_P: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "Graph_Invariants E P Q"   and "Q a" shows "P a"
Orbit_Relation_memE: fixes G :: "('c \<Rightarrow> 'c) set"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "transformation_group G S"   and "(x, y) \<in> transformation_group.Orbit_Relation G S" obtains \<alpha> :: "'c \<Rightarrow> 'c" where "\<alpha> \<in> G" and "x \<in> S" and "y = \<alpha> x"
abs_eq: fixes xa :: "nat \<Rightarrow>\<^sub>0 nat"   and x :: "'a" shows "monom xa x = MPoly (monomial x xa)"
R'_wf: shows "is_pref_profile (R(i := Ri'))"
bind_SE'_def: fixes f :: "'a \<Rightarrow> ('c \<times> 'a) option"   and g :: "'a \<Rightarrow> ('b \<times> 'a) option" shows "f ;- g = _bind_SE _ f g"
is_processT8: fixes P :: "'b process" shows "\<forall>s X. s \<in> D P \<longrightarrow> (s, X) \<in> F P"
singular_chain_boundary_alt: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain (Suc p) X c" shows "singular_chain p X (chain_boundary (Suc p) c)"
path_entry_loop: fixes G :: "nat graph_rec"   and p :: "nat list"   and n :: "nat" assumes "cfg_doms G"   and "path_entry (g_E G) p n"   and "n \<in> set p" shows "\<exists>p'. path_entry (g_E G) p' n \<and> n \<notin> set p'"
nb: fixes v :: "'a" assumes "v \<in> dom s" shows "s v = (\<phi>\<^sub>S\<inverse> \<Psi> \<phi>\<^sub>S \<Psi> s) v"
poly2_def: fixes p :: "'c poly poly"   and x :: "'c"   and y :: "'c" shows "poly2 p x y = poly (poly p [:y:]) x"
D_eq_D: fixes x :: "'s"   and y :: "'t"   and x' :: "'s" assumes "R x y"   and "R x' y" shows "distr (K.T x) (stream_space (count_space UNIV)) (smap f) = distr (K.T x') (stream_space (count_space UNIV)) (smap f)"
bij_betw_hom_Hom_3: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a"   and "ide b"   and "f \<in> hom a b" shows "MkArr (Dom a) (Dom b) (Map f) = f"
topology_generated_by_topspace: fixes S :: "'c set set" shows "topspace (topology_generated_by S) = \<Union> S"
"5_1": shows "f T2 \<subseteq> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t T1"
dgrad_dummy_def: shows "dgrad_dummy = (SOME d. dickson_grading d)"
absolutely_integrable_linear: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and h :: "'d \<Rightarrow> 'e" assumes "f absolutely_integrable_on s"   and "bounded_linear h" shows "h \<circ> f absolutely_integrable_on s"
int: fixes m :: "'b"   and n :: "'b" assumes "{m, n} \<subseteq> N \<and> m \<noteq> n" shows "B1 m \<inter> B1 n = {}"
ur: shows "\<exists>!x. root_cond (abs_int_poly (reflect_poly p), inverse r, inverse l) x"
image_mset_cong2: fixes M :: "'a multiset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and N :: "'a multiset" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x"   and "M = N" shows "image_mset f M = image_mset g N"
valid_bind'_cong: fixes f :: "'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e"   and g :: "'e \<Rightarrow> ('d \<times> 'e) option"   and M :: "'e \<Rightarrow> (bool \<times> 'e) option" assumes "f \<sigma> = g \<sigma>" shows "(\<sigma> \<Turnstile> f ;- M) = (\<sigma> \<Turnstile> g ;- M)"
transfer_action: fixes s :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state"   and a :: "'m node_action"   and s' :: "'s net_state" assumes "s \<in> reachable (closed (pnet np n)) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n"   and "(s, a, s') \<in> automaton.trans (closed (pnet np n))" obtains \<sigma>' :: "nat \<Rightarrow> 'g" and \<zeta>' :: "'l net_state" where "((\<sigma>, \<zeta>), a, \<sigma>', \<zeta>') \<in> automaton.trans (oclosed (opnet onp n))" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')"
B: shows "geodesic_segment_between (geodesic_subsegment G x s t) (geodesic_segment_param G x s) (geodesic_segment_param G x t)"
elims_2: fixes x :: "'b ugraph"   and xa :: "'b"   and xb :: "('b \<times> 'b) list"   and xc :: "'b" assumes "path x xa xb xc" obtains g :: "'b ugraph" and u :: "'b" and v :: "'b" where "x = ga__" and "xa = u__" and "xb = []" and "xc = v" and "u__ = v"   | g :: "'b ugraph" and u :: "'b" and e :: "'b \<times> 'b" and ps :: "('b \<times> 'b) list" and w :: "'b" where "x = g" and "xa = u" and "xb = e # ps" and "xc = w" and "\<exists>v. e = (u, v) \<and> e \<in> edges g \<and> path g v ps w"
l_cancel_one': fixes x :: "nat \<Rightarrow> 'a"   and a :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "a \<in> carrier P" shows "(x = x \<oplus>\<^bsub>P\<^esub> a) = (a = \<zero>\<^bsub>P\<^esub>)"
subcls_C_Object: fixes P :: "'b prog"   and C :: "char list"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool" assumes "is_class P C"   and "wf_prog wf_md P" shows "P \<turnstile> C \<preceq>\<^sup>* Object"