hcju/isabelleps huggingface dataset:
R: fixes s :: "real" assumes "s \<in> {t..u}" shows "discrete_evolution (euler_increment f) (t + h) t (x t) + (h\<^sup>2 / 2) *\<^sub>R f' (s, x s) (1, f s (x s)) \<in> R"
U2: fixes i :: "nat" assumes "i < length U" shows "(T ! i, U ! i) \<in> timpl_closure' c \<and> (S ! i, U ! i) \<in> timpl_closure' c"
equals_cod_if_value_is_ide: assumes "C.ide a" shows "\<alpha>.map = \<alpha>.G.map"
span_subset_spanI: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> ideal B" shows "ideal A \<subseteq> ideal B"
hcomp_iso_in_hom: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c"   and \<nu> :: "'c"   and h :: "'c"   and k :: "'c" assumes "\<guillemotleft>\<mu> : f \<cong>\<^sub>C g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<cong>\<^sub>C k\<guillemotright>"   and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" shows "\<guillemotleft>\<nu> \<star>\<^sub>C \<mu> : h \<star>\<^sub>C f \<cong>\<^sub>C k \<star>\<^sub>C g\<guillemotright>"
map_removeAll_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and xs :: "'a list" assumes "inj_on f (insert x (set xs))" shows "map f (removeAll x xs) = removeAll (f x) (map f xs)"
IH_2: fixes u :: "'a set list"   and w :: "nat \<Rightarrow> 'a set" assumes "suspendable \<xi>'" shows "(u \<frown> w) \<Turnstile>\<^sub>n \<xi>' = w \<Turnstile>\<^sub>n \<xi>'"
ptensor_mat_carrier: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "ptensor_mat m1 m2 \<in> carrier_mat d0 d0"
sets_SUP_cong: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and N :: "'d \<Rightarrow> 'e measure" assumes "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (Sup (M ` I)) = sets (Sup (N ` I))"
approx_add_mono1: fixes b :: "'a star"   and c :: "'a star"   and d :: "'a star" assumes "b \<approx> c" shows "d + b \<approx> d + c"
succss_closed: fixes X :: "'a set" assumes "nodeAbs ` succss X \<subseteq> nodeAbs ` X"   and "X \<subseteq> {x. isNode x}" shows "nodeAbs ` reachable X = nodeAbs ` X"
in_mono: fixes A :: "'h set"   and B :: "'h set"   and x :: "'h" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
call_dtermsl_empty: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> dtermsl p" shows "not_call p"
mk_moeibus_id: fixes a :: "complex" shows "mk_moebius a 0 0 a = id_moebius"
Enc_Hash: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg" shows "Enc X Y \<noteq> Hash X'"
ibind_strict_1: fixes bind :: "'a \<Rightarrow> (unit \<Rightarrow> 'b) \<Rightarrow> 'b"   and return :: "unit \<Rightarrow> 'a"   and ASSERT :: "bool \<Rightarrow> 'a"   and ASSUME :: "bool \<Rightarrow> 'a"   and f :: "unit \<Rightarrow> 'b" assumes "generic_Assert bind return ASSERT ASSUME" shows "bind bot f = bot"
insert_compr: fixes a :: "'d"   and B :: "'d set" shows "insert a B = {x. x = a \<or> x \<in> B}"
f_the_inv_into_f: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and y :: "'e" assumes "inj_on f A"   and "y \<in> f ` A" shows "f (the_inv_into A f y) = y"
assms_3: shows "\<forall>x\<in>X. \<not> isConst (I x)"
Sretr_sym_trans: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "sym theta \<and> trans theta" shows "trans (Sretr theta)"
simps_2: fixes a :: "'b"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.remove1Prio a (t # bq) = (if SkewBinomialHeapStruc.prio t = a then bq else t # SkewBinomialHeapStruc.remove1Prio a bq)"
write1_def: fixes c :: "'e \<Rightarrow> 'c"   and a :: "'d \<Rightarrow> 'e"   and P :: "('c, 'd) action" shows "c`!`a \<rightarrow> P \<equiv> oPrefix (\<lambda>A. c (a A)) P"
is_rbt_rbtreeify: fixes kvs :: "('a \<times> 'c) list" assumes "sorted (map fst kvs)"   and "distinct (map fst kvs)" shows "is_rbt (rbtreeify kvs)"
learn_cpE_2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option" assumes "learn_cp \<psi> U s"   and "f \<in> U" shows "\<exists>i n\<^sub>0. \<psi> i = f \<and> (\<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i)"
CS_def: fixes c :: "com" shows "CS c = Collecting.lfp (step UNIV) c"
global_def: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool" shows "global P \<equiv> \<lambda>(\<sigma>, uu_). P \<sigma>"
weakCongSimE: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Rel :: "('e \<times> ('d, 'e, 'f) psi \<times> ('d, 'e, 'f) psi) set"   and Q :: "('d, 'e, 'f) psi"   and Q' :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.weakCongSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P Rel Q"   and "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> Q (\<tau> \<prec> Q')" obtains P' :: "('d, 'e, 'f) psi" where "(P, P') \<in> {(P, P'). env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<tau> \<prec> P')}\<^sup>+" and "(\<Psi>, P', Q') \<in> Rel"
multiplicity_prime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p" shows "factorial_semiring.multiplicity times one p p = 1"
proper_sub_design_axioms: shows "proper_sub_design \<U> \<A> \<V> \<B>"
one_closed: shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
simps_1: shows "Common.enat_of_option None = \<infinity>"
isomorphic_sym: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<simeq> G\<^sub>2" shows "G\<^sub>2 \<simeq> G\<^sub>1"
evalCPS_a_def: fixes l :: "lambda" shows "\<aPR> l = (let ve = {}.; \<beta> = Map.empty; f = \<aA> (L l) \<beta> ve in \<aF>\<cdot>(Discr (the_elem f, [{AStop}], ve, \<abinit>)))"
exhaust: fixes y :: "('a, 'b) IType" obtains x1 :: "'a" where "y = IObj x1"   | x2 :: "'b" where "y = IMor x2"   | x3 :: "bool" where "y = IBool x3"
lookup_tensors_ht_l0': fixes j :: "nat"   and r0 :: "nat"   and "is" :: "nat list"   and M :: "nat" assumes "j < r0"   and "is \<lhd> [M, M]" shows "lookup (tensors_from_net (witness' r0 [M]) $ j) is = (if is = [j, j] then 1 else 0)"
subNEq: fixes s :: "real" assumes "s \<in> {0..<1}"   and "s = 0 \<or> Re (g s) = Re z"   and "\<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z"   and "finite_ReZ_segments (subpath 0 s g) z" shows "P"
pre: shows "(l = PAodv-:6 \<or> l = PAodv-:7 \<or> l = PRrep-:0 \<or> l = PRrep-:1) \<and> sip (\<sigma> i) \<noteq> dip (\<sigma> i) \<longrightarrow> dip (\<sigma> i) \<in> kD (rt (\<sigma> (sip (\<sigma> i)))) \<and> dsn (\<sigma> i) \<le> nsqn (rt (\<sigma> (sip (\<sigma> i)))) (dip (\<sigma> i)) \<and> (nsqn (rt (\<sigma> (sip (\<sigma> i)))) (dip (\<sigma> i)) = dsn (\<sigma> i) \<longrightarrow> the (dhops (rt (\<sigma> (sip (\<sigma> i)))) (dip (\<sigma> i))) \<le> hops (\<sigma> i) \<or> the (flag (rt (\<sigma> (sip (\<sigma> i)))) (dip (\<sigma> i))) = Aodv_Basic.inv)"
sourcesD_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "a \<in> weak_composition.sources V H \<mu>" shows "partial_magma.ide V a"
gcsort_add_input: fixes xs :: "'c list" shows "add_inv (length xs) (0, [length xs], xs)"
hoare_tripleD_2: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set"   and \<sigma> :: "heap option"   and r :: "'d" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P"   and "run c (Some h) \<sigma> r" shows "(the_state \<sigma>, new_addrs h as (the_state \<sigma>)) \<Turnstile> Q r"
assms: fixes v :: "i" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
case_prodI2': fixes p :: "'e \<times> 'f"   and c :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "\<And>a b. (a, b) = p \<Longrightarrow> c a b x" shows "(case p of (a, b) \<Rightarrow> c a b) x"
not_less_zero: fixes n :: "'q multiset" shows "\<not> n \<subset># {#}"
order_antisym: fixes x :: "'d"   and y :: "'d" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
getlocs_setlocs_lemma: fixes s :: "state"   and s' :: "state"   and Y :: "vname"   and k :: "nat" shows "loc (setlocs s (loc s')[Y::=k]) = loc (s'[Y::=k])"
simps_14: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (Imp p q) = (Ifm bbs bs p \<longrightarrow> Ifm bbs bs q)"
WT_gpv_inline1_1: fixes out :: "'call'"   and rpv :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "Inr (out, rpv, rpv') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>" shows "out \<in> outs_\<I> \<I>'"
contra_subsetD: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A \<subseteq> B"   and "c \<notin> B" shows "c \<notin> A"
h_plus_OUT_eq_IN: fixes i :: "nat"   and x :: "'v" assumes "enum_v (Suc i) = x" shows "d_OUT (h_plus (Suc i)) x = d_IN (h_plus i) x"
processed_alt: fixes a :: "'e"   and b :: "'e"   and xs :: "'e list"   and ps :: "('e \<times> 'e) list" shows "processed (a, b) xs ps = (a \<in> set xs \<and> b \<in> set xs \<and> (a, b) \<notin>\<^sub>p set ps)"
Abs_swap_fresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "good X"   and "fresh xs x' X" shows "Abs xs x X = Abs xs x' (X #[x' \<and> x]_xs)"
content_gcd_content: fixes f :: "'a poly"   and g :: "'a poly" shows "content (gcd f g) = gcd (content f) (content g)"
bounded_insert: fixes x :: "'c"   and S :: "'c set" shows "bounded (insert x S) = bounded S"
insert_map_inject2: fixes i :: "nat"   and x :: "'c"   and f :: "nat \<Rightarrow> 'c"   and y :: "'c"   and g :: "nat \<Rightarrow> 'c" assumes "insert_map i x f = insert_map i y g" shows "f = g"
is_path_undir_split_distinct: fixes G :: "('c, 'd) graph"   and v :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and v' :: "'c"   and a :: "'c"   and w :: "'d"   and b :: "'c" assumes "valid_graph G"   and "is_path_undir G v p v'"   and "(a, w, b) \<in> set p \<or> (b, w, a) \<in> set p" shows "\<exists>p' p'' u u'. is_path_undir G v p' u \<and> is_path_undir G u' p'' v' \<and> length p' < length p \<and> length p'' < length p \<and> (u \<in> {a, b} \<and> u' \<in> {a, b}) \<and> (a, w, b) \<notin> set p' \<and> (b, w, a) \<notin> set p' \<and> (a, w, b) \<notin> set p'' \<and> (b, w, a) \<notin> set p''"
d: shows "0 < a - ennreal (Norm N (x' - x))"
lset_lzip: fixes xs :: "'c llist"   and ys :: "'d llist" shows "lset (lzip xs ys) = {(lnth xs n, lnth ys n) |n. enat n < min (llength xs) (llength ys)}"
tendsto_null_sum: fixes I :: "'c set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e"   and F :: "'d filter" assumes "\<And>i. i \<in> I \<Longrightarrow> ((\<lambda>x. f x i) \<longlongrightarrow> (0::'e)) F" shows "((\<lambda>i. sum (f i) I) \<longlongrightarrow> (0::'e)) F"
circle_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real" shows "circle i j d \<equiv> R2 i j \<and> circle_axioms d"
square_free_m_smultI: fixes f :: "int poly"   and ci :: "int"   and c :: "int" assumes "p.square_free_m f"   and "p.M (ci * c) = 1" shows "p.square_free_m (Polynomial.smult c f)"
indecomposable_imp_type: fixes \<gamma> :: "V"   and X :: "bool \<Rightarrow> V set" assumes "indecomposable \<gamma>"   and "\<And>b. ordertype (X b) VWF \<le> \<gamma>"   and "\<And>b. small (X b)"   and "\<And>b. X b \<subseteq> ON"   and "elts \<gamma> \<subseteq> \<Union> (range X)" shows "\<exists>b. ordertype (X b) VWF = \<gamma>"
uminus_lit_swap: fixes a :: "'a literal"   and b :: "'a literal" shows "(a = - b) = (- a = b)"
bind_game_le_inv_q: fixes \<A> :: "nat \<times> nat \<Rightarrow> (nat \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" shows "spmf (rivest_commit.bind_game \<A>) True \<le> 1 / real q"
left_refines_AC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<sqsubseteq> (a \<Squnion> b)"
preserves_reflects_arr: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.arr (Span.src f) = Span.arr f"
extendf_dterm_sem_continuous: fixes f' :: "('sf + 'sz, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "dfree f'"   and "is_interp I" shows "continuous_on UNIV (\<lambda>x. dterm_sem (extendf I x) f' \<nu>)"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s1)) \<and> t2 = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s2))"
freeIn_substInfinitesimalQuadratic_fm: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and F :: "atom fm" assumes "var \<notin> vars a"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "var \<notin> vars d" shows "freeIn var (substInfinitesimalQuadratic_fm var a b c d F)"
bounded_clinear_zero: shows "bounded_clinear (\<lambda>x. 0::'b)"
lossless_return_pmf_iff: fixes x :: "'c option" shows "lossless_spmf (return_pmf x) = (x \<noteq> None)"
map_eq_Cons_conv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and y :: "'c"   and ys :: "'c list" shows "(map f xs = y # ys) = (\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys)"
ex_sup: fixes x :: "'b"   and y :: "'b" shows "\<exists>sup. is_sup x y sup"
eq: fixes x :: "'a" assumes "x \<in> S"   and "x \<noteq> a" shows "f x = g x"
quotientP_def: fixes P :: "'b \<Rightarrow> bool"   and \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "P /// \<phi> \<equiv> S2P (Collect P // P2R \<phi>)"
tree_map_of_works: fixes pres :: "('b \<times> 'a) list"   and x :: "'b" shows "lookup (M.tree_map_of' RBT_Set.empty pres) x = AList_Upd_Del.map_of pres x"
intros_1: fixes M :: "'a measure"   and N :: "'a measure" assumes "space M \<subset> space N" shows "M \<le> N"
is_weak_ranking_code: fixes xs :: "'b set list" shows "is_weak_ranking xs = is_weak_ranking_aux {} xs"
dvd_abs_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "'b"   and k :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "dvd.dvd times m (abs k) = dvd.dvd times m k"
sub2_divisor: fixes k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub2 k l (r, A) = (r', A')"   and "l < k"   and "k \<le> dim_row A"   and "l < dim_col A"   and "r \<noteq> (0::'a)" shows "r' \<noteq> (0::'a)"
isometry_on_complete_image: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f"   and "complete X" shows "complete (f ` X)"
points_index_pair_rep_num: fixes B :: "'b set multiset"   and x :: "'b"   and y :: "'b" assumes "\<And>b. b \<in># B \<Longrightarrow> x \<in> b" shows "int (B index {x, y}) = B rep y"
eval_eqI: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "VPar t u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "\<lbrace>t\<rbrace> = \<lbrace>u\<rbrace>"
Pxxy_def: fixes x :: "'b \<times> 'b \<times> 'c" shows "Pxxy x = (if x \<in> (\<lambda>x. (X x, X x, Y x)) ` space M then prob ((\<lambda>x. (X x, X x, Y x)) -` {x} \<inter> space M) else 0)"
map_inv_f_f: fixes f :: "'c \<Rightarrow> 'd option"   and x :: "'c"   and y :: "'d" assumes "inj_on f (dom f)"   and "f x = Some y" shows "map_inv f y = Some x"
braun_ssa_CFG_SSA_Transformed: shows "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry defs uses defs' uses' phis' var"
succ_finite: fixes G :: "('a, 'b) graph"   and v :: "'a" assumes "finite (edges G)" shows "finite (succ G v)"
r_3: shows "(flow0 x \<circ> s \<circ> r) \<longlonglongrightarrow> l"
adopt_node_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> adopt_node_locs parent owner_document document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
skl1_inv4I: fixes x :: "skl1_state" assumes "\<And>Rb A B gnx Ni. 0 < signalsResp x (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) \<Longrightarrow> guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress x Rb = Some {xny, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni" shows "x \<in> skl1_inv4"
plus_prod_def: fixes m1 :: "'a \<times> 'b"   and m2 :: "'a \<times> 'b" shows "m1 + m2 \<equiv> (fst m1 + fst m2, snd m1 + snd m2)"
mkFin: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "\<not> final f" shows "P g (makeFaceFinal f g)"
tmmul_nb0: fixes t :: "tm"   and i :: "poly" assumes "tmbound0 t" shows "tmbound0 (tmmul t i)"
A5_DAcc_level0: shows "DAcc level0 sA5 = {}"
C1: fixes \<Psi>' :: "'b"   and Q :: "('a, 'b, 'c) psi" shows "(\<Psi>', Q, Q) \<in> Rel"
sup_norm_pres: fixes H :: "'c set"   and h :: "'c \<Rightarrow> real"   and c :: "('c \<times> real) set set"   and M :: "('c \<times> real) set set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "graph H h = \<Union> c"   and "M = norm_pres_extensions E p F f"   and "c \<in> chains M" shows "\<forall>x\<in>H. h x \<le> p x"
completeness_nat: fixes Cs :: "fterm literal set set" assumes "finite Cs"   and "\<forall>C\<in>Cs. finite C"   and "\<forall>F G. \<not> eval\<^sub>c\<^sub>s F G Cs" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
filterlim_realpow_sequentially_gt1: fixes x :: "'a" assumes "1 < norm x" shows "filterlim ((^) x) at_infinity sequentially"
eSubstAbs_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and A :: "'d"   and ys :: "'varSort"   and eY :: "'c withERR"   and y :: "'var" assumes "\<forall>us s. \<not> igWlsAbs MOD (us, s) A" shows "eSubstAbs MOD ys eY y (OK A) = ERR"
neq_Nil_conv: fixes xs :: "'e list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
assms_2: shows "B \<in> decomps\<^sub>e\<^sub>s\<^sub>t S N \<I>"
simps_3: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and v :: "nat" shows "TS M2 M1 \<Omega> V m (Suc (Suc v)) = TS M2 M1 \<Omega> V m (Suc v) \<union> C M2 M1 \<Omega> V m (Suc (Suc v))"
upd_cs_snoc_Call: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and cs :: "'edge list"   and as :: "'edge list" assumes "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "upd_cs cs (as @ [a]) = a # upd_cs cs as"
IH: fixes y :: "V"   and z :: "V" assumes "y \<in> elts ya__"   and "lift x y = lift x z" shows "y = z"
orderE: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" obtains "a = min a b"
subset_set_code: fixes xs :: "'c list"   and ys :: "'c list" shows "(set xs \<subseteq> set ys) = list_all (\<lambda>x. x \<in> set ys) xs"
add_0: fixes a :: "'a" shows "(0::'a) + a = a"
eq_cbox: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(cbox a b = cbox c d) = (cbox a b = {} \<and> cbox c d = {} \<or> a = c \<and> b = d)"
assms_2: shows "\<forall>n. f n \<in> S"
hermitean_mk_circline: fixes A :: "real"   and B :: "complex"   and D :: "real" shows "hermitean (cor A, B, cnj B, cor D)"
prems_4: shows "ops = before @ (oid, ref) # op_list @ [oper]"
distinct_25: fixes x3 :: "fresh_t"   and x5 :: "msg" shows "Nonce x3 \<noteq> Hash x5"
step_imp_unlabeled_step: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(a, b, c) \<in> ars" shows "(a, c) \<in> unlabel ars"
MGFn_Call: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and e :: "expr"   and ps :: "expr list"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and mn :: "mname"   and pTs' :: "ty list" assumes "\<forall>C sig. G,A\<turnstile>{=:n} In1l (Methd C sig)\<succ> {G\<rightarrow>}"   and "G,A\<turnstile>{=:n} In1l e\<succ> {G\<rightarrow>}"   and "G,A\<turnstile>{=:n} In3 ps\<succ> {G\<rightarrow>}"   and "wf_prog G" shows "G,A\<turnstile>{=:n} In1l ({accC,statT,mode}e\<cdot>mn( {pTs'}ps))\<succ> {G\<rightarrow>}"
indep_sets_cong: fixes I :: "'c set"   and J :: "'c set"   and F :: "'c \<Rightarrow> 'a set set"   and G :: "'c \<Rightarrow> 'a set set" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> F i = G i" shows "indep_sets F I = indep_sets G J"
proj_Cons: fixes qs :: "'a list"   and S :: "'a set"   and a :: "'a"   and as :: "'a list" assumes "Lxy qs S = a # as"   and "a \<in> S" obtains pre :: "'a list" and suf :: "'a list" where "qs = pre @ [a] @ suf" and "\<And>x. x \<in> S \<Longrightarrow> x \<notin> set pre" and "Lxy suf S = as"
ordLeq_finite_Field: fixes r :: "('a \<times> 'a) set"   and s :: "('b \<times> 'b) set" assumes "r \<le>o s"   and "finite (Field s)" shows "finite (Field r)"
simps_1: fixes i :: "nat" shows "prefixes i 0 = [i]"
le_lt_comparator_of_1: shows "le_of_comp comparator_of = (\<le>)"
intro: fixes x :: "'b \<Longrightarrow> 'c" assumes "weak_lens x"   and "wb_lens_axioms x" shows "wb_lens x"
init\<pi>_def: shows "init\<pi> \<equiv> Map.empty"
assms_2: shows "AE \<omega> in M. \<not> P \<omega>"
independent_length_eq_dimension: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and n :: "nat"   and E :: "'c set"   and Us :: "'c list" assumes "ring R"   and "subfield K R"   and "ring.dimension R n K E"   and "ring.independent R K Us"   and "set Us \<subseteq> E" shows "(length Us = n) = (ring.Span R K Us = E)"
rel_funD: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and x :: "'a"   and y :: "'b" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
copermIL: fixes ys :: "('a, 'b, 'c) event list"   and xs :: "('a, 'b, 'c) event list"   and z :: "('a, 'b, 'c) event llist" assumes "ys <~~> xs"   and "t = prepend xs z" shows "coperm (prepend ys z) t"
distinct_7: fixes x51 :: "bexp"   and x52 :: "com" shows "com.SKIP \<noteq> WHILE x51 DO x52"
red_unionI1: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "punit.red F p q" shows "punit.red (F \<union> G) p q"