hcju/isabelleps huggingface dataset:
assume_D: fixes \<sigma> :: "'d"   and P :: "'d \<Rightarrow> bool"   and M :: "'d \<Rightarrow> (bool \<times> 'd) option" assumes "\<sigma> \<Turnstile> _bind_SE _ (assume\<^sub>S\<^sub>E P) M" shows "\<exists>\<sigma>. P \<sigma> \<and> (\<sigma> \<Turnstile> M)"
absCase_Abs_subst: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and f :: "'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and x :: "'var" assumes "isInBar (xs, s)"   and "wls s X"   and "compatAbsSubst xs s f" shows "absCase xs s f (Abs xs x X) = f x X"
norm_le_l1: fixes x :: "'c" shows "norm x \<le> (\<Sum>b\<in>Basis. \<bar>inner x b\<bar>)"
InrI: fixes b :: "'c"   and B :: "'c set"   and A :: "'d set" assumes "b \<in> B" shows "Inr b \<in> A <+> B"
res_def: fixes range :: "'b set" shows "rorc.res range = RES (rorc.rnd_oracle range \<oplus>\<^sub>O rorc.rnd_oracle range) Map.empty"
per_supset_compare: fixes p1 :: "('a \<times> 'a) set"   and p2 :: "('a \<times> 'a) set"   and x1 :: "'a"   and x2 :: "'a" assumes "(p1, p2) \<in> local.per_supset_rel"   and "x1 \<in> Domain p2"   and "x2 \<in> Domain p2" shows "per_compare p1 x1 x2 = per_compare p2 x1 x2"
pdivides_imp_splitted: fixes p :: "'a list"   and q :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "q \<in> carrier (poly_ring R)"   and "q \<noteq> []"   and "splitted q"   and "p pdivides q" shows "splitted p"
iso_points_inv_map: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "inv_into \<V> bij_map ` \<V>' = \<V>"
linear_imp_differentiable: fixes f :: "'a \<Rightarrow> 'b"   and net :: "'a filter" assumes "linear f" shows "f differentiable net"
dprefI: fixes X :: "'a set"   and Y :: "'a set" assumes "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> Pd (Xd x)" shows "dpref X Y"
ccw'_mirror_point0: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real"   and w :: "real \<times> real" shows "ccw' (mirror_point x y) z w = ccw' y (mirror_point x z) (mirror_point x w)"
major: shows "\<tau>Exec_movesr ci P t es h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"
wens_mono: fixes A :: "'b set"   and B :: "'b set"   and F :: "'b program"   and act :: "('b \<times> 'b) set" assumes "A \<subseteq> B" shows "wens F act A \<subseteq> wens F act B"
le_cInf_iff: fixes S :: "'a multiset set"   and a :: "'a multiset" assumes "S \<noteq> {}"   and "subset_mset.bdd_below S" shows "(a \<subseteq># Inf S) = (\<forall>x\<in>S. a \<subseteq># x)"
meet_right: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_lower_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> y \<sqsubseteq>\<^bsub>L\<^esub> y"
intyness_simps_6: fixes num :: "num" shows "numeral num = real (numeral num)"
altc_optionT_def: shows "altc = altc_option altc"
vec_of_poly_n_poly_of_vec: fixes g :: "'a vec"   and n :: "nat" assumes "dim_vec g = n" shows "vec_of_poly_n (poly_of_vec g) n = g"
kruskal_spanning: fixes g :: "'a" shows "[symmetric g]   f := bot; h := g; WHILE h \<noteq> bot INV {kruskal_spanning_invariant f g h}  VAR {\<lambda>(e, f, h). card {x. regular x \<and> x \<le> - - h}}  DO e := minarc h; IF e \<le> - forest_components f THEN f := f \<sqinter> - (top * e * f\<^sup>T\<^sup>\<star>) \<squnion> (f \<sqinter> top * e * f\<^sup>T\<^sup>\<star>)\<^sup>T \<squnion> e  ELSE SKIP FI; h := h \<sqinter> - e \<sqinter> - e\<^sup>T OD   [spanning_forest f g]"
norm_fpxs_def: fixes f :: "'b fpxs" shows "norm f = (if f = 0 then 0 else 2 powr - real_of_rat (fpxs_val f))"
SeqConstP_imp_SeqTermP: fixes s :: "tm"   and k :: "tm"   and t :: "tm" shows "{SeqConstP s k t} \<turnstile> SeqTermP s k t"
glb_LEX_lexmin: fixes C :: "(nat \<Rightarrow> 'a) set" assumes "chain_on (LEX P) C (SEQ A)"   and "C \<noteq> {}" shows "glb (LEX P) C (lexmin C)"
ex_least_nat_less: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n"   and "\<not> P 0" shows "\<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (Suc k)"
consistent_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "consistent A b = (\<exists>x. is_solution x A b)"
divmod_BitM_2_eq: fixes m :: "num" shows "divmod (Num.BitM m) (num.Bit0 num.One) = (numeral m - 1, 1)"
Sup_pres_map_dual_var: fixes f :: "'c \<Rightarrow> 'd" shows "Sup_pres f = Inf_pres (\<partial>\<^sub>F f)"
S_def: shows "S \<equiv> \<lambda>x xa. \<Sum>m\<in>{0<..nat \<lfloor>log (real xa) (2 * real x)\<rfloor>}. \<lfloor>2 * real x / real xa ^ m\<rfloor> - 2 * \<lfloor>real x / real xa ^ m\<rfloor>"
two_pred_simps_2: shows "two_pred\<cdot>once = none"
decseq_const: fixes k :: "'c" shows "decseq (\<lambda>x. k)"
cont_if: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and c :: "bool" assumes "cont luba orda lubb ordb f"   and "cont luba orda lubb ordb g" shows "cont luba orda lubb ordb (\<lambda>x. if c then f x else g x)"
f4: fixes a :: "'a"   and aa :: "'a" shows "ad (ad (ad a \<cdot> aa)) = ad (ad aa + a)"
one_eq: shows "\<one>\<^bsub>units_of R\<^esub> = 1"
addsE: fixes b :: "'a"   and a :: "'a" assumes "b adds a" obtains k :: "'a" where "a = b + k"
meets_asym: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a" assumes "class.interval meets \<I>"   and "meets p q" shows "\<not> meets q p"
CK_nf_neg_def: fixes f :: "'b set \<Rightarrow> 'b set" shows "CK_nf_neg f = (\<exists>g. CK_nf_pos g \<and> f = C \<circ> g)"
get_def: shows "get \<equiv> \<lambda>s. ((s, s), False)"
knows_Spy_Notes: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" shows "knows Spy (Notes A X # evs) = (if A \<in> bad then insert X (knows Spy evs) else knows Spy evs)"
intro\<^sub>3: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "lc \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> lc'" shows "lc \<R>\<^sup>u\<^bsub>\<Gamma>\<^esub> lc'"
execn_to_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(OK x2 = OK y2) = (x2 = y2)"
extractable_def: fixes implem :: "chan \<Rightarrow> msg"   and K :: "msg set"   and I :: "msg set" assumes "semivalid_implem implem" shows "semivalid_implem.extractable implem K I \<equiv> {implem (Insec A B M) |A B M. implem (Insec A B M) \<in> I} \<union> {implem (Auth A B M) |A B M. implem (Auth A B M) \<in> I} \<union> {implem (Confid A B M) |A B M. implem (Confid A B M) \<in> I \<and> (A, B) \<in> broken K} \<union> {implem (Secure A B M) |A B M. implem (Secure A B M) \<in> I \<and> (A, B) \<in> broken K}"
assoc_closed: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "Arr f \<and> B.ide f"   and "Arr g \<and> B.ide g"   and "Arr h \<and> B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "Arr \<a>\<^sub>B[f, g, h]"
ht_hash_ht_distinct_in_dom_unique_value: fixes l :: "('a \<times> 'b) list list"   and k :: "'a" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l"   and "k \<in> dom (map_of (concat l))" shows "\<exists>!v. (k, v) \<in> set (concat l)"
le_refl: fixes x :: "'b" shows "x \<sqsubseteq> x"
swap_cs_Recv_Snapshot: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "cs e i = cs e' i"
cone_alt_def: fixes Vs :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> carrier_vec n" shows "gram_schmidt.cone n Vs = {x. \<exists>Ws. set Ws \<subseteq> Vs \<and> x \<in> gram_schmidt.cone_list n Ws}"
mk_linear_orders_distinct: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and r :: "'a list"   and C :: "'a set"   and B :: "'a set" assumes "f_range_on A f"   and "r \<in> mk_linear_orders f C B"   and "B \<subseteq> A" shows "distinct r"
bigo_iff: fixes g1 :: "'a \<Rightarrow> real"   and g2 :: "'a \<Rightarrow> real"   and h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "g1 \<in> G"   and "g2 \<in> G"   and "h1 \<in> H"   and "h2 \<in> H" shows "((\<lambda>x. g1 x * h1 x) \<in> O[F](\<lambda>x. g2 x * h2 x)) = (g1 \<in> o[F](g2) \<or> g1 \<in> \<Theta>[F](g2) \<and> h1 \<in> O[F](h2))"
dom_cod: fixes f :: "'c" assumes "Cop.arr f" shows "Cop.dom (Cop.cod f) = Cop.cod f"
in_closure_of: fixes x :: "'b"   and X :: "'b topology"   and S :: "'b set" shows "(x \<in> X closure_of S) = (x \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<in> S \<and> y \<in> T)))"
edge_succ_wv: shows "edge_succ HM (rev_H vw) = rev_H uw"
\<phi>_\<psi>: fixes b :: "'c"   and a :: "'c"   and x :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "C.ide b"   and "C.ide a"   and "x \<in> Hom.set (b, a)" shows "(S.UP \<circ> inl) (\<psi> (b, a) x) = x"
id_2: shows "mat\<^sub>h (D * C) = mat\<^sub>h D * mat\<^sub>h C"
listrel_eq_len: fixes xs :: "'g list"   and ys :: "'h list"   and r :: "('g \<times> 'h) set" assumes "(xs, ys) \<in> listrel r" shows "length xs = length ys"
holds_Stream: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and s :: "'a stream" shows "holds P (x ## s) = P x"
red_supsetE: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "A \<sqsupset>p B" obtains p :: "'t \<Rightarrow>\<^sub>0 'c" where "is_red A p" and "\<not> is_red B p"
set_contI: fixes f :: "'b \<Rightarrow> 'c set" assumes "\<And>Y. chain Y \<Longrightarrow> f (\<Squnion>i. Y i) = \<Union> (f ` range Y)" shows "cont f"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "suffixes (x # xs) = suffixes xs @ [x # xs]"
less_eq_ereal_def: fixes x :: "ereal"   and y :: "ereal" shows "(x \<le> y) = (x < y \<or> x = y)"
insert_DiffM2: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "M - {#x#} + {#x#} = M"
analyzed_is_all_analyzed_in: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set" shows "(\<forall>t\<in>M. analyzed_in t M) = analyzed M"
prj_in_homs_2: shows "\<guillemotleft>\<p>\<^sub>1[\<mu>.leg0, H\<nu>H\<pi>\<rho>.leg1] : H\<mu>H\<nu>H\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<mu>.apex\<guillemotright>"
islimpt_approachable_le: fixes x :: "'b"   and S :: "'b set" shows "x islimpt S = (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x \<le> e)"
axioms_2: fixes G :: "('c, 'd) graph" assumes "connected_graph G" shows "connected_graph_axioms G"
f_chain_in_rtrancl: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and R :: "('b \<times> 'b) set" assumes "m \<le> n"   and "\<forall>i\<in>{m..<n}. (f i, f (Suc i)) \<in> R" shows "(f m, f n) \<in> R\<^sup>*"
T_slow_median_def: fixes xs :: "'a list" shows "T_slow_median xs = T_slow_select ((length xs - 1) div 2) xs + 1"
DisjI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] \<or> [\<psi> in v]" shows "[\<phi> \<^bold>\<or> \<psi> in v]"
to_fract_uminus: fixes x :: "'b" shows "to_fract (- x) = - to_fract x"
assms_3: shows "\<KK> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
fundfoldpair_fundchamber_in_half_chamber_system_f: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs" shows "C0 \<in> f \<turnstile> \<C>"
assms_4: shows "\<forall>z\<in>set ys. r y z"
SeqE: fixes \<pi> :: "char list \<Rightarrow> com option"   and c1 :: "com"   and c2 :: "com"   and s1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and s3 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (c1;; c2, s1) \<Rightarrow> s3" obtains s\<^sub>2 :: "char list \<Rightarrow> int \<Rightarrow> int" where "\<pi>: (c1, s1) \<Rightarrow> s\<^sub>2" and "\<pi>: (c2, s\<^sub>2) \<Rightarrow> s3"
nth_enumerate_eq: fixes m :: "nat"   and xs :: "'g list"   and n :: "nat" assumes "m < length xs" shows "enumerate n xs ! m = (n + m, xs ! m)"
monom_Suc: fixes a :: "'b"   and n :: "nat" shows "monom a (Suc n) = pCons (0::'b) (monom a n)"
assms_1: shows "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u))"
get_root_node_root_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_root_node ptr \<rightarrow>\<^sub>r root" shows "root |\<in>| object_ptr_kinds h"
refl: fixes S :: "('a, 'b) eq_object_scheme"   and x :: "'a" assumes "equivalence S"   and "x \<in> carrier S" shows "x .=\<^bsub>S\<^esub> x"
Sup_eqI: fixes A :: "'a set"   and x :: "'a" assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"   and "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y" shows "Sup A = x"
Always_Constrains_weaken: fixes F :: "'b program"   and C :: "'b set"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> Always C"   and "F \<in> A Co A'"   and "C \<inter> B \<subseteq> A"   and "C \<inter> A' \<subseteq> B'" shows "F \<in> B Co B'"
elims_3: fixes x :: "sheap"   and xa :: "'a exp list" assumes "\<not> iconfs x xa" obtains sh :: "sheap" and e :: "'a exp" and es :: "'a exp list" where "x = sh" and "xa = e # es" and "\<not> (case val_of e of None \<Rightarrow> iconf sh e \<and> \<not> sub_RIs es | \<lfloor>v\<rfloor> \<Rightarrow> iconfs sh es)"
types_exist_lfilled_weak: fixes k :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and lfilled :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "Lfilled k lholed es lfilled"   and "\<S>\<bullet>\<C> \<turnstile> lfilled : ts _> ts'" shows "\<exists>t1s t2s \<C>' arb_label arb_return. \<S>\<bullet>\<C>\<lparr>label := arb_label, return := arb_return\<rparr> \<turnstile> es : t1s _> t2s"
mat_of_cols_carrier_2: fixes n :: "nat"   and vs :: "'a vec list" shows "dim_row (mat_of_cols n vs) = n"
well_def_sdp: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "well_def prog" shows "sub_distrib_pconj prog"
prems_5: shows "\<S>\<bullet>\<C> \<turnstile> [Local n i vls es] : ts _> ts'"
op_amd_empty_def: shows "op_amd_empty \<equiv> op_map_empty"
monoid_add_isom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "monoid_add_isom hom \<equiv> inj_monoid_add_hom hom \<and> bijective hom"
Tensor_left_id: fixes xs :: "int poly fract list list" shows "[[1]] \<otimes> xs = xs"
getFresh: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and V :: "'a set" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "finite V" shows "Generic_Syntax.getFresh var V \<in> var \<and> Generic_Syntax.getFresh var V \<notin> V"
elims_2: fixes x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "Derivation x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = a__" and "xa = []" and "xb = b__" and "a__ = b__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<exists>x. Derives1 a (fst d) (snd d) x \<and> Derivation x D b"
prems_4: shows "fcard G^V + length G^E = Suc k + fcard S"
less_eq_finite_2_def: fixes x :: "finite_2"   and y :: "finite_2" shows "(x \<le> y) = (x = y \<or> x < y)"
lunit_char_3: fixes a :: "'c" assumes "C.ide a" shows "\<exists>!f. \<guillemotleft>f : \<I>\<^sub>C \<otimes>\<^sub>C a \<rightarrow>\<^sub>C a\<guillemotright> \<and> \<I>\<^sub>C \<otimes>\<^sub>C f = (\<iota>\<^sub>C \<otimes>\<^sub>C a) \<cdot>\<^sub>C C.assoc' \<I>\<^sub>C \<I>\<^sub>C a"
abs_eq_iff: fixes x :: "'a \<Rightarrow> 'a"   and y :: "'a \<Rightarrow> 'a" shows "(abs_endofun x = abs_endofun y) = (x = y)"
left_idem: fixes a :: "'b"   and b :: "'b" shows "a + (a + b) = a + b"
prOp_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('d, 'c) sAbs"   and SEM :: "('index, 'bindex, 'd, 'sort, 'opSym, 'c, 'e) semDom_scheme" shows "prOp g gA SEM \<equiv> \<forall>delta inp binp val. wlsInp delta inp \<and> wlsBinp delta binp \<and> sWlsVal SEM val \<longrightarrow> g (Op delta inp binp) val = sOp SEM delta (lift (\<lambda>X. g X val) inp) (lift (\<lambda>A. gA A val) binp)"
scale_one: fixes x :: "'a" shows "(1::'a) * x = x"
it_to_list_def: fixes it :: "'h \<Rightarrow> ('i \<Rightarrow> bool) \<Rightarrow> ('j \<Rightarrow> 'j list \<Rightarrow> 'j list) \<Rightarrow> 'k list \<Rightarrow> 'g"   and s :: "'h" shows "it_to_list it s \<equiv> it s (\<lambda>_. True) (\<lambda>x l. l @ [x]) []"
methd_Object: fixes G :: "prog" assumes "wf_prog G" shows "methd G Object = table_of (map (\<lambda>(s, m). (s, Object, m)) Object_mdecls)"
sinh_real_eq_iff: fixes x :: "real"   and y :: "real" shows "(sinh x = sinh y) = (x = y)"
distinct_61: fixes x6 :: "'a OCL_Types.type"   and x7 :: "'a OCL_Types.type" shows "OrderedSet x6 \<noteq> Bag x7"
orelse_def: fixes x :: "'b option"   and y :: "'b option" shows "Quickcheck_Exhaustive.orelse x y = (case x of None \<Rightarrow> y | Some x' \<Rightarrow> Some x')"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and ustate_wrapper :: "('b, 'c) ustate_wrapper" shows "P (case ustate_wrapper of USW1 x \<Rightarrow> f1 x | USW2 x \<Rightarrow> f2 x) = ((\<forall>x1. ustate_wrapper = USW1 x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. ustate_wrapper = USW2 x2 \<longrightarrow> P (f2 x2)))"
lcons_def: shows "lcons \<equiv> \<Lambda> a llist. llist_abs\<cdot>(sinr\<cdot>(:up\<cdot>a, up\<cdot>llist:))"
slpa_split_1: 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 cs :: "'a list"   and as :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.same_level_path_aux kind get_return_edges cs as"   and "as = xs @ ys"   and "CFG.upd_cs kind cs xs = []" shows "CFG.same_level_path_aux kind get_return_edges cs xs"
openI: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" shows "open S"
weakPsiCongInputPres: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a" assumes "\<And>Tvec. length xvec = length Tvec \<Longrightarrow> \<Psi> \<rhd> P[xvec::=Tvec] \<approx> Q[xvec::=Tvec]" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<doteq> M\<lparr>\<lambda>*xvec N\<rparr>.Q"
verts3_def: fixes G :: "('b, 'c) pre_digraph" shows "verts3 G \<equiv> {v \<in> verts G. 2 < in_degree G v}"
NSsummable_def: fixes f :: "nat \<Rightarrow> real" shows "NSsummable f = (\<exists>s. f NSsums s)"
pass\<^sub>1_size: fixes hs :: "'b heap list" shows "size_hps (pass\<^sub>1 hs) = size_hps hs"
llength_eq_enat_lfiniteD: fixes xs :: "'b llist"   and n :: "nat" assumes "llength xs = enat n" shows "lfinite xs"
step'_def: shows "step' = Step (\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some (update S x (aval' e S))) (\<lambda>b. inv_bval' b True)"
enforce_spmf_K_False: fixes p :: "'d spmf" shows "enforce_spmf (\<lambda>_. False) p = return_pmf None"
fv_subst1_def: fixes s :: "('b \<times> form) list" shows "fv_subst1 s = \<Union> (fv ` snd ` set s)"
at_begin_fst_bwtn_Bk: fixes as :: "nat"   and am :: "nat list"   and q :: "nat"   and aaa :: "cell list"   and ires :: "cell list" assumes "at_begin_fst_bwtn (as, am) (q, aaa, []) ires" shows "at_begin_fst_bwtn (as, am) (q, aaa, [Bk]) ires"
Base_foo_def: shows "Base_foo = (foo_sig, \<lparr>access = Public, static = False, pars = [z], resT = Class Base, mbody = \<lparr>lcls = [], stmt = Return (!! z)\<rparr>\<rparr>)"
J_ivl: fixes x :: "'a" assumes "x \<in> D" shows "J0 \<le> J x u"
elims: fixes x :: "fml option"   and xa :: "fml option"   and y :: "fml option" assumes "Ando x xa = y" obtains \<phi> :: "fml" and \<psi> :: "fml" where "x = Afml \<phi>" and "xa = Afml \<psi>__" and "y = Afml (\<phi> && \<psi>__)"   | \<psi> :: "fml option" where "x = undeff" and "xa = \<psi>" and "y = undeff"   | v :: "fml" where "x = Afml v" and "xa = undeff" and "y = undeff"
fresh_shallow: fixes x :: "'a"   and e :: "Syntax.term" assumes "atom x \<sharp> e" shows "atom x \<sharp> \<lparr>e\<rparr>"
LeastI2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
derivations1_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.derivations1 \<NN> \<TT> \<RR> = {(u, v) |u v. CFG.derives1 \<NN> \<TT> \<RR> u v}"
omega_sum_unfold_coind: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<omega> \<le> (x\<^sup>\<star> \<cdot> y)\<^sup>\<omega> + (x\<^sup>\<star> \<cdot> y)\<^sup>\<star> \<cdot> x\<^sup>\<omega>"
weak_unit_iff_self_target: fixes b :: "'a" shows "weak_unit b = (b \<in> targets b)"
mask_out_sub_mask: fixes x :: "'b word"   and n :: "nat" shows "x AND NOT (mask n) = x - (x AND mask n)"
exhaust: fixes y :: "'b rule" obtains x1 :: "'b match_expr" and x2 :: "action" where "y = Rule x1 x2"
stutter_sampler_between: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat" assumes "stutter_sampler f \<sigma>"   and "f k \<le> n"   and "n < f (Suc k)" shows "\<sigma> n = \<sigma> (f k)"
intros_1: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "matched n [] n"