hcju/isabelleps huggingface dataset:
case_4: fixes f1 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'd"   and f2 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and f4 :: "'e \<Rightarrow> 'd"   and f5 :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'd"   and x4 :: "'e" shows "(case Snapshot x4 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f4 x4"
1: shows "f \<otimes> f = f \<otimes> \<I>"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> [$C v1, $C v2, $e] : ts _> ts'"
lunit_char_3: fixes a :: "'d" assumes "D.ide a" shows "\<exists>!f. \<guillemotleft>f : \<I>\<^sub>1 \<otimes>\<^sub>D a \<rightarrow>\<^sub>D a\<guillemotright> \<and> \<I>\<^sub>1 \<otimes>\<^sub>D f = (\<iota>\<^sub>1 \<otimes>\<^sub>D a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<I>\<^sub>1, \<I>\<^sub>1, a]"
comp_reduce: fixes f :: "'a"   and g :: "'a"   and k :: "'a"   and h :: "'a" assumes "f \<cdot> g = k"   and "C.seq f g"   and "C.seq g h" shows "f \<cdot> g \<cdot> h = k \<cdot> h"
distinct_87: fixes x31 :: "Syntax.var"   and x32 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Lam x31 x32 \<noteq> Unroll x14"
simps_7: fixes I :: "\<I>"   and \<phi> :: "'a MFOTL.formula" shows "MFOTL.future_reach (formula.Next I \<phi>) = MFOTL.future_reach \<phi> + right I + 1"
someI: fixes P :: "'h \<Rightarrow> bool"   and x :: "'h" assumes "P x" shows "P (Eps P)"
leg_simps_1: shows "C.arr HfHgh.dom.leg0"
r_psi_def: shows "r_psi = Cn 2 r_ifz [recf.Id 2 1, recf.Id 2 0, Cn 2 r [Cn 2 r_dec [recf.Id 2 1]]]"
Obj_pbeta_subst: fixes L :: "char list set"   and t :: "sterm"   and t' :: "sterm"   and f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type"   and f' :: "Label \<Rightarrow> sterm option"   and l :: "Label" assumes "finite L"   and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> t' = \<sigma>[s,p] t'')"   and "Obj f T => Obj f' T"   and "lc (Obj f T)"   and "body t" shows "Obj (f(l \<mapsto> t)) T => Obj (f'(l \<mapsto> t')) T"
mult_to_poly: fixes c :: "'b"   and d :: "'b" shows "[:c:] * [:d:] = [:c * d:]"
B_def: shows "B \<equiv> \<Union>s\<in>A\<^sup>\<star>. {infsuff A s}"
stutter_equiv_trans: fixes \<rho> :: "nat \<Rightarrow> 'f"   and \<sigma> :: "nat \<Rightarrow> 'f"   and \<tau> :: "nat \<Rightarrow> 'f" assumes "\<rho> \<approx> \<sigma>"   and "\<sigma> \<approx> \<tau>" shows "\<rho> \<approx> \<tau>"
conc_star_star: fixes A :: "'a list set" shows "star A @@ star A = star A"
closed_segment_iv_subset_domain: fixes t :: "real" assumes "t \<in> T" shows "{t0--t} \<subseteq> T"
cltn2_compose_left_abs: fixes M :: "((real, 3) vec, 3) vec"   and A :: "cltn2" assumes "invertible M" shows "cltn2_compose (cltn2_abs M) A = cltn2_abs (M ** cltn2_rep A)"
state_cfg_corec: fixes s :: "'a"   and d :: "'b \<Rightarrow> 'a pmf"   and c :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'b" shows "state (cfg_corec s d c x) = s"
gamma3_smooth: shows "gamma3 piecewise_C1_differentiable_on {0..1}"
names_def: fixes H :: "(('c, 'b) fm list \<times> 'b) set" shows "names H \<equiv> {uu_. \<exists>ps i. uu_ = i \<and> (ps, i) \<in> H}"
c: fixes x :: "real" assumes "x \<in> {0..1} - s" shows "D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b)) = F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
Pd_refl: fixes x :: "'x"   and d :: "'d" assumes "x \<in> Field (Pd d)" shows "(x, x) \<in> Pd d"
Rg: shows "g \<in> {..n + m} \<rightarrow> carrier R"
obj_def': fixes a :: "'b" shows "D.obj a = (D.arr a \<and> trg\<^sub>D a = a)"
L2_set_def: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "L2_set f A = sqrt (\<Sum>i\<in>A. (f i)\<^sup>2)"
mset_zero_iff: fixes x :: "'k list" shows "(mset x = {#}) = (x = [])"
subnetsOfAdr_def: fixes a :: "'b" shows "subnetsOfAdr a = {x. a \<sqsubset> x}"
compatibleI_with_proj: fixes G :: "'a pair_pre_digraph"   and H :: "'a pair_pre_digraph" shows "compatible (with_proj G) (with_proj H)"
Gateway_L8: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "\<forall>j\<le>2 * d. lose (t + j) = [False]"   and "ack t = [sending_data]"   and "\<forall>t3\<le>t + d. a t3 = []" shows "\<forall>x\<le>d + d. ack (t + x) = [sending_data]"
not_zero_le_neg_numeral: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq zero (uminus (numeral.numeral one plus n))"
left_options: fixes l :: "game zet"   and r :: "game zet" shows "left_options (Game l r) = l"
distinct_mset_remdups_mset_id: fixes C :: "'a multiset" assumes "distinct_mset C" shows "remdups_mset C = C"
dom_simp: fixes f :: "'b \<times> 'b" assumes "B.VxV.arr f" shows "B.VxV.dom f = (B.dom (fst f), B.dom (snd f))"
functor_comp_def_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and G :: "('b, 'h, 'd, 'i, 'f, 'j, 'k) Functor_scheme" shows "F \<approx>>;;; G \<equiv> Functor F \<and> Functor G \<and> CatDom G = CatCod F"
val_Zp_not_equal_imp_notequal_3: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "val_Zp x \<noteq> val_Zp y" shows "x \<oplus> y \<noteq> \<zero>"
has_finite_products: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "nat set" assumes "category_with_finite_products C"   and "finite I" shows "category.has_products C I"
strict_implies_order: fixes b :: "'k multiset"   and a :: "'k multiset" assumes "b \<subset># a" shows "b \<subseteq># a"
has_integral_open_interval: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and a :: "'a"   and b :: "'a" shows "(f has_integral y) (box a b) = (f has_integral y) (cbox a b)"
ith_typing: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and es :: "exp list"   and h :: "exp"   and t :: "exp list"   and Cs :: "nat list" assumes "CT;\<Gamma> \<turnstile>+ (es @ h # t) : Cs" shows "CT;\<Gamma> \<turnstile> h : Cs ! length es"
trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "border a b"   and "border b c" shows "border a c"
sup_commute: fixes sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "'x"   and y :: "'x" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
simps_2: fixes n :: "nat" shows "C (Suc n) = D n div A n"
asym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubset> x"   and "x \<in> X"   and "y \<in> X" shows "False"
MakeCatCompDef: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" shows "f \<approx>>\<^bsub>MakeCat C\<^esub> g = f \<approx>>\<^bsub>C\<^esub> g"
that: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "{u \<in> V. real_of_int m \<le> u x} = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n"   and "normalized M" shows "thesis"
disj_matrices_add_le_x: fixes A :: "'a matrix"   and B :: "'a matrix"   and C :: "'a matrix" assumes "disj_matrices A B"   and "disj_matrices B C" shows "(B + A \<le> C) = (A \<le> C \<and> B \<le> 0)"
hash: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" assumes "merge_on A h bo m" shows "bo \<le> vimage2p h h (=)"
HcompNml_assoc: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "Nml t"   and "Nml u"   and "Nml v"   and "Src t = Trg u"   and "Src u = Trg v" shows "(t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> v"
assms: shows "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr h = Some h'"
F: shows "\<F> g' = {f\<^sub>1} \<union> set (replace f [f\<^sub>2] (faces g))"
GOAL_INDICATION_def: fixes uu :: "'a" shows "\<paragraph>uu \<equiv> True"
ennreal_sqrt2: shows "ennreal (sqrt 2) = ennsqrt 2"
2: shows "\<forall>y\<in>A. (minim A, y) \<in> r"
mono_update: fixes a :: "'b"   and a' :: "'b"   and S :: "'b Abs_State.st"   and S' :: "'b Abs_State.st"   and x :: "char list" assumes "a \<sqsubseteq> a'"   and "S \<sqsubseteq> S'" shows "update S x a \<sqsubseteq> update S' x a'"
IH: fixes y :: "nat"   and c :: "int" assumes "y < l"   and "c \<noteq> 0"   and "\<bar>c\<bar> < p ^ y"   and "poly_mod.dvdm (p ^ y) u (smult c f)" shows "u dvdm f"
rec_t_eq_steps: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'a"   and stp :: "nat" assumes "tm_wf0 tp" shows "trpl_code (steps0 (Suc 0, Bk \<up> l, <lm>) tp stp) = rec_exec rec_conf [code tp, bl2wc (<lm>), stp]"
arc_def: fixes e :: "'a \<times> 'a"   and uv :: "'a \<times> 'a" shows "G.arc e uv \<equiv> e \<in> parcs G \<and> fst e = fst uv \<and> snd e = snd uv"
induct: fixes P :: "'a aexp \<Rightarrow> 'a aexp \<Rightarrow> bool"   and a0 :: "'a aexp"   and a1 :: "'a aexp" assumes "\<And>l1 l2. P (L l1) (L l2)"   and "\<And>l1 v. P (L l1) (V v)"   and "\<And>l1 v va. P (L l1) (Plus v va)"   and "\<And>l1 v va. P (L l1) (Minus v va)"   and "\<And>l1 v va. P (L l1) (Times v va)"   and "\<And>v1 l1. P (V v1) (L l1)"   and "\<And>v1 v2. P (V v1) (V v2)"   and "\<And>v1 v va. P (V v1) (Plus v va)"   and "\<And>v1 v va. P (V v1) (Minus v va)"   and "\<And>v1 v va. P (V v1) (Times v va)"   and "\<And>e1 e2 l2. P (Plus e1 e2) (L l2)"   and "\<And>e1 e2 v2. P (Plus e1 e2) (V v2)"   and "\<And>e1 e2 e1' e2'. \<lbrakk>P e1 e1'; P e2 e2'\<rbrakk> \<Longrightarrow> P (Plus e1 e2) (Plus e1' e2')"   and "\<And>e1 e2 v va. P (Plus e1 e2) (Minus v va)"   and "\<And>e1 e2 v va. P (Plus e1 e2) (Times v va)"   and "\<And>e1 e2 e1' e2'. \<lbrakk>P e1 e1'; P e2 e2'\<rbrakk> \<Longrightarrow> P (Minus e1 e2) (Minus e1' e2')"   and "\<And>e1 e2 e1' e2'. P (Minus e1 e2) (Times e1' e2')"   and "\<And>e1 e2 v. P (Minus e1 e2) (L v)"   and "\<And>e1 e2 v. P (Minus e1 e2) (V v)"   and "\<And>e1 e2 v va. P (Minus e1 e2) (Plus v va)"   and "\<And>e1 e2 e1' e2'. \<lbrakk>P e1 e1'; P e2 e2'\<rbrakk> \<Longrightarrow> P (Times e1 e2) (Times e1' e2')"   and "\<And>e1 e2 v. P (Times e1 e2) (L v)"   and "\<And>e1 e2 v. P (Times e1 e2) (V v)"   and "\<And>e1 e2 v va. P (Times e1 e2) (Plus v va)"   and "\<And>e1 e2 v va. P (Times e1 e2) (Minus v va)" shows "P a0 a1"
V0_in_FIN: shows "V\<^sub>0 \<in> FIN"
cong_mod_left: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo b a) c a = unique_euclidean_semiring.cong modulo b c a"
interchange_same_rows: fixes A :: "(('b, 'c) vec, 'd) vec"   and a :: "'d" shows "interchange_rows A a a = A"
AlwaysD: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> Always A" shows "Init F \<subseteq> A \<and> F \<in> Stable A"
X_\<G>\<F>_Y_\<F>\<G>: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set" assumes "X \<subseteq> subformulas\<^sub>\<mu> \<phi>"   and "Y \<subseteq> subformulas\<^sub>\<nu> \<phi>"   and "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<Sigma>\<^sub>1)"   and "\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<Pi>\<^sub>1)" shows "X \<subseteq> \<G>\<F> \<phi> w \<and> Y \<subseteq> \<F>\<G> \<phi> w"
proj_fm: fixes J :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'c \<Rightarrow> 'b"   and x :: "'b"   and m :: "'b \<Rightarrow> 'd" assumes "function_to_finmap J f f'"   and "x \<in> J" shows "(function_to_finmap.fm J f f' m)\<^sub>F (f x) = m x"
linorder_chain: fixes Y :: "'b set" shows "Complete_Partial_Order.chain (\<le>) Y"
1: shows "2 * sqrt (1 - (cmod \<langle>z,w\<rangle>)\<^sup>2 / (\<langle>z\<rangle>\<^sup>2 * \<langle>w\<rangle>\<^sup>2)) = 2 * sqrt ((\<langle>z\<rangle>\<^sup>2 * \<langle>w\<rangle>\<^sup>2 - (cmod \<langle>z,w\<rangle>)\<^sup>2) / (\<langle>z\<rangle>\<^sup>2 * \<langle>w\<rangle>\<^sup>2))"
le_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B Le C D \<or> C D Le A B"
im_eq: fixes A :: "'b set" shows "(+) a ` A = {x. x - a \<in> A}"
non_empty_cycle_root_move_root_forward_cycle: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "non_empty_cycle_root r x"   and "point q"   and "r \<le> x\<^sup>\<star> ; q" shows "x ; q \<noteq> 0 \<and> x\<^sup>T ; q \<noteq> 0"
S_sym: fixes T :: "'w set" assumes "order.greater_eq S T" shows "order.greater_eq T (uminus ` T)"
weak_partial_order_top_axioms: shows "weak_partial_order_top (fpl \<P> \<H>)"
SBHWriteVolatile: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Write True a (D, f) A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, sb @ [Write\<^sub>s\<^sub>b True a (D, f) (f \<theta>) A L R W], m, True, \<O>, \<R>, \<S>)"
indicator_simps_1: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "chi S x = (1::'b)"
sum_iff: fixes u :: "V"   and A :: "V"   and B :: "V" shows "(u \<in>\<^sub>\<circ> A \<Uplus> B) = ((\<exists>x. x \<in>\<^sub>\<circ> A \<and> u = ZFC_Cardinals.Inl x) \<or> (\<exists>y. y \<in>\<^sub>\<circ> B \<and> u = ZFC_Cardinals.Inr y))"
simps_6: fixes e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "E (EIf e1 e2 e3) \<rho> = {v. \<exists>n. VNat n \<in> E e1 \<rho> \<and> (n = 0 \<longrightarrow> v \<in> E e3 \<rho>) \<and> (n \<noteq> 0 \<longrightarrow> v \<in> E e2 \<rho>)}"
dom_right_map_add: fixes x :: "'c"   and g :: "'c \<Rightarrow> 'd option"   and f :: "'c \<Rightarrow> 'd option" assumes "x \<notin> dom g"   and "x \<in> dom f" shows "(f ++ g) x = f x"
Cons_1: assumes "\<And>kv. kv \<in> set xs \<Longrightarrow> f kv = g kv" shows "map_pair f xs = map_pair g xs"
hb: shows "\<Gamma> \<turnstile> t \<Leftrightarrow> v : T\<^sub>1'"
uniq_zero: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "ell_field_class.add a p\<^sub>1 p\<^sub>2 = p\<^sub>2" shows "p\<^sub>1 = Infinity"
hyps_1: fixes i :: "nat" assumes "n = j - i"   and "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "msgs (S t i) cid \<noteq> []"   and "last (msgs (S t i) cid) = Marker"   and "ps (S t i) p \<noteq> None"   and "channel cid = Some (p, q)"   and "\<forall>k. i \<le> k \<and> k < j \<and> regular_event (t ! k) \<longrightarrow> occurs_on (t ! k) \<noteq> p" shows "msgs (S t j) cid = [] \<or> msgs (S t j) cid \<noteq> [] \<and> last (msgs (S t j) cid) = Marker"
quasi_inverse_simps_2: fixes f :: "'a" assumes "equivalence_map f" shows "ide f\<^sup>~"
\<Gamma>_in_hom: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<Gamma> a : \<rho>\<^sub>0 a \<Rightarrow>\<^sub>D \<sigma>\<^sub>0 a\<guillemotright>"
Int_lower2: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B \<subseteq> B"
num_def: shows "num = fst (quot_of_fract x) * snd (quot_of_fract y) + snd (quot_of_fract x) * fst (quot_of_fract y)"
index_swap_distinct: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "distinct xs"   and "Suc n < length xs" shows "index (swap n xs) x = (if x = xs ! n then Suc n else if x = xs ! Suc n then n else index xs x)"
s1: shows "s1 = atomize_poly (idx_pm_of_pm bs s1')"
angle_partition: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C" shows "Acute A B C \<or> Per A B C \<or> Obtuse A B C"
fpos_def: shows "fpos = (\<lambda>w. max 0 (f w))"
abs_valid_operator_5': fixes name :: "char list"   and pre :: "(nat \<times> nat) list"   and eff :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "wf_operator (name, pre, eff, cost)" shows "list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') pre) pre"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "x \<notin> A" shows "prod g (insert x A) = g x * prod g A"
convol_o: fixes f :: "'h \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and h :: "'e \<Rightarrow> 'h" shows "<f, g> \<circ> h = <f \<circ> h, g \<circ> h>"
at_within_interior: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> interior S" shows "at x within S = at x"
avg_def: fixes x :: "rat"   and y :: "rat" shows "avg x y = x / 2 + y / 2"
instr_correct: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "instr list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and stk :: "val list"   and loc :: "val list"   and pc :: "nat"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"   and "\<lfloor>\<sigma>'\<rfloor> = exec (P, None, h, (stk, loc, C, M, pc) # frs)"   and "P,\<Phi> |- (None, h, (stk, loc, C, M, pc) # frs) [ok]" shows "P,\<Phi> |- \<sigma>' [ok]"
mult_le_cancel_left_neg: 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less c zero" shows "less_eq (times c a) (times c b) = less_eq b a"
antisym_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "Quickcheck_Examples.antisym R \<equiv> \<forall>x y. R x y \<longrightarrow> R y x \<longrightarrow> x = y"
l8_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C" shows "Tarski_neutral_dimensionless.Per Bet Cong C B A"
comm_diagrams: fixes U :: "'c set"   and V :: "'c set"   and x :: "'d" assumes "is_open\<^sub>Y U"   and "is_open\<^sub>Y V"   and "V \<subseteq> U"   and "x \<in> \<O>\<^sub>Y U" shows "(im_sheaf_morphisms U V \<circ> \<phi>\<^sub>f U) x = (\<phi>\<^sub>f V \<circ> \<rho>\<^sub>Y U V) x"
\<tau>_3: shows "continuous_on (ball p \<delta>) \<tau>"
sint_sbintrunc': fixes bin :: "int" shows "sint (word_of_int bin) = signed_take_bit (LENGTH('b) - 1) bin"
simps_1: fixes name :: "name"   and vs :: "value list" shows "value_to_sterm (Vconstr name vs) = name $$ map value_to_sterm vs"
primes_omega_eq_0_iff: fixes n :: "nat" shows "(primes_omega n = 0) = (n = 0 \<or> n = 1)"
rec_max_triangle_def: shows "rec_max_triangle = (let cond = CN rec_not [CN rec_le [CN rec_triangle [recf.Id 2 0], recf.Id 2 1]] in CN (rec_max1 cond) [recf.Id 1 0, recf.Id 1 0])"
rev_ImageI: fixes a :: "'c"   and A :: "'c set"   and b :: "'d"   and r :: "('c \<times> 'd) set" assumes "a \<in> A"   and "(a, b) \<in> r" shows "b \<in> r `` A"
Re_neq: shows "\<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z"
inc_rel: shows "incseq (\<lambda>n x. F n x * indicator (\<Union>n. (T ^^ n) --` A) x)"
finite_Domain: fixes r :: "('a \<times> 'b) set" assumes "finite r" shows "finite (Domain r)"
ide_compE: fixes g :: "'c"   and f :: "'c" assumes "C.ide (g \<cdot>\<^sub>C f)" obtains "C.seq g f" and "C.seq f g" and "g \<cdot>\<^sub>C f = C.dom f" and "g \<cdot>\<^sub>C f = C.cod g"
GR': fixes p :: "('a, 'b) fm"   and a :: "'a"   and q :: "('a, 'b) fm" assumes "\<turnstile> \<^bold>\<not> p\<langle>\<^bold>\<dagger> a []/0\<rangle> \<^bold>\<longrightarrow> q"   and "a \<notin> params {p, q}" shows "\<turnstile> \<^bold>\<not> \<^bold>\<forall> p \<^bold>\<longrightarrow> q"
fmfilter_cong: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x y. m $$ x = Some y \<Longrightarrow> P x = Q x" shows "fmfilter P m = fmfilter Q m"
insort_split: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "(x \<in> set (insort y xs)) = (x = y \<or> x \<in> set xs)"
starts_of_def: shows "starts_of \<equiv> fst \<circ> snd"
orbit_stab_fun_is_well_defined0: fixes x :: "'c"   and R :: "'a set"   and g1 :: "'a"   and g2 :: "'a" assumes "x \<in> E"   and "R \<in> rcosets stabilizer G \<phi> x"   and "g1 \<in> R"   and "g2 \<in> R" shows "\<phi> (inv g1) x = \<phi> (inv g2) x"
Sup_le_iff: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'d set"   and b :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"