hcju/isabelleps huggingface dataset:
homeomorphism_on_inverse: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "homeomorphism_on X f" shows "homeomorphism_on (f ` X) (inv_into X f)"
continuous_within_compose3: fixes f :: "'f \<Rightarrow> 'd"   and x :: "'f"   and g :: "'d \<Rightarrow> 'e"   and s :: "'f set" assumes "isCont g (f x)"   and "continuous (at x within s) f" shows "continuous (at x within s) (\<lambda>x. g (f x))"
wr: shows "well_ordered_set (P \<union> {p}) (\<sqsubseteq>)"
len_acts: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list" assumes "BIT_Off acts qs init" shows "length acts = length qs"
foldD_closed: fixes A :: "'b set"   and e :: "'a" assumes "finite A"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D (\<cdot>) e A \<in> D"
aliquot_sum_def: fixes n :: "nat" shows "aliquot_sum n \<equiv> \<Sum> (properdiv_set n)"
substitutes_function: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm" shows "\<exists>!X. substitutes A x M X"
ub_cost: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<exists>k>0. \<forall>l s. P l s \<longrightarrow> e' s \<le> k * e s"   and "\<turnstile>\<^sub>1 {P} c { e' \<Down> Q}" shows "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}"
12: shows "taq {x. poly p x = 0} q \<equiv> \<Sum>y\<in>{x. poly p x = 0 \<and> 0 < poly q x} \<union> {x. poly p x = 0 \<and> poly q x < 0}. Sturm_Tarski.sign (poly q y)"
degree_scaleR_pdevs: fixes r :: "real"   and x :: "'b pdevs" shows "degree (scaleR_pdevs r x) = (if r = 0 then 0 else degree x)"
partition_between_ref_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and lo :: "nat"   and hi :: "nat"   and xs0 :: "'c list" shows "partition_between_ref R h lo hi xs0 = ASSERT (hi < length xs0 \<and> hi < length xs0 \<and> lo \<le> hi) \<bind> (\<lambda>_. choose_pivot3 R h xs0 lo hi \<bind> (\<lambda>k. ASSERT (k < length xs0) \<bind> (\<lambda>_. RETURN (swap xs0 k hi) \<bind> (\<lambda>xs. ASSERT (length xs = length xs0) \<bind> (\<lambda>_. partition_main R h lo hi xs)))))"
domain_axioms_def: fixes R :: "('c, 'd) ring_scheme" shows "domain_axioms R \<equiv> \<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<forall>a b. a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub> \<longrightarrow> a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>)"
not_\<omega>_le_of_nat: fixes n :: "nat" shows "\<not> \<omega> \<le> of_nat n"
closed_subsets_ideal_aux: fixes \<aa> :: "'a set"   and \<bb> :: "'a set"   and x :: "'a set" assumes "ideal \<aa> R (+) (\<cdot>) \<zero> \<one>"   and "ideal \<bb> R (+) (\<cdot>) \<zero> \<one>"   and "pr_ideal R x (+) (\<cdot>) \<zero> \<one>"   and "\<aa> \<subseteq> x \<or> \<bb> \<subseteq> x" shows "ideal_gen_by_prod \<aa> \<bb> \<subseteq> x"
face: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> \<G>"   and "Y \<in> \<G>" shows "X \<inter> Y face_of X"
greatest_fixpoint_unique: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_fixpoint f" shows "\<exists>!x. is_greatest_fixpoint f x"
r2f_def: fixes R :: "('c \<times> 'd) set" shows "\<F> R = bd\<^sup>-\<^sub>\<F> ((``) R)"
assms_6: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a = \<tau> \<or> (\<exists>d. a = i:deliver(d)) \<longrightarrow> U (\<sigma> i) (\<sigma>' i))"
norm_summable_fps: fixes z :: "'a"   and f :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f" shows "summable (\<lambda>n. norm (fps_nth f n * z ^ n))"
simplePath_length_less_V: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "u \<in> cf.V"   and "cf.isSimplePath u p v" shows "length p < card cf.V"
f_dvd_hqh: shows "f dvd h ^ CARD('a) - h"
prj'_chine_composite_2: shows "cods.Prj\<^sub>1\<^sub>0 \<cdot> \<mu>\<nu>\<pi>.chine = \<nu>.chine \<cdot> doms.Prj\<^sub>1\<^sub>0"
set_intros_2: fixes y :: "('c, 'd) Term.term"   and x22 :: "('c, 'd) Term.term list"   and ya :: "'c"   and x21 :: "'c" assumes "y \<in> set x22"   and "ya \<in> funs_term y" shows "ya \<in> funs_term (Fun x21 x22)"
ba_bot_def: fixes z :: "'a" shows "bot = - (z \<squnion> - z)"
awalkE: fixes u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "H.awalk u p v" obtains "set (H.awalk_verts u p) \<subseteq> verts (app_iso hom G)" and "set p \<subseteq> arcs (app_iso hom G)" and "H.cas u p v" and "H.awhd u p = u" and "H.awlast u p = v"
arb_uvp: shows "\<forall>n w. cls_val_process Mkt2 arb_pf n w = prices Mkt2 pos_stock n w * diff_inv - pr n w"
degree_pCons_eq: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "degree (pCons a p) = Suc (degree p)"
3: fixes xx :: "'var"   and tt :: "'trm" assumes "xx \<in> FvarsT t" shows "(tt, xx) \<notin> set txs"
tr_intro: fixes \<pi>l :: "('state \<times> 'aprop set) stream list" shows "tr \<pi>l"
AsParallelSkips: fixes Qs :: "'d set"   and as :: "(('d, 'e, 'f) ann \<times> 'd set \<times> 'd set) list"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and Q :: "'d set"   and A :: "'d set"   and cs :: "('d, 'e, 'f) com list" assumes "Qs = \<Inter> (set (map (\<lambda>a. fst (snd a)) as))" shows "assertionsR \<Gamma> \<Theta> Q A (AnnPar as) (Parallel cs) Qs"
F5: shows "\<forall>x\<^sub>3 x\<^sub>1. (0::'a) \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1"
flip_conv: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and b :: "'c"   and a :: "'b" shows "flip f b a = f a b"
moebius_ocircline_opposite_ocircline: fixes M :: "moebius"   and H :: "ocircline" shows "moebius_ocircline M (opposite_ocircline H) = opposite_ocircline (moebius_ocircline M H)"
simps: fixes a1 :: "v sem_env"   and a2 :: "exp0"   and a3 :: "(v, v) result" shows "cupcake_evaluate_single a1 a2 a3 = ((\<exists>env cn es rs vs v0. a1 = env \<and> a2 = Con cn es \<and> a3 = Rval v0 \<and> do_con_check (c env) cn (length es) \<and> list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs \<and> sequence_result rs = Rval vs \<and> build_conv (c env) cn (rev vs) = Some v0) \<or> (\<exists>env cn es. a1 = env \<and> a2 = Con cn es \<and> a3 = Rerr (Rabort Rtype_error) \<and> \<not> do_con_check (c env) cn (length es)) \<or> (\<exists>env cn es rs err. a1 = env \<and> a2 = Con cn es \<and> a3 = Rerr err \<and> do_con_check (c env) cn (length es) \<and> list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs \<and> sequence_result rs = Rerr err) \<or> (\<exists>env n v0. a1 = env \<and> a2 = Var n \<and> a3 = Rval v0 \<and> nsLookup (sem_env.v env) n = Some v0) \<or> (\<exists>env n. a1 = env \<and> a2 = Var n \<and> a3 = Rerr (Rabort Rtype_error) \<and> nsLookup (sem_env.v env) n = None) \<or> (\<exists>env n e. a1 = env \<and> a2 = Fun n e \<and> a3 = Rval (Closure env n e)) \<or> (\<exists>env es rs vs env' e bv. a1 = env \<and> a2 = App Opapp es \<and> a3 = bv \<and> list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs \<and> sequence_result rs = Rval vs \<and> do_opapp (rev vs) = Some (env', e) \<and> cupcake_evaluate_single env' e bv) \<or> (\<exists>env es rs vs. a1 = env \<and> a2 = App Opapp es \<and> a3 = Rerr (Rabort Rtype_error) \<and> list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs \<and> sequence_result rs = Rval vs \<and> do_opapp (rev vs) = None) \<or> (\<exists>env es rs err op0. a1 = env \<and> a2 = App op0 es \<and> a3 = Rerr err \<and> list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs \<and> sequence_result rs = Rerr err) \<or> (\<exists>env e v0 pes e' uu_ env' bv. a1 = env \<and> a2 = Mat e pes \<and> a3 = bv \<and> cupcake_evaluate_single env e (Rval v0) \<and> cupcake_match_result (c env) v0 pes Bindv = Rval (e', uu_, env') \<and> cupcake_evaluate_single (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) e' bv) \<or> (\<exists>env e v0 pes err. a1 = env \<and> a2 = Mat e pes \<and> a3 = Rerr err \<and> cupcake_evaluate_single env e (Rval v0) \<and> cupcake_match_result (c env) v0 pes Bindv = Rerr err) \<or> (\<exists>env e err pes. a1 = env \<and> a2 = Mat e pes \<and> a3 = Rerr err \<and> cupcake_evaluate_single env e (Rerr err)))"
closure_mono: fixes S :: "'d set"   and T :: "'d set" assumes "S \<subseteq> T" shows "closure S \<subseteq> closure T"
T_def: shows "T = (\<lambda>a. S \<inter> R a)"
STAR_cos_zero: shows "(*f* cos) 0 = 1"
induct: fixes x :: "'b monoid list \<times> ('b \<Rightarrow> 'b) list"   and P :: "'b monoid list \<times> ('b \<Rightarrow> 'b) list \<Rightarrow> bool" assumes "exact_seq x"   and "\<And>G1 G2 f. group_hom G1 G2 f \<Longrightarrow> P ([G2, G1], [f])"   and "\<And>G K l g q H h. \<lbrakk>exact_seq (G # K # l, g # q); P (G # K # l, g # q); Group.group H; h \<in> hom G H; kernel G H h = g ` carrier K\<rbrakk> \<Longrightarrow> P (H # G # K # l, h # g # q)" shows "P x"
case_prodE: fixes c :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and p :: "'k \<times> 'l" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'k" and y :: "'l" where "p = (x, y)" and "c x y"
SD_preorderD_2: fixes q :: "'alt pmf"   and p :: "'alt pmf" assumes "SD (Pareto R) q p" shows "q \<in> lotteries_on alts"
cvalidt_postD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t"   and "s \<in> P"   and "t \<notin> Fault ` F" shows "t \<in> Normal ` Q \<union> Abrupt ` A"
assms: shows "\<forall>x\<in>vars\<^sub>l\<^sub>s L. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
prod_ide_eq: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "local.prod a b = HF'.prod a b"
valid_insert: fixes \<Delta> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty"   and \<Gamma> :: "(name \<times> ty) list" assumes "valid (\<Delta> @ [(x, T)] @ \<Gamma>)" shows "valid (\<Delta> @ \<Gamma>)"
intros_1: fixes \<sigma> :: "'tp"   and c :: "('c, 'd) lit list" assumes "infTp \<sigma>" shows "\<sigma> \<turnstile> c"
matchs_tl: fixes fps :: "(char list \<times> pat) list"   and l :: "char list"   and t :: "trm"   and fs :: "(char list \<times> trm) list"   and ts :: "trm list" assumes "\<turnstile> fps [\<rhd>] (l, t) \<Colon> fs \<Rightarrow> ts"   and "fps\<langle>l\<rangle>\<^sub>? = \<bottom>" shows "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts"
poly_map_poly: fixes p :: "'a poly"   and x :: "'a" shows "poly (map_poly hom p) (hom x) = hom (poly p x)"
sym_mpoly_altdef: fixes A :: "nat set"   and k :: "nat" shows "sym_mpoly A k = (\<Sum>X | X \<subseteq> A \<and> card X = k. MPoly_Type.monom (monom_of_set X) (1::'a))"
fst_conv: fixes x1 :: "'ag"   and x2 :: "'ah" shows "fst (x1, x2) = x1"
recv_marker_other_channels_not_shrinking: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and ev :: "('a, 'b, 'c) event"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and c' :: "('a, 'b, 'c) configuration"   and i' :: "nat" assumes "distributed_system channel"   and "ev = RecvMarker i p q"   and "distributed_system.next channel trans send recv c ev c'" shows "(length (msgs c i') \<le> length (msgs c' i')) = (i \<noteq> i')"
cond_and_distr: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and b :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool"   and S :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<and> Q \<triangleleft> b \<triangleright>  R \<and> S) = ((P \<triangleleft> b \<triangleright>  R) \<and> (Q \<triangleleft> b \<triangleright>  S))"
dirichlet_prod_def: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "f \<star> g = (\<lambda>n. \<Sum>d | d dvd n. f d * g (n div d))"
single_one_def: shows "single_one \<equiv> \<lambda>l k A i j. (i \<notin> {k, l} \<longrightarrow> A $$ (i, k) = (0::'b)) \<and> A $$ (l, k) = (1::'b)"
IP: fixes k :: "'d"   and it :: "'d set"   and v :: "'a"   and \<sigma> :: "'c" assumes "k \<in> dom m - it"   and "invar v"   and "m k = Some (\<alpha> v)"   and "it \<subseteq> dom m"   and "I it \<sigma>" shows "I (insert k it) (f (k, v) \<sigma>)"
natural_transformation_axioms: shows "natural_transformation J (\<cdot>) D D' \<tau>"
dres_order_simps_4: fixes r :: "'b" shows "dRETURN r \<noteq> dSUCCEED"
is_Sconst_def: fixes sterm :: "sterm" shows "is_Sconst sterm = (\<exists>x1. sterm = Sconst x1)"
closerToD_def: shows "closerToD \<equiv> {B \<in> folding.\<C>. order.greater (chamber_distance B C) (chamber_distance B D)}"
elims_3: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" assumes "\<not> BsSkewBinomialHeapStruc.elem_invar x" obtains e :: "'a" and a :: "'b" and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" where "x = BsSkewBinomialHeapStruc.Element e a q" and "\<not> ((\<forall>x. x \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)) \<longrightarrow> a \<le> BsSkewBinomialHeapStruc.eprio x \<and> BsSkewBinomialHeapStruc.elem_invar x) \<and> SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q))"
Match: fixes P :: "pi"   and V :: "Early_Semantics.residual"   and b :: "name" assumes "P \<longmapsto>\<^sub>e V" shows "[b\<frown>b]P \<longmapsto>\<^sub>e V"
that: fixes g :: "'b \<Rightarrow> 'a" assumes "homeomorphism (f ` S) S g f" shows "thesis"
vector_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "is_vector y" shows "(x \<cdot> y\<^sup>\<smile>) ; z = x ; (y \<cdot> z)"
qSwap_preserves_qGood1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "qGood X" shows "qGood (X #[[x \<and> y]]_zs)"
simplicial_chain_0: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set" shows "simplicial_chain p S 0"
P_def: shows "P = (\<lambda>l. \<forall>\<^sub>F r in at_top. cindexE (- r) r f = real_of_int l)"
extend_inv: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "extend (\<lambda>(x, u). inv h x) = project (\<lambda>(x, u). h x) UNIV"
rep_eq: fixes x :: "uint16" shows "Rep_uint64 (uint64_of_uint16 x) = ucast (Rep_uint16 x)"
simps_1: fixes x :: "'a" shows "opaque (Opaque x) = [x]"
adj_iso2: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "mono g"
nff_\<alpha>_upd: fixes i :: "nat"   and l :: "'a list"   and dflt :: "'a"   and v :: "'a" assumes "i < length l" shows "nff_\<alpha> dflt (l[i := v]) = (nff_\<alpha> dflt l)(i := v)"
empty_iff: fixes c :: "'f" shows "(c \<in> {}) = False"
gR_downset_def: fixes x :: "'c" shows "x\<down> \<equiv> {y. y \<preceq>\<^sub>R x}"
coeff_monom_mult: fixes m' :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and q :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (MPoly_Type.monom m' a * q) (m' + m) = a * MPoly_Type.coeff q m"
eval_psi_aux1: shows "psi 0 = ln (real Numeral1)"
eWls_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and s :: "'sort"   and X :: "'c" shows "eWls MOD s (OK X) = igWls MOD s X"
finitesum_base_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and I :: "'h set"   and f :: "'h \<Rightarrow> 'e set" shows "finitesum_base M I f = \<Union> {f i |i. i \<in> I}"
nemp_concat_hull_1: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<noteq> \<epsilon>"   and "us \<in> lists G\<^sub>+" shows "concat us \<in> \<langle>G\<rangle>"
CK_nf_real_card: shows "card ((\<lambda>f. f RRR) ` {f. CK_nf f}) = 14"
lm027: fixes P1 :: "'c set set"   and P2 :: "'c set set" assumes "\<Union> P1 \<inter> \<Union> P2 = {}"   and "is_non_overlapping P1"   and "is_non_overlapping P2" shows "is_non_overlapping (P1 \<union> P2)"
hoare_diagram: fixes D :: "'a \<times> 'a \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" assumes "dmono D"   and "\<forall>w i j. \<Turnstile> X w i{| D (i, j) |}Sup_less X w j" shows "\<Turnstile> \<Squnion>range X{| pt D |}(\<Squnion>range X \<sqinter> - grd (step D))"
run_alt_nondet: fixes m1 :: "('c, 'b) nondetT"   and m2 :: "('c, 'b) nondetT" shows "run_nondet (alt_nondet m1 m2) = Monad_Overloading.bind (run_nondet m1) (\<lambda>A. Monad_Overloading.bind (run_nondet m2) (\<lambda>B. Monad_Overloading.return (A + B)))"
Le_le_LtI: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "Le a \<le> Lt b"
uv_sigma_1: shows "(U, x) \<in> Sigma I \<FF>"
maximal_existence_flow_1: fixes x :: "real \<Rightarrow> 'a"   and K :: "real set"   and t0 :: "real"   and x0 :: "'a" assumes "(x solves_ode f) K S"   and "x t0 = x0"   and "is_interval K"   and "t0 \<in> K"   and "K \<subseteq> T" shows "K \<subseteq> existence_ivl t0 x0"
assms_2: fixes i :: "nat" shows "e i \<in> {- 1..1}"
induct: fixes P :: "pexpr \<Rightarrow> nat \<Rightarrow> pexpr \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "pexpr"   and a1 :: "nat"   and a2 :: "pexpr"   and a3 :: "nat" assumes "\<And>e n e1 e2 m. \<lbrakk>P e n e1 m; isin e n e1 m = None \<Longrightarrow> P e n e2 m; \<And>x2 x y. \<lbrakk>isin e n e1 m = Some x2; (x, y) = x2; x \<noteq> 0\<rbrakk> \<Longrightarrow> P e x e2 m\<rbrakk> \<Longrightarrow> P e n (PExpr2 (PMul e1 e2)) m"   and "\<And>e n e' k m. (k \<noteq> 0 \<Longrightarrow> P e n e' (k * m)) \<Longrightarrow> P e n (PExpr2 (PPow e' k)) m"   and "\<And>e n e' m. P (PExpr1 e) n (PExpr1 e') m"   and "\<And>e n e' m. P (PExpr2 e) n (PExpr1 e') m" shows "P a0 a1 a2 a3"
known_ptr_defs: fixes ptr :: "(_) object_ptr" shows "ShadowRootClass.known_ptr ptr = (DocumentClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> is_shadow_root_ptr x))"
samequot_termination: fixes as :: "'a"   and r :: "'b rexp" assumes "while_option fst (rderiv_and_add as) (True, [\<guillemotleft>r\<guillemotright>]) = None" shows "False"
ptr_kinds_M_ptr_kinds_returns_result: fixes h :: "'a"   and ptr_kinds :: "'a \<Rightarrow> 'c fset"   and f :: "'c list \<Rightarrow> ('a, exception, 'b) prog"   and x :: "'b" shows "h \<turnstile> Heap_Error_Monad.bind (l_ptr_kinds_M.a_ptr_kinds_M ptr_kinds) f \<rightarrow>\<^sub>r x = h \<turnstile> f (sorted_list_of_set (fset (ptr_kinds h))) \<rightarrow>\<^sub>r x"
assms_2: shows "arr \<nu> \<and> b \<in> sources \<nu> \<and> c \<in> targets \<nu>"
CatExp'_def: fixes A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" shows "CatExp' A B \<equiv> \<lparr>Obj = {F. Ftor F : A \<longrightarrow> B}, Mor = {\<eta>. NatTrans \<eta> \<and> NTCatDom \<eta> = A \<and> NTCatCod \<eta> = B}, Dom = NTDom, Cod = NTCod, Id = IdNatTrans, Comp = (\<bullet>)\<rparr>"
UN_Un: fixes M :: "'d \<Rightarrow> 'c set"   and A :: "'d set"   and B :: "'d set" shows "\<Union> (M ` (A \<union> B)) = \<Union> (M ` A) \<union> \<Union> (M ` B)"
f_shrink_Cons: fixes k :: "nat"   and xs :: "'b message_af list"   and ys :: "'b message_af list" assumes "0 < k"   and "length xs = k" shows "(xs @ ys) \<div> k = last_message xs # ys \<div> k"
row_smult: fixes i :: "nat"   and A :: "'c mat"   and k :: "'c" assumes "i < dim_row A" shows "row (k \<cdot>\<^sub>m A) i = k \<cdot>\<^sub>v row A i"
authTicket_crypt_authK: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbIV" shows "authK \<in> authKeys evs"
Dsemi_def: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "Dsemi prod \<equiv> \<forall>x y z. prod (prod x y) z = prod x (prod y z)"
check_all_const_deg_prop: fixes l :: "real poly list" shows "(check_all_const_deg l = True) = (\<forall>p\<in>set l. degree p = 0)"
right_neutral: fixes a :: "'c \<times> 'd" shows "times_prod' a one_prod = a"
Cons_path2: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and n :: "'b"   and ns :: "'b list"   and m :: "'b"   and n' :: "'b" assumes "graph_path \<alpha>e \<alpha>n invar inEdges'"   and "graph_path_base.path2 \<alpha>n invar inEdges' g n ns m"   and "n' \<in> set (graph_path_base.predecessors inEdges' g n)" shows "graph_path_base.path2 \<alpha>n invar inEdges' g n' (n' # ns) m"
intro: fixes U\<^sub>1 :: "'a set"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "preorder_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1"   and "order_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2" shows "preorder_order_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1 U\<^sub>2 le\<^sub>2 ls\<^sub>2"
span_Un: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set"   and T :: "'e set" assumes "module scale" shows "module.span scale (S \<union> T) = {x + y |x y. x \<in> module.span scale S \<and> y \<in> module.span scale T}"
pair_sym_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs sG"   and "w \<notin> pverts sG" shows "pair_sym_digraph (subdivide sG e w)"
Ch_CH_irc_idem: fixes h :: "'c"   and A :: "'a set" shows "Ch h (CH A) = Ch h A"
ide_MkIde: fixes A :: "bool" assumes "A \<in> {False, True}" shows "ide (MkIde A)"
symbolic_run_interp_cons_morph: fixes \<gamma> :: "'b constr"   and \<Gamma> :: "'b constr list" shows "\<lbrakk> \<gamma> \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<gamma> # \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
card_Union_le_sum_card: fixes U :: "'b set set" assumes "\<forall>u\<in>U. finite u" shows "card (\<Union> U) \<le> sum card U"
DenyAllowDisj: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and p :: "int" assumes "dom (C (AllowPortFromTo a b p)) \<noteq> {}" shows "dom (C (DenyAllFromTo a b)) \<inter> dom (C (AllowPortFromTo a b p)) \<noteq> {}"
a_host_shadow_root_rel_shadow_root: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_option :: "(_) shadow_root_ptr option"   and shadow_root :: "(_) shadow_root_ptr" assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r shadow_root_option" shows "(shadow_root_option = Some shadow_root) = ((cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> host_shadow_root_rel h)"