hcju/isabelleps huggingface dataset:
obs_consistent_behaviours: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec"   and bc :: "'f list"   and ba :: "'e list" assumes "obs_consistent R pi Sa Sc"   and "bc \<in> beh Sc"   and "ba \<in> beh Sa"   and "(ba, bc) \<in> seq_lift R" shows "map pi (map (obs Sc) bc) = map (obs Sa) ba"
Nml_HcompNml_1: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Nml t"   and "E.Nml u"   and "E.Src t = E.Trg u" shows "E.Nml (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
inv_one: shows "inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> \<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> = \<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"
build_append: fixes w :: "'b list"   and a :: "'b"   and u :: "'b list"   and v :: "nat \<Rightarrow> 'b" shows "(w @ a # u) \<frown> v = w \<frown> a ## u \<frown> v"
add_mset_eq_singleton_iff: fixes x :: "'b"   and M :: "'b multiset"   and y :: "'b" shows "(add_mset x M = {#y#}) = (M = {#} \<and> x = y)"
chine_composites_1: shows "Chn (\<mu> \<star> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr>) = chine_hcomp \<mu> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr>"
allVars_in_allDefs: fixes v :: "'val"   and g :: "'g" assumes "v \<in> allVars g" shows "\<exists>n\<in>set (\<alpha>n g). v \<in> allDefs g n"
reachable_target_elim: fixes q :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set"   and p :: "nat \<times> 'c" assumes "q \<in> greachable A w V p" obtains r :: "(nat \<times> 'c) list" where "gpath A w V r p" and "q = gtarget r p"
w_simps_6: shows "cod \<mu>.chine = \<mu>.chine"
split_mset_set: fixes C :: "'b set"   and C1 :: "'b set"   and C2 :: "'b set" assumes "C = C1 \<union> C2"   and "C1 \<inter> C2 = {}"   and "finite C1"   and "finite C2" shows "mset_set C = mset_set C1 + mset_set C2"
linorder_from_keys_permute: fixes g :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "g permutes A" shows "linorder_from_keys A (f \<circ> g) = map_relation A g (linorder_from_keys A f)"
periodN_rev: fixes w :: "'b list"   and p :: "nat" assumes "periodN w p" shows "periodN (rev w) p"
HNatInfinite_add: fixes x :: "nat star"   and y :: "nat star" assumes "x \<in> HNatInfinite" shows "x + y \<in> HNatInfinite"
not_less_eq_funE: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and c :: "nat"   and n :: "nat" assumes "\<not> f \<lesssim> g"   and "0 < c" obtains m :: "nat" where "n < m" and "c * g m < f m"
product_nn_integral_insert_rev: fixes I :: "'d set"   and i :: "'d"   and f :: "('d \<Rightarrow> 'e) \<Rightarrow> ennreal" assumes "finite I"   and "i \<notin> I"   and "f \<in> borel_measurable (Pi\<^sub>M (insert i I) (\<lambda>_. lborel))" shows "integral\<^sup>N (Pi\<^sub>M (insert i I) (\<lambda>_. lborel)) f = \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x(i := y)) \<partial>Pi\<^sub>M I (\<lambda>_. lborel) \<partial>lborel"
assms: shows "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)"
convex_on_ereal_alt_mem: fixes C :: "'b set"   and f :: "'b \<Rightarrow> ereal"   and x :: "'b"   and y :: "'b"   and m :: "real" assumes "convex C"   and "Lower_Semicontinuous.convex_on C f"   and "x \<in> C"   and "y \<in> C"   and "0 \<le> m"   and "m \<le> 1" shows "f (m *\<^sub>R x + (1 - m) *\<^sub>R y) \<le> ereal m * f x + (1 - ereal m) * f y"
SourcesA9_L0: shows "Sources level0 sA9 = {sA6, sA7, sA8, sA9}"
mu_times_mu2: shows "mu * mu2 = monomial (lc (q g1 \<odot> g1) / lc g2) (punit.lt (q g2))"
simps_1: fixes val :: "'b"   and l :: "nat"   and i :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_3_c_inner_loop val l i 0 A = A"
decseqD: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "decseq f"   and "i \<le> j" shows "f j \<le> f i"
AssignIdx_retv_def: fixes x :: "char list"   and i :: "aexp"   and rv :: "char list" shows "AssignIdx_retv x i rv \<equiv> x[i] ::= V rv"
intro: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "monoid M composition unit" shows "left_translations_of_monoid M composition unit"
lnth_LCons: fixes x :: "'c"   and xs :: "'c llist"   and n :: "nat" shows "lnth (LCons x xs) n = (case n of 0 \<Rightarrow> x | Suc n' \<Rightarrow> lnth xs n')"
module_pair_with_def: fixes plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f" shows "module_pair_with plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> module_with plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> module_with plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
mm_equiv_intro: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and lc\<^sub>1 :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and lc\<^sub>2 :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "abs.strong_low_bisim_mm \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "abs.mm_equiv_abv lc\<^sub>1 lc\<^sub>2"
cqt_5_mod: fixes \<psi> :: "\<kappa> \<Rightarrow> \<o>"   and \<tau> :: "\<kappa>" assumes "SimpleExOrEnc \<psi>" shows "[[\<psi> \<tau> \<^bold>\<rightarrow> (\<^bold>\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= \<tau>)]]"
the_dflt_Z_eq: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool"   and m :: "'c option"   and d :: "'c" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar" shows "(the_default (0::'c) m = d) = (m = None \<and> d = (0::'c) \<or> m = Some d)"
iT_Div_conv_count: fixes d :: "nat"   and I :: "nat set" assumes "0 < d" shows "I \<oslash> d = {k. I \<inter> [k * d\<dots>,d - Suc 0] \<noteq> {}}"
comp_ide_arr: fixes b :: "'a"   and f :: "'a" assumes "S.ide b"   and "S.seq b f" shows "b \<cdot>\<^sub>S f = f"
shead_def: shows "shead \<equiv> sfst oo sscase\<cdot>\<bottom>\<cdot>ID oo slist_rep"
ex_cross_ratio: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and c :: "complex_homo" assumes "u \<noteq> v"   and "u \<noteq> w"   and "v \<noteq> w" shows "\<exists>z. cross_ratio z u v w = c"
CAs0_to_CAs: shows "CAs0 \<cdot>\<cdot>cl \<eta>s0 = CAs"
flush_all_until_volatile_write_append_Ghost_commute: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and \<D>' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>', sb @ [Ghost\<^sub>s\<^sub>b A L R W], \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ts m"
reflexive_axioms: shows "reflexive X (\<sqsubseteq>)"
cong: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "web.separating_of_bipartite \<Gamma> \<equiv> web.separating_of_bipartite \<Gamma>"
rt_eq: shows "return_time p2 (poincare_map p1 x0) + return_time p1 x0 = return_time p2 x0"
smap_shift: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and s :: "'b stream" shows "smap f (xs @- s) = map f xs @- smap f s"
parabolic_fixed_point_2: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "Gromov_extension f (neutral_fixed_point f) = neutral_fixed_point f"
game_semi_forgetful_bad_def: shows "game_semi_forgetful_bad = seed_gen \<bind> (\<lambda>seed. exec_gpv (semi_forgetful_RO seed) \<A> (Map.empty, False) \<bind> (\<lambda>x. return_spmf (snd x)))"
units_group: shows "Group.group (units_of (add_monoid R))"
some_equality: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(SOME x. P x) = a"
is_sup_least: fixes x :: "'a"   and y :: "'a"   and s :: "'a"   and z :: "'a" assumes "is_sup x y s"   and "x \<sqsubseteq> z"   and "y \<sqsubseteq> z" shows "s \<sqsubseteq> z"
summatory_totient_asymptotics: shows "(\<lambda>x. sum_upto (\<lambda>n. real (totient n)) x - 3 / pi\<^sup>2 * x\<^sup>2) \<in> O(\<lambda>x. x * ln x)"
Cons_2: shows "\<alpha> \<in># r|\<beta> # as|"
daMPMk: fixes k :: "nat" assumes "k < n" shows "adjoint (M k) * P k * M k \<in> carrier_mat d d"
C_slide: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.C_algebra plus times one zero less_eq less star" shows "times (star (times x y)) x = times x (star (times y x))"
scott_continuousE: fixes f :: "'a \<Rightarrow> 'a" assumes "local.scott_continuous f" obtains "f ` A \<subseteq> A" and "\<And>X s. \<lbrakk>X \<subseteq> A; directed X (\<sqsubseteq>); X \<noteq> {}; extreme_bound A (\<sqsubseteq>) X s\<rbrakk> \<Longrightarrow> extreme_bound A (\<sqsubseteq>) (f ` X) (f s)"
divisor_count_pos: fixes n :: "nat" assumes "0 < n" shows "0 < divisor_count n"
that: fixes M :: "'a dfa" assumes "dfa M"   and "dfa.language M = L"   and "card (states M) = card (UNIV // eq_app_right L)" shows "thesis"
lem1: fixes x :: "'b"   and y :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x \<noteq> y" shows "(x, M = y, N) = (x \<in># N \<and> M = y, N - (x, {#}))"
scale_sum_right: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "module scale" shows "scale a (sum f A) = (\<Sum>x\<in>A. scale a (f x))"
disc_43: fixes x4 :: "'c set" shows "is_Abs (Abs x4)"
SN: shows "SN (S'\<^sup>* O R' O S'\<^sup>*)"
descent: fixes r :: "'rule"   and s :: "'sequent"   and sl :: "'sequent fset"   and S :: "'structure" assumes "r \<in> R"   and "eff r s sl"   and "S \<in> structure"   and "\<not> sat S s" shows "\<exists>s' S'. s' |\<in>| sl \<and> S' \<in> structure \<and> \<not> sat S' s' \<and> (\<forall>v v' b. (v, b, v') \<in> \<delta> s r s' \<longrightarrow> (\<sigma> (v', S'), \<sigma> (v, S)) \<in> Ord \<and> (b \<longrightarrow> \<sigma> (v', S') \<noteq> \<sigma> (v, S)))"
zero_leastI: fixes x :: "'a" shows "x \<succ> (0::'a)"
assms_5: shows "is_a_intersec Q B C' B' C"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and r :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "identity_arrow_of_spans C r" shows "identity_arrow_in_span_bicategory C prj0 prj1 r"
singleton: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "semilattice_fset f" shows "semilattice_fset.F f {|x|} = x"
order_base_def: fixes A :: "'b set" shows "order_base A = (\<Union>x\<in>A. {{y \<in> A. x \<le> y}})"
image_chamber_system: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplex.chamber_system (f \<turnstile> X) = f \<turnstile> ChamberComplex.chamber_system X"
disc_iff_2: fixes x :: "('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "(\<not> Resumption.resumption.is_Done (x \<bind> f)) = (\<not> Resumption.resumption.is_Done x \<or> \<not> (Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x)))))"
in_hom_char: fixes f :: "'d"   and a :: "'d"   and b :: "'d" shows "D'.in_hom f a b = (D'.arr a \<and> D'.arr b \<and> D'.arr f \<and> \<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>)"
card'_def: shows "card' \<equiv> map_fun Rep_nat_int id card"
simps: fixes rest1 :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme"   and rest2 :: "('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'h spmf"   and a1 :: "'a spmf"   and a2 :: "'h spmf" shows "trace_rest_closure rest1 rest2 IA IU p q a1 a2 = (\<exists>tr. a1 = trace_rest_aux rest1 p tr \<and> a2 = trace_rest_aux rest2 q tr \<and> set tr \<subseteq> IA \<times> UNIV <+> IU \<times> UNIV)"
"case": fixes f :: "('b \<Rightarrow> bool) \<Rightarrow> 'a"   and x :: "'b \<Rightarrow> bool" shows "(case Pred x of Pred x \<Rightarrow> f x) = f x"
locked_in_vf_same: fixes v_f :: "process \<Rightarrow> val option"   and v :: "val"   and w :: "val" assumes "locked_in_vf v_f v"   and "locked_in_vf v_f w" shows "v = w"
monotone_emeasure_spmf: shows "monotone (ord_spmf (=)) (\<le>) (\<lambda>p. emeasure (measure_spmf p))"
ballE: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
cn_merge_gs_tl_app: fixes gs :: "(abc_inst list \<times> nat \<times> nat) list"   and g :: "abc_inst list \<times> nat \<times> nat"   and pstr :: "nat" shows "cn_merge_gs (gs @ [g]) pstr = cn_merge_gs gs pstr [+] cn_merge_gs [g] (pstr + length gs)"
wf_operators_2: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "wf_ast_problem problem" shows "\<forall>\<pi>\<in>set (ast_problem.ast\<delta> problem). ast_problem.wf_operator problem \<pi>"
norm_sum: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (comm_monoid_add.sum plus zero f A) \<le> (\<Sum>i\<in>A. norm (f i))"
sel_empt: shows "\<forall>i<n. S_M S M (CAs ! i) = {#}"
bounded_check_eqv_def: shows "bounded_check_eqv \<equiv> Presb.bounded.check_eqv"
consec_assoc2: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "consec i j \<and> consec (i \<squnion> j) k \<longrightarrow> consec i (j \<squnion> k)"
comm_mult_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (r' |\<^bsub>rel\<^esub> s') \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s)"
assms: shows "prove_nonneg prnt i p (slp_of_fas [fa]) XSS"
obtain_fresh: fixes x :: "'b" obtains a :: "'a" where "atom a \<sharp> x"
exp_less_one_iff: fixes x :: "real" shows "(exp x < 1) = (x < 0)"
decisive_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set" assumes "decisive_on A f"   and "B \<subseteq> A"   and "B \<noteq> {}" shows "f B \<noteq> {}"
block_stable: fixes j :: "nat"   and i :: "nat"   and t :: "nat" assumes "j < e_length (block i t)"   and "j \<noteq> gap i t" shows "e_nth (block i t) j = e_nth (block i (Suc t)) j"
conseq_1: shows "\<exists>k>0. \<forall>l s. P' l s \<longrightarrow> e s \<le> k * e' s \<and> (\<forall>t. \<exists>l'. P l' s \<and> (Q l' t \<longrightarrow> Q' l t))"
inj_swap: fixes b :: "'a"   and i :: "'a" shows "inj (swap b i)"
wf_rexp: shows "\<forall>x\<in>valid_ENC n ` FO. local.wf n x"
ego2_s_init: shows "ego2.s 0 = ego.q \<delta>"
reversepath_simps_4: fixes g :: "real \<Rightarrow> 'b" shows "pathfinish (reversepath g) = pathstart g"
absCase_def: fixes xs :: "'varSort"   and s :: "'sort"   and f :: "'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" shows "absCase xs s f A \<equiv> if wlsAbs (xs, s) A then f (fst (castAbs xs s A)) (snd (castAbs xs s A)) else undefined"
r_coset_subset_G: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c" assumes "Group.monoid G"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x \<subseteq> carrier G"
update_convs_4: fixes getConflictFlag' :: "bool \<Rightarrow> bool"   and getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getConflictFlag_update getConflictFlag' \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag' getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr>"
a: shows "fst (punit.lt (rep_list p), p) \<in> keys (rep_list (snd (punit.lt (rep_list p), p)))"
map_gpv_conv_map_gpv': fixes f :: "'k \<Rightarrow> 'n"   and g :: "'l \<Rightarrow> 'o" shows "map_gpv f g = map_gpv' f g id"
integrable_const: fixes M :: "'a measure"   and a :: "'b" assumes "finite_measure M" shows "integrable M (\<lambda>x. a)"
tendsto_rabs_zero: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> 0) F" shows "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
cases: fixes a1 :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list"   and a2 :: "'a"   and a3 :: "String.literal"   and a4 :: "'a list"   and a5 :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" assumes "unique_operation a1 a2 a3 a4 a5" obtains operations :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list" and \<tau> :: "'a" and name :: "String.literal" and \<pi> :: "'a list" and oper :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" where "a1 = operations" and "a2 = \<tau>" and "a3 = name" and "a4 = \<pi>" and "a5 = oper" and "operation' operations \<tau> name \<pi> oper" and "\<not> operation_not_unique operations \<tau> name \<pi> oper"
simps_3: fixes A :: "pol"   and x :: "nat"   and B :: "pol" shows "sqr (PX A x B) = mkPX (sqr A) (x + x) (sqr B) \<langle>+\<rangle> mkPX (Pc 2 \<langle>*\<rangle> A \<langle>*\<rangle> mkPinj 1 B) x (Pc 0)"
mu_below_nu: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_fixpoint f"   and "has_greatest_fixpoint f" shows "\<mu> f \<le> \<nu> f"
lookup_dflt_eq_lookup_pair: fixes ko :: "'a nat_term_order"   and xs :: "('a \<times> 'b) list" assumes "ko_ntm.oalist_inv_raw ko xs" shows "lookup_dflt xs = lookup_pair_ko_ntm ko xs"
w_in_hom_1: shows "\<guillemotleft>\<pi>.chine : src s\<^sub>0 \<rightarrow> src t\<^sub>0\<guillemotright>"
disconnected_nodes_eq_h2: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "doc_ptr \<noteq> 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>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr" shows "h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h' \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
d_IN_current_of_bipartite: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "('a + 'a) \<times> ('a + 'a) \<Rightarrow> ennreal"   and x :: "'a" assumes "web \<Gamma>"   and "current (bipartite_web_of \<Gamma>) f" shows "d_IN (web.current_of_bipartite \<Gamma> f) x = d_IN f (Inr x) - f (Inl x, Inr x)"
finitesum_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and I :: "'h set"   and f :: "'h \<Rightarrow> 'e set" shows "finitesum M I f = {x. \<exists>n g. g \<in> {j. j \<le> n} \<rightarrow> finitesum_base M I f \<and> x = \<Sigma>\<^sub>e M g n}"
28338: fixes x :: "'a"   and y :: "'a" shows "- (- - (x \<sqinter> y) \<sqinter> x) = - (x \<sqinter> y)"
proj_Cons_2: fixes n :: "'d"   and a :: "'e"   and A :: "('d strand_label \<times> 'e) list" shows "proj n ((\<star>, a) # A) = (\<star>, a) # proj n A"
simplex_minus_1: fixes S :: "'b set" shows "(- 1 simplex S) = (S = {})"
aux2: fixes n :: "nat"   and sch :: "type_scheme"   and S :: "nat \<Rightarrow> typ" assumes "new_tv n sch" shows "subst_to_scheme (\<lambda>k. if n \<le> k then BVar (k - n) else FVar k) (bound_typ_inst S sch) = bound_scheme_inst (subst_to_scheme (\<lambda>k. if n \<le> k then BVar (k - n) else FVar k) \<circ> S) sch"
prems_3: shows "hd (filter (\<lambda>x. f a = f x) (x # xs)) = a"
quotient_eq_iff_same_a_r_cos: fixes I :: "'a list set"   and a :: "'a list"   and b :: "'a list" assumes "ideal I (K [X])"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "(a \<ominus>\<^bsub>K [X]\<^esub> b \<in> I) = (I +>\<^bsub>K [X]\<^esub> a = I +>\<^bsub>K [X]\<^esub> b)"
homotopic_loops_trans: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a"   and r :: "real \<Rightarrow> 'a" assumes "homotopic_loops s p q"   and "homotopic_loops s q r" shows "homotopic_loops s p r"