hcju/isabelleps huggingface dataset:
ex_trimming: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "countable \<^bold>E\<^bsub>\<Gamma>\<^esub>"   and "\<And>x. weight \<Gamma> x \<noteq> \<top>" shows "\<exists>g. trimming \<Gamma> f g"
assms_1: fixes h :: "heap"   and as :: "nat set" assumes "P (h, as)" shows "in_range (h, as)"
last_ConsL: fixes xs :: "'g list"   and x :: "'g" assumes "xs = []" shows "last (x # xs) = x"
B_meas: fixes n :: "nat" shows "B n \<in> sets M"
simps_19: fixes v :: "'a exp"   and e :: "'a exp" shows "init_switch (throw v) e = throw v"
moebius_pt_eq_I: fixes M :: "moebius"   and z1 :: "complex_homo"   and z2 :: "complex_homo" assumes "moebius_pt M z1 = moebius_pt M z2" shows "z1 = z2"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "src\<^sub>C (g \<cdot>\<^sub>C f) = src\<^sub>C g \<cdot>\<^sub>C src\<^sub>C f"
enat_le_plus_same_1: fixes x :: "enat"   and y :: "enat" shows "x \<le> x + y"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "R.seq g f" shows "R.cod (R.comp g f) = R.cod g"
expand_update: fixes f :: "'d \<Rightarrow> 'c set"   and x :: "'d"   and S :: "'c set" assumes "f x \<noteq> {}" shows "expand (f(x := S)) = (\<Union>y\<in>S. (\<lambda>g. g(x := y)) ` expand f)"
simps_1: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'a" shows "prg \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = prg"
simps_3: fixes e :: "'c"   and a :: "'d"   and t :: "('c, 'd) SkewBinomialTree"   and t' :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" shows "local.insert e a (t # t' # bq) = (if rank t \<noteq> rank t' then Node e a 0 [] # t # t' # bq else skewlink e a t t' # bq)"
pmf_nonneg: fixes p :: "'a pmf"   and x :: "'a" shows "0 \<le> pmf p x"
Vset_succ_TC: fixes x :: "V"   and k :: "V"   and u :: "V" assumes "x \<in> elts (Vset (ZFC_in_HOL.succ k))"   and "u \<sqsubset> x" shows "u \<in> elts (Vset k)"
scalingD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and c :: "real"   and x :: "'a" assumes "scaling t"   and "sound P"   and "0 \<le> c" shows "c * t P x = t (\<lambda>x. c * P x) x"
that: shows "c \<subseteq>\<^sub>g Catch c1 c2"
edges_of_walk_empty_2: fixes v :: "'a" shows "edges_of_walk [v] = {}"
word_sless_alt: fixes a :: "'b word"   and b :: "'b word" shows "(a <s b) = (sint a < sint b)"
SIod_self: fixes D :: "'c Order" assumes "Order D" shows "D = SIod D (carrier D)"
sinvar_no_offending: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" assumes "sinvar G nP" shows "SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP = {}"
allg: fixes upds :: "('a \<times> 'd) set"   and A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'e"   and xs :: "('a \<times> 'f \<times> 'g) list"   and l'' :: "'a \<Rightarrow> 'b"   and q :: "'f \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>l s n x. x \<in> fst ` upds \<Longrightarrow> A (l(x := n)) = A l"   and "lesvars xs \<subseteq> fst ` upds" shows "A (ListUpdate l'' xs q) = A l''"
set_selsort: fixes xs :: "'a list" shows "set (selsort xs) = set xs"
e_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and i :: "'a" assumes "x \<in> cbox (0::'a) One"   and "y \<in> cbox (0::'a) One"   and "z \<in> cbox (0::'a) One"   and "i \<in> Basis"   and "norm (x - z) < e \<and> norm (y - z) < e \<and> label x i \<noteq> label y i" shows "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
eventually_le_at_bot: fixes c :: "'a" shows "\<forall>\<^sub>F x in at_bot. x \<le> c"
rc: shows "root_cond (p, l, r) (real_of_1 (p, l, r))"
rep_eq: fixes x :: "'b"   and xa :: "'b fset" shows "fset (finsert x xa) = insert x (fset xa)"
le_top: fixes x :: "'a" shows "x \<le> \<top> \<cdot> x"
ms_a: shows "mset (zip xsa ysa) = mset (zip xs ys)"
inf_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B = Collect ((\<lambda>x. x \<in> A) \<sqinter> (\<lambda>x. x \<in> B))"
TTreeAnalysisCardinalityHeap_axioms: shows "TTreeAnalysisCardinalityHeap Texp Aexp Aheap Theap"
power_radical: fixes a :: "'b fps"   and r :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat" assumes "a $ 0 \<noteq> (0::'b)" shows "(r (Suc k) (a $ 0) ^ Suc k = a $ 0) = (fps_radical r (Suc k) a ^ Suc k = a)"
in_set_zipE: fixes x :: "'d"   and y :: "'e"   and xs :: "'d list"   and ys :: "'e list" assumes "(x, y) \<in> set (zip xs ys)" obtains "x \<in> set xs" and "y \<in> set ys"
LID\<^sub>LI_1: fixes s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr"   and r :: "'a" assumes "s = (\<sigma>, \<tau>, e)"   and "r \<in> LID\<^sub>S \<sigma>" shows "r \<in> LID\<^sub>L s"
gcs_plus_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and u :: "'c"   and s :: "'c"   and t :: "'c" assumes "class.ulcs_powerprod plus minus zero lcs" shows "lcs_powerprod.gcs plus minus lcs (plus u s) (plus u t) = plus u (lcs_powerprod.gcs plus minus lcs s t)"
subspaceI: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale"   and "(0::'b) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> scale c x \<in> S" shows "module.subspace scale S"
Betas_fresh: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and x :: "name" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* t"   and "atom x \<sharp> s" shows "atom x \<sharp> t"
prems_1: shows "\<Union> (set (A # As)) \<subseteq> carrier"
elims_2: fixes x :: "'a"   and xa :: "'b list"   and xb :: "'a" assumes "G.cas x xa xb" obtains u :: "'a" and v :: "'a" where "x = u__" and "xa = []" and "xb = v__" and "u__ = v__"   | u :: "'a" and e :: "'b" and es :: "'b list" and v :: "'a" where "x = u" and "xa = e # es" and "xb = v" and "tail G e = u \<and> G.cas (head G e) es v"
comb_cong: fixes x :: "'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> real" assumes "\<And>z. z \<in> B \<Longrightarrow> x z = y z" shows "comb x = comb y"
ref_AIFI: fixes \<Phi> :: "bool"   and S :: "'c nres"   and R :: "('c \<times> 'd) set"   and S' :: "'d nres" assumes "\<Phi> \<Longrightarrow> S \<le> \<Down> R S'" shows "S \<le> \<Down> R (if \<Phi> then S' else FAIL)"
apex_unique_up_to_equivalence: fixes \<rho>' :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg s \<rho>' f' g'" shows "\<exists>w w' \<phi> \<psi> \<theta> \<nu> \<theta>' \<nu>'. internal_equivalence w' w \<psi> \<phi> \<and> \<guillemotleft>w : src s\<^sub>0 \<rightarrow> src f'\<guillemotright> \<and> \<guillemotleft>w' : src f' \<rightarrow> src s\<^sub>0\<guillemotright> \<and> \<guillemotleft>\<theta> : f' \<star> w \<Rightarrow> s\<^sub>0\<guillemotright> \<and> \<guillemotleft>\<nu> : s\<^sub>1 \<Rightarrow> g' \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> \<sigma> = (s \<star> \<theta>) \<cdot> \<a>[s, f', w] \<cdot> (\<rho>' \<star> w) \<cdot> \<nu> \<and> \<guillemotleft>\<theta>' : s\<^sub>0 \<star> w' \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<nu>' : g' \<Rightarrow> s\<^sub>1 \<star> w'\<guillemotright> \<and> local.iso \<nu>' \<and> \<rho>' = (s \<star> \<theta>') \<cdot> \<a>[s, s\<^sub>0, w'] \<cdot> (\<sigma> \<star> w') \<cdot> \<nu>'"
unstream_butlast_trans_Some: fixes g :: "('e, 'f) generator"   and b :: "'e"   and s :: "'f" shows "unstream (butlast_trans g) (Some b, s) = butlast (b # unstream g s)"
intros_1: fixes trn :: "('state, 'act, 'out) trans" assumes "validTrans trn" shows "valid2 [trn]"
cases: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a" assumes "Domainp r a" obtains aa :: "'a" and b :: "'b" where "a = aa" and "r aa b"
infsetsum_Sigma: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and f :: "'c \<times> 'd \<Rightarrow> 'e" assumes "countable A"   and "\<And>i. countable (B i)"   and "f abs_summable_on Sigma A B" shows "infsetsum f (Sigma A B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B x. f (x, y))"
bounded_subset_cbox_symmetric: fixes S :: "'a set" assumes "bounded S" obtains a :: "'a" where "S \<subseteq> cbox (- a) a"
lower_asymptotic_density_intersection: fixes A :: "nat set"   and B :: "nat set" shows "lower_asymptotic_density A + lower_asymptotic_density B \<le> lower_asymptotic_density (A \<inter> B) + 1"
is_run_def: fixes G :: "('c, 'd) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'c" shows "graph_defs.is_run G r \<equiv> r 0 \<in> g_V0 G \<and> ipath (g_E G) r"
store_local_label_empty_1: fixes i :: "nat"   and \<S> :: "s_context"   and s :: "s" assumes "i < length (s_inst \<S>)"   and "store_typing s \<S>" shows "label (s_inst \<S> ! i) = []"
comp_types: fixes B :: "'o"   and C :: "'o"   and A :: "'o" shows "(\<bullet>) \<in> Hom B C \<rightarrow> Hom A B \<rightarrow> Hom A C"
simps_1: fixes x :: "char list"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "\<epsilon> x \<cdot>\<^sub>t \<sigma> = \<sigma> x"
Units_eq: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "Units G = carrier G"
to_effect_def: fixes op :: "('c, 'd) sas_plus_operator" shows "SAS_Plus_Representation.to_effect op \<equiv> map (\<lambda>(v, a). (v, a)) (effect_of op)"
SuccI: fixes kl :: "'b list"   and k :: "'b"   and Kl :: "'b list set" assumes "kl @ [k] \<in> Kl" shows "k \<in> BNF_Greatest_Fixpoint.Succ Kl kl"
wf_subcls1: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "wf_prog wf_md P" shows "wf ((subcls1 P)\<inverse>)"
some_limit_cone_def: shows "some_limit_cone = (SOME \<chi>. local.limit_cone some_limit \<chi>)"
nonneg: fixes xy :: "'a option \<times> 'a option" shows "0 \<le> (case xy of (None, None) \<Rightarrow> pmf q None | (None, Some y') \<Rightarrow> spmf q y' - spmf p y' | (Some x', None) \<Rightarrow> 0 | (Some x', Some y') \<Rightarrow> if x' = y' then spmf p x' else 0)"
group_hom_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" shows "group_hom G H h \<equiv> Group.group G \<and> Group.group H \<and> group_hom_axioms G H h"
tickFree_implies_front_tickFree_single: fixes t :: "'a event list"   and a :: "'a event" assumes "tickFree t" shows "front_tickFree (t @ [a])"
simps_2: fixes s :: "'b \<times> 'b"   and ss :: "('b \<times> 'b) list"   and x :: "'b" shows "preprm_apply (s # ss) x = swp_apply s (preprm_apply ss x)"
hfun_hcomp: fixes A :: "hf"   and B :: "hf"   and F :: "hf"   and C :: "hf"   and G :: "hf" assumes "hfun A B F"   and "hfun B C G" shows "hfun A C (hcomp G F)"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "functor A B F"   and "faithful_functor_axioms A F" shows "faithful_functor A B F"
Map_via_comp: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Map f = (\<lambda>x\<in>Dom f. Map (f \<cdot> MkArr {U} (Dom (local.dom f)) (\<lambda>_\<in>{U}. DOWN (UP x))) U)"
progress: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and p' :: "'a"   and is' :: "instr list" assumes "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')" shows "p' \<noteq> p \<or> is' \<noteq> []"
sign_simps_23: fixes ag :: "'a"   and bg :: "'a"   and n :: "nat" shows "(ag * bg) ^ n = ag ^ n * bg ^ n"
map_eq_append_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'a list"   and zs :: "'a list" shows "(map f xs = ys @ zs) = (\<exists>ys' zs'. map f ys' = ys \<and> map f zs' = zs \<and> xs = ys' @ zs')"
mult_mat_vec_carrier: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "A *\<^sub>v v \<in> carrier_vec nr"
f: fixes A :: "'a set" assumes "A \<in> sets M" shows "\<integral>\<^sup>+ x. f x * indicator A x \<partial>M = \<integral>\<^sup>+ x. f' x * indicator A x \<partial>M"
zip_rev: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys" shows "zip (rev xs) (rev ys) = rev (zip xs ys)"
zero: fixes ia :: "nat"   and ja :: "nat" assumes "ia < n"   and "k \<le> ia"   and "ja < k" shows "B $$ (ia, ja) = (0::'a)"
desargues_config_rot_CCW: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "desargues_config B C A B' C' A' N P M R"
has_derivative_sequence: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and x0 :: "'a"   and l :: "'b" assumes "convex S"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_derivative f' n x) (at x within S)"   and "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"   and "x0 \<in> S"   and "(\<lambda>n. f n x0) \<longlonglongrightarrow> l" shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) \<longlonglongrightarrow> g x \<and> (g has_derivative g' x) (at x within S)"
mult_right_zero: fixes x :: "'c" shows "bot_homomorphism ((\<sqinter>) x)"
ODList_toList: fixes xs :: "'b odlist" shows "ODList (toList xs) = xs"
tab_is_tabulation: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r r.tab (tab\<^sub>0 r) (tab\<^sub>1 r)"
hyp_1: assumes "mat (row_length M) (length M) M" shows "mat (row_length M * length v) (length M) (vec_mat_Tensor v M)"
sum_fsmult_distrib: fixes m :: "'v"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'f" assumes "m \<in> V"   and "finite A"   and "f ` A \<subseteq> UNIV" shows "sum f A \<sharp>\<cdot> m = (\<Sum>a\<in>A. f a \<sharp>\<cdot> m)"
compact_Union: fixes S :: "'b set set" assumes "finite S"   and "\<And>T. T \<in> S \<Longrightarrow> compact T" shows "compact (\<Union> S)"
simps_2: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" shows "timpls_transformable_to' TI (Fun f T) (Fun g S) = ((f = g \<or> is_Abs f \<and> is_Abs g \<and> in_trancl TI (the_Abs f) (the_Abs g)) \<and> list_all2 (timpls_transformable_to' TI) T S)"
Some_in_opt: fixes x :: "'b"   and A :: "'b set" shows "(Some x \<in> opt A) = (x \<in> A)"
map_ident: fixes t :: "'a list" shows "map (\<lambda>x. x) t = t"
is_valid_operator_op: shows "is_valid_operator_sas_plus \<Psi> op"
ultraE: fixes T :: "'b set set"   and F :: "'b set set"   and A :: "'b set" assumes "ultra\<^bsub>T\<^esub> F"   and "A \<subseteq> carrier\<^bsub>T\<^esub>" obtains "A \<in> F"   | "carrier\<^bsub>T\<^esub> - A \<in> F"
pow_generator_eq_iff_cong: fixes G :: "('a, 'b) cyclic_group_scheme"   and x :: "nat"   and y :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "finite (carrier G)" shows "(\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> x = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> y) = [x = y] (mod order G)"
real_vector_affinity_eq: fixes m :: "real"   and x :: "'b"   and c :: "'b"   and y :: "'b" assumes "m \<noteq> 0" shows "(m *\<^sub>R x + c = y) = (x = y /\<^sub>R m - c /\<^sub>R m)"
has_complex_derivative_uniform_sequence: fixes S :: "complex set"   and f :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and f' :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "open S"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially" shows "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. f' n x) \<longlonglongrightarrow> g' x"
b_init: fixes s :: "'a"   and T :: "('a, 'b) TS_scheme" assumes "s \<in> init T" shows "[s] \<in> beh T"
diff_self: fixes x :: "'a" assumes "x \<in> E" shows "x - x = (0::'a)"
takeWhile_takes_all: fixes xs :: "'c list"   and m :: "nat"   and P :: "'c \<Rightarrow> bool" assumes "length xs = m" shows "(m \<le> length (takeWhile P xs)) = Ball (set xs) P"
rev_swap: fixes xs :: "'f list"   and ys :: "'f list" shows "(rev xs = ys) = (xs = rev ys)"
is_natural_1: fixes f :: "'c \<times> 'c" assumes "C.CC.arr f" shows "FoT\<^sub>C.map f \<cdot>\<^sub>D \<phi> (C.CC.dom f) = \<phi> f"
affected_fixate_trivial: fixes a :: "'b"   and f :: "'b perm" assumes "a \<notin> affected f" shows "affected (fixate a f) = affected f"
find_Some_imp_lookup_Some: fixes P :: "'a \<Rightarrow> bool"   and xs :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" assumes "List.find (\<lambda>(k, uu_). P k) xs = Some (k, v)" shows "lookup P xs = Some v"
encode_problem_parallel_complete_vi_c: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and k :: "nat"   and v :: "'b" assumes "is_valid_problem_strips \<Pi>"   and "is_parallel_solution_for_problem \<Pi> \<pi>"   and "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>)" shows "valuation_for_plan \<Pi> \<pi> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)"
operational_semantics_trans_generalized: fixes n :: "nat"   and \<C>\<^sub>1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and \<C>\<^sub>2 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and m :: "nat"   and \<C>\<^sub>3 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" assumes "\<C>\<^sub>1 \<hookrightarrow>\<^bsup>n\<^esup> \<C>\<^sub>2"   and "\<C>\<^sub>2 \<hookrightarrow>\<^bsup>m\<^esup> \<C>\<^sub>3" shows "\<C>\<^sub>1 \<hookrightarrow>\<^bsup>n + m\<^esup> \<C>\<^sub>3"
convex_hull_convex_hull_listD: fixes A :: "'a vec set"   and x :: "'a vec" assumes "A \<subseteq> carrier_vec n"   and "x \<in> convex_hull A" shows "\<exists>as. set as \<subseteq> A \<and> x \<in> convex_hull_list as"
bij_1: shows "bij_betw f (uverts G\<^sub>2) (uverts G\<^sub>2')"
free_module_mem: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and m :: "'f"   and n :: "'h"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "n \<in> carrier N"   and "FM\<^bsub>R\<^esub> P M N" shows "(m, n) \<in> carrier P"
simps_46: fixes x3 :: "'b" shows "set1_xstate (Fault x3) = {}"
normalize_prime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "normalize p = p"
in_sccs_verts_conv_reachable: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set" assumes "wf_digraph G" shows "(S \<in> pre_digraph.sccs_verts G) = (S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u))"
top_edge_is_compos_of_scal_and_g1: shows "(\<lambda>x. twoC (1, x)) = (\<lambda>x. (g1 x, x)) \<circ> (\<lambda>x. a + (b - a) * x)"
simps_2: fixes x :: "'a"   and s :: "'a \<Rightarrow> value option" shows "aval (V x) s = s x"
S5: shows "b_least2 f x y \<le> b_least2 f x (Suc y)"
simps_1: shows "ReZ \<i>\<^sub>\<int> = 0"
perm: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs <~~> ys" shows "P xs = P ys"
beukers_nn_integral1_def: fixes r :: "nat"   and s :: "nat" shows "beukers_nn_integral1 r s = \<integral>\<^sup>+x\<in>{0<..<1} \<times> {0<..<1}. (case x of (x, y) \<Rightarrow> ennreal (- ln (x * y) / (1 - x * y) * x ^ r * y ^ s))\<partial>lborel"
remove_key_sum: fixes k :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'c" shows "remove_key k f + monomial (lookup f k) k = f"
simps_1: fixes uu :: "'b" shows "comlist \<lbrace> uu \<rbrace> = []"
DF_N: fixes k :: "nat"   and i :: "nat" shows "DF k i \<subseteq> N \<inter> {0<..}"
vsv_vimageI2: fixes a :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda A f)" shows "VLambda A f\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (VLambda A f)"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (F f) = Cop.arr f"
InvariantGetReasonIsReasonAfterExhaustiveUnitPropagate: fixes state :: "State" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))" shows "let state' = exhaustiveUnitPropagate state in InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
silent_move_from_def: fixes trsys :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and \<tau>move :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and s0 :: "'a"   and s1 :: "'a"   and s2 :: "'a" shows "\<tau>trsys.silent_move_from trsys \<tau>move s0 s1 s2 = ((\<tau>trsys.silent_move trsys \<tau>move)\<^sup>*\<^sup>* s0 s1 \<and> \<tau>trsys.silent_move trsys \<tau>move s1 s2)"
prv_imp_cnjI: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp \<phi> (imp \<chi> (cnj \<phi> \<chi>)))"
nontrivial_ring: assumes "carrier R \<noteq> {\<zero>}" shows "\<zero> \<noteq> \<one>"
nat_pow_Suc: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'a" shows "[Suc n] \<cdot>\<^bsub>P\<^esub> x = [n] \<cdot>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> x"
inj_on_insert: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and A :: "'a set" shows "inj_on f (insert a A) = (inj_on f A \<and> f a \<notin> f ` (A - {a}))"
psubset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "order.greater (insert x B) A = (if x \<in> B then order.greater B A else if x \<in> A then order.greater B (A - {x}) else order.greater_eq B A)"
integrable_on_cmult_right: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set"   and c :: "real" assumes "f integrable_on s" shows "(\<lambda>x. f x * of_real c) integrable_on s"