hcju/isabelleps huggingface dataset:
TR_submodule: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "submodule R P (TR\<^bsub>R\<^esub> M N P)"
double_diff: fixes A :: "'q set"   and B :: "'q set"   and C :: "'q set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "B - (C - A) = A"
assms: shows "s \<in> deriv c r \<rightarrow> v"
from_hma_to_hma\<^sub>m: fixes A :: "'a mat" assumes "A \<in> carrier_mat CARD('b) CARD('c)" shows "Mod_Type_Connect.from_hma\<^sub>m (Mod_Type_Connect.to_hma\<^sub>m A) = A"
join_2: shows "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"
F: shows "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
poly_of_const_def: shows "UP.poly_of_const = (\<lambda>k. UP.normalize [k])"
A: shows "\<exists>xi. xi \<in> Gromov_boundary \<and> Gromov_extension f xi = xi \<and> additive_strength f xi < 0"
rep_eq: fixes x :: "mini_alg_unique" shows "Rep_mini_alg_unique (mau_inverse x) = ma_inverse (Rep_mini_alg_unique x)"
mm_equiv_elim: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and c\<^sub>1 :: "'c"   and mds :: "Mode \<Rightarrow> 'a set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and c\<^sub>2 :: "'c"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_security_init.mm_equiv dma \<C>_vars eval" obtains \<R> :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set" where "sifum_security_init.strong_low_bisim_mm dma \<C>_vars eval \<R>" and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> \<R>"
simps_5: fixes f :: "'d \<Rightarrow> 'c"   and p :: "('b, 'd) fm" shows "sub f (\<^bold>\<diamond> p) = (\<^bold>\<diamond> sub f p)"
elims_2: fixes x :: "'b infty"   and xa :: "'b infty" assumes "x \<le> xa" obtains uv_ :: "'b infty" where "x = uv_" and "xa = Infty"   | a :: "'b" and b :: "'b" where "x = Num a" and "xa = Num b" and "a \<le> b"
simps_7: fixes x3 :: "dB"   and x1 :: "nat" shows "Abs x3 \<noteq> Var x1"
lc_setI: fixes f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "f \<in> F"   and "f \<noteq> 0" shows "lcf f \<in> punit.lc_set F"
that: fixes f :: "nat \<Rightarrow> nat" assumes "pivot_fun (row_echelon A) f (dim_col (row_echelon A))" shows "thesis"
insertion_code: fixes i :: "nat \<Rightarrow> 'a"   and mp :: "'a mpoly" shows "insertion i mp = (\<Sum>m\<in>monomials mp. MPoly_Type.coeff mp m * (\<Prod>k\<in>keys m. i k ^ lookup m k))"
zero_order_2: fixes n :: "'c multiset" shows "(n \<subseteq># {#}) = (n = {#})"
simp1: fixes y :: "brick list"   and xs :: "wall"   and x :: "brick list" assumes "codomain_block y \<noteq> 0"   and "is_tangle_diagram xs \<and> is_tangle_diagram (basic (make_vert_block (nat (codomain_block y)))) \<longrightarrow> is_tangle_diagram (xs \<otimes> basic (make_vert_block (nat (codomain_block y))))" shows "is_tangle_diagram (x * xs) \<and> is_tangle_diagram (basic y) \<longrightarrow> is_tangle_diagram (x * xs \<otimes> basic y)"
that: fixes as :: "nat \<Rightarrow> 'a" assumes "A.is_run as"   and "\<forall>i. R (as i) (cs i)" shows "thesis"
assms_6: shows "l' \<in> labels \<Gamma> q"
wf_measure: fixes f :: "'b \<Rightarrow> nat" shows "wf (measure f)"
c: shows "column i (fst (QR_decomposition A)) = column ia (fst (QR_decomposition A))"
divide: fixes biG :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and biI :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and biH :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b" assumes "subdivision biG biI"   and "subdivision_step (fst biI) (snd biI) (fst biH) (snd biH) (u, v, w) (uv, uw, vw)" shows "subdivision biG biH"
irreducible_def: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" shows "irreducible G a = (a \<notin> Units G \<and> (\<forall>b\<in>carrier G. properfactor G b a \<longrightarrow> b \<in> Units G))"
frameWeaken: fixes F :: "'b frame"   and G :: "'b frame" shows "F \<hookrightarrow>\<^sub>F F \<otimes>\<^sub>F G"
something_disjoint: fixes x :: "i" assumes "x \<noteq> u" shows "\<exists>v. \<not> O v x"
iNextStrong_iEx_conv: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNextStrong t0 I P = (\<diamond> t {inext t0 I} \<down>> t0. P t)"
bind_ask2: fixes m :: "'m"   and f :: "'a option \<Rightarrow> 'r \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
plus_uint8_def: shows "(+) \<equiv> map_fun Rep_uint8 (map_fun Rep_uint8 Abs_uint8) (+)"
WT_converter_comp: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and \<I>'' :: "('l, 'm) \<I>"   and conv' :: "('j, 'k, 'l, 'm) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>', \<I>'' \<turnstile>\<^sub>C conv' \<surd>" shows "\<I>, \<I>'' \<turnstile>\<^sub>C conv \<odot> conv' \<surd>"
extremum_unique: fixes a :: "'b" shows "(\<top> \<le> a) = (a = \<top>)"
C_3: shows "\<forall>l s. P l s \<longrightarrow> pre C Q l s"
hyps: fixes G :: "nat \<times> ('a \<times> nat \<times> nat) set" assumes "edges (graph_of G) \<subseteq> L \<times> UNIV" shows "edges (graph_of (mk_chain sel Rules G i)) \<subseteq> L \<times> UNIV"
prime_nat_def: fixes p :: "nat" shows "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> m = 1 \<or> m = p))"
dom_char: fixes f :: "'b" shows "J.dom f = (if f \<in> Collect J0.arr \<union> Collect J1.arr then f else J.null)"
poly_add_replicate_zero'_2: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and n :: "nat" assumes "ring R"   and "set p \<subseteq> carrier R" shows "ring.poly_add R (replicate n \<zero>\<^bsub>R\<^esub>) p = ring.normalize R p"
indep_finite: fixes A :: "'a set" assumes "indep A" shows "finite A"
assms_2: shows "f2 \<in> o[F](f1)"
piecewise_differentiable_D2: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "g1 +++ g2 piecewise_differentiable_on {0..1}"   and "pathfinish g1 = pathstart g2" shows "g2 piecewise_differentiable_on {0..1}"
concl_of_def: fixes inference :: "'b inference" shows "concl_of inference = (case inference of Infer x1 x2 x3 \<Rightarrow> x3)"
prj1_simps_3: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "cod \<p>\<^sub>1[f, g] = local.dom f"
disj_notE_PL: fixes A :: "'a set"   and q :: "'a"   and p :: "'a" assumes "A \<turnstile> q"   and "A \<turnstile> p .\<or>. .\<not> q" shows "A \<turnstile> p"
set_empty_eq: fixes xo :: "'d option" shows "(set_option xo = {}) = (xo = None)"
index_less: fixes x :: "'e"   and xs :: "'e list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "index xs x < n"
PiE_restrict: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "f \<in> Pi\<^sub>E A B" shows "restrict f A = f"
CH_XH_largest: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "CH (XH_largest ds X) = X"
n_absorb1: fixes x :: "'a"   and y :: "'a" shows "!x \<oplus> !x \<cdot> !y = !x"
assms_4: fixes x :: "'a"   and A :: "'a trm"   and T :: "PreSimplyTyped.type" assumes "x \<notin> S"   and "P S A" shows "P S (Fn x T A)"
graphEqImage: fixes x :: "'c"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> X" shows "graph X f ,, x = f x"
rep_list_monomial: fixes u :: "'t"   and c :: "'b" assumes "distinct fs" shows "rep_list (monomial c u) = (local.punit.monom_mult c (pp_of_term u) (fs ! component_of_term u) when component_of_term u < length fs)"
case_eq_if: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and tm :: "'b tm" shows "(case tm of TM x xa \<Rightarrow> f x xa) = f (tm.val tm) (un_TM2 tm)"
Abs_IF1_inverse: fixes y :: "'d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool" assumes "y \<in> car_init1 undefined" shows "Rep_IF1 (Abs_IF1 y) = y"
Jmp: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and jump :: "jump" shows "E,dt\<Turnstile>Jmp jump\<Colon>\<surd>"
domsdisj: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and A :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and y :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "p \<noteq> []"   and "\<And>x s. s \<in> set p \<and> x \<in> dom (Cp A) \<Longrightarrow> x \<notin> dom (Cp s)"   and "y \<in> dom (Cp A)" shows "y \<notin> dom (Cp (list2FWpolicy p))"
\<W>_montone: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and P :: "(symbol \<times> character list) list set"   and Q :: "(symbol \<times> character list) list set"   and Doc :: "character list"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "P \<subseteq> Q" shows "LocalLexing.\<W> \<NN> \<TT> \<RR> \<SS> Lex Doc P k \<subseteq> LocalLexing.\<W> \<NN> \<TT> \<RR> \<SS> Lex Doc Q k"
exists_split_characterisations_6: shows "(\<exists>x. x \<sqinter> x\<^sup>T = (1::'a) \<and> linear x) = (\<exists>x. asymmetric x \<and> strict_linear x)"
divides_mult: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b c"   and "algebraic_semidom.coprime times one a b" shows "dvd.dvd times (times a b) c"
matrix_eq: fixes A :: "(('b, 'c) vec, 'd) vec"   and B :: "(('b, 'c) vec, 'd) vec" shows "(A = B) = (\<forall>x. A *v x = B *v x)"
less_imp_insert_lepoll: fixes A :: "'a set"   and B :: "'b set"   and a :: "'a" assumes "A \<prec> B" shows "insert a A \<lesssim> B"
sign_changes_singleton: fixes x :: "'a" shows "sign_changes [x] = 0"
reachable_mono: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H (del_vert u)" shows "u \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> v"
floor_divide_lower: fixes q :: "'b"   and p :: "'b" assumes "(0::'b) < q" shows "of_int \<lfloor>p / q\<rfloor> * q \<le> p"
CardinalityPrognosisSafe_axioms: shows "CardinalityPrognosisSafe prognosis cHeap Aheap Aexp"
add_tvars'_tvs_pre: fixes t :: "Core.term"   and acc :: "(variable \<times> String.literal set) list" shows "set (add_tvars' t acc) = set acc \<union> tvs t"
fbox_iso: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'b set" assumes "P \<le> Q" shows "|F] P \<le> |F] Q"
neq_le_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.order less_eq less"   and "a \<noteq> b"   and "less_eq a b" shows "less a b"
isPath_fwd_cases: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "Graph.isPath c s p t" obtains "p = []" and "t = s"   | p' :: "(nat \<times> nat) list" and u :: "nat" where "p = (s, u) # p'" and "(s, u) \<in> Graph.E c" and "Graph.isPath c u p' t"
IH: assumes "finite I" shows "sup_measure.F id I \<le> sup_measure.F id (I \<union> J)"
assms_1: shows "continuous_on (cbox (0::'a) One) f"
Cons_3: shows "list_all (\<lambda>i. i < length ls \<and> ls ! i = ts) (is @ [i])"
elims: fixes x :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and y :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "start_proof x = y" obtains S :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" where "x = S" and "y = ([S], S)"
Value_InitStatus_HAInitValue: fixes A :: "('b, 'c, 'a) hierauto" shows "Value (InitStatus A) = HAInitValue A"
pt_bij4: fixes pi :: "('e \<times> 'e) list"   and x :: "'d"   and y :: "'d" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)"   and "pi \<bullet> x = pi \<bullet> y" shows "x = y"
aaa: shows "a = sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<i])"
b: fixes f :: "'a \<Rightarrow> real \<times> real" shows "(\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f"
swap_commute: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and zs' :: "'varSort"   and x :: "'var"   and y :: "'var"   and x' :: "'var"   and y' :: "'var" assumes "good X"   and "zs \<noteq> zs' \<or> {x, y} \<inter> {x', y'} = {}" shows "X #[x \<and> y]_zs #[x' \<and> y']_zs' = X #[x' \<and> y']_zs' #[x \<and> y]_zs"
neg_eq_TT_iff: fixes x :: "tr" shows "(neg\<cdot>x = TT) = (x = FF)"
store_store_ids_imply_messages_same: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e1 :: "'b"   and mo :: "'a"   and r :: "'a"   and e2 :: "'b" assumes "xs prefix of j"   and "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"   and "(r, Store e2 i r) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
random_bst_singleton: fixes x :: "'a" shows "random_bst {x} = return_pmf \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle>"
simps_6: fixes r :: "(bool \<times> 'b) rexp" shows "final (Star r) = final r"
null_sets_distr_iff: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and A :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "(A \<in> null_sets (distr M N f)) = (f -` A \<inter> space M \<in> null_sets M \<and> A \<in> sets N)"
le_ASSUMEI: fixes \<Phi> :: "bool"   and M :: "'ma"   and M' :: "'ma" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> bind (ASSUME \<Phi>) (\<lambda>_. M')"
l2_inv5_init: shows "init l2 \<subseteq> l2_inv5"
Cdom2: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "x \<in> dom (Cp b)" shows "Cp (a \<oplus> b) x = Cp b x"
REF_R_BRK: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi> it \<sigma>"   and "\<Phi>' it' \<sigma>'"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "it \<noteq> {}"   and "it' \<noteq> {}"   and "\<not> c \<sigma>"   and "\<not> c' \<sigma>'" shows "(\<sigma>, \<sigma>') \<in> R'"
llist_of_tllist_inverse: fixes b :: "('a, 'b) tllist" shows "tllist_of_llist (terminal b) (llist_of_tllist b) = b"
invariance_of_domain_sphere_affine_set: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and r :: "real"   and a :: "'a" assumes "continuous_on S f"   and "inj_on f S"   and "f ` S \<subseteq> T"   and "r \<noteq> 0"   and "affine T"   and "aff_dim T < int DIM('a)"   and "openin (top_of_set (sphere a r)) S" shows "openin (top_of_set T) (f ` S)"
join_conv_bind_option: fixes x :: "'c option option" shows "join_option x = x \<bind> id"
Accessible_left_lang_eq: fixes q :: "hf" shows "Accessible.left_lang q = left_lang q"
union_inter_neutral: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B"   and "\<forall>x\<in>A \<inter> B. g x = (1::'b)" shows "prod g (A \<union> B) = prod g A * prod g B"
gim: shows "g ` span ((+) (- b) ` T) = span ((+) (- a) ` S)"
execs_elim_cases_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>[],css,s\<rangle> \<Rightarrow> t" obtains "css = []" and "t = s"   | nrms :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" and sa :: "'e" and abrs :: "('e, 'd, 'f) com list" where "css = (nrms__, abrs__) # cssa__" and "s = Normal sa__" and "\<Gamma>\<turnstile> \<langle>nrms__,cssa__,Normal sa__\<rangle> \<Rightarrow> t"   | abrs :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" and sa :: "'e" and nrms :: "('e, 'd, 'f) com list" where "css = (nrmsa__, abrsa__) # cssaa__" and "s = Abrupt sa" and "\<Gamma>\<turnstile> \<langle>abrsa__,cssaa__,Normal sa\<rangle> \<Rightarrow> t"   | nrms :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" and f :: "'f" and abrs :: "('e, 'd, 'f) com list" where "css = (nrmsb__, abrsb__) # cssab__" and "s = Fault f" and "\<Gamma>\<turnstile> \<langle>nrmsb__,cssab__,Fault f\<rangle> \<Rightarrow> t"   | nrms :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" and abrs :: "('e, 'd, 'f) com list" where "css = (nrms, abrs) # cssa" and "s = Stuck" and "\<Gamma>\<turnstile> \<langle>nrms,cssa,Stuck\<rangle> \<Rightarrow> t"
boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "sup b c \<le> a" obtains "b \<le> a" and "c \<le> a"
MakeFaceFinal_facesAt_distinct: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "facesAt_distinct (makeFaceFinal f g)"
singletonE: fixes b :: "'b"   and a :: "'b" assumes "b \<in> {a}" obtains "b = a"
of_weak_ranking_permute': fixes f :: "'b \<Rightarrow> 'b"   and xs :: "'b set list" assumes "f permutes \<Union> (set xs)" shows "map_relation f (of_weak_ranking xs) = of_weak_ranking (map ((`) (inv f)) xs)"
wlsAbs: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" shows "wlsAbs (xs, s) A = (isInBar (xs, s) \<and> (\<exists>x X. wls s X \<and> A = Abs xs x X))"
strict_iff_order: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subset># b) = (a \<subseteq># b \<and> a \<noteq> b)"
ABP_def: shows "ABP = (\<lambda>l. AB l ^ (n (l - 1) - n i) * f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1)))"
element_ptr_casts_commute3: fixes node_ptr :: "(_) node_ptr" assumes "is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" shows "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>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) = node_ptr"
p\<^sub>0_simps: shows "trg f\<^sub>0gh\<^sub>1.p\<^sub>0 = src (tab\<^sub>1 g \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)"
assms: shows "{$$} \<turnstile> v, Lam x vP', vV : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
assms_3: shows "reduced_row_echelon_form_upt_k A k"
span_span: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set" assumes "module scale" shows "module.span scale (module.span scale A) = module.span scale A"
is_natural_1: fixes f :: "'a" assumes "C.arr f" shows "C.map f \<cdot> C.map (C.dom f) = C.map f"
option_rel_Some2: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e option"   and y :: "'f" shows "rel_option A x (Some y) = (\<exists>x'. x = Some x' \<and> A x' y)"
orderI: fixes a :: "'c"   and b :: "'c" assumes "a = min a b" shows "a \<le> b"
map_2: fixes f :: "'d \<Rightarrow> 'b"   and x2 :: "'d" shows "map_option f (Some x2) = Some (f x2)"
that: fixes y :: "'a" assumes "y \<in> C"   and "\<not> y \<le> x" shows "thesis"
set_integral_Un_AE: fixes A :: "'a set"   and B :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M"   and "set_integrable M A f"   and "set_integrable M B f" shows "set_lebesgue_integral M (A \<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f"
elims: fixes x :: "'b constr list"   and y :: "'b run set" assumes "\<lbrakk>\<lbrakk> x \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = y" obtains "x = []" and "y = {\<rho>. True}"   | \<gamma> :: "'b constr" and \<Gamma> :: "'b constr list" where "x = \<gamma> # \<Gamma>" and "y = \<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"
allow_pfun_def: fixes pf :: "'e \<Rightarrow> 'f option" shows "A\<^sub>p pf \<equiv> \<lambda>x. case pf x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>allow y\<rfloor>"
inf_closed_set_def: fixes X :: "'b set" shows "inf_closed_set X = (\<forall>x\<in>X. \<forall>y\<in>X. x \<sqinter> y \<in> X)"
preorder_on_def: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "Order_Predicates.preorder_on carrier le \<equiv> ((\<forall>x y. le x y \<longrightarrow> x \<in> carrier) \<and> (\<forall>x y. le x y \<longrightarrow> y \<in> carrier)) \<and> (\<forall>x. x \<in> carrier \<longrightarrow> le x x) \<and> (\<forall>x y z. le x y \<longrightarrow> le y z \<longrightarrow> le x z)"
relhoare_abstract_UN: fixes pre :: "('a \<times> 'b) set"   and Ra :: "'c \<Rightarrow> ('a \<times> 'a) set"   and x :: "'c"   and Rc :: "('b \<times> 'b) set"   and post :: "('a \<times> 'b) set" assumes "{pre} Ra x, Rc {> post}" shows "{pre} \<Union> (range Ra), Rc {> post}"
reflp_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "reflp_on P A = (\<forall>a\<in>A. P a a)"
tok_id_hash_rel: fixes n :: "nat"   and m :: "nat" assumes "n \<in> bad_tok_prik" shows "s\<^sub>0 \<Turnstile> tok_id_hash n m"
sg2: shows "\<forall>t. buffer t = (if dt t = [] then fin_inf_append [[]] buffer t else dt t)"
isOK_I: fixes m :: "'d + 'e"   and x :: "'e" assumes "m = Inr x" shows "isOK m"
X: shows "simple_distributed M (\<lambda>x. (X x, Y x)) Px"
le_cases: fixes x :: "'t"   and y :: "'t" obtains "x \<preceq>\<^sub>t y"   | "y \<preceq>\<^sub>t x"
Tpows_v_def: shows "Tpows_v = map (\<lambda>S. S v) Tpows"
intensional_simps_28: fixes P :: "'b \<Rightarrow> bool" shows "P \<or> \<not> P \<equiv> #True"
S6Reply: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MClkReply memCh crCh cst p \<and> HNext rmhist p \<and> $S6 rmhist p \<and> unchanged (e p, r p, m p) \<longrightarrow> S1 rmhist p$"
invh_baldR_invc: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invh l"   and "invh r"   and "bheight l = bheight r + 1"   and "invc l" shows "invh (baldR l a r) \<and> bheight (baldR l a r) = bheight l"
FunctorExt_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" shows "FunctorExt F \<equiv> (##) F \<in> extensional (mor\<^bsub>CatDom F\<^esub>)"