hcju/isabelleps huggingface dataset:
set_insort_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less" shows "set (linorder.insort_insert_key less_eq (\<lambda>x. x) x xs) = insert x (set xs)"
js_equals_upt_k: fixes js :: "nat list"   and k :: "nat" assumes "js <~~> [0..<k + 1]"   and "\<And>i xs ys. \<lbrakk>i < k; js = xs @ ys; xs \<noteq> []; i = last xs\<rbrakk> \<Longrightarrow> i + 1 \<in> set ys" shows "js = [0..<k + 1]"
ordertype_VWF_inc_eq: fixes A :: "V set"   and \<pi> :: "V \<Rightarrow> V" assumes "A \<subseteq> ON"   and "\<pi> ` A \<subseteq> ON"   and "small A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x < y\<rbrakk> \<Longrightarrow> \<pi> x < \<pi> y" shows "tp (\<pi> ` A) = tp A"
unique_decomposition: fixes e :: "('a, 'b, 'c) expr"   and \<E>\<^sub>1 :: "('a, 'b, 'c) cntxt"   and r\<^sub>1 :: "('a, 'b, 'c) expr"   and \<E>\<^sub>2 :: "('a, 'b, 'c) cntxt"   and r\<^sub>2 :: "('a, 'b, 'c) expr" assumes "decompose e \<E>\<^sub>1 r\<^sub>1"   and "decompose e \<E>\<^sub>2 r\<^sub>2" shows "\<E>\<^sub>1 = \<E>\<^sub>2 \<and> r\<^sub>1 = r\<^sub>2"
simps_1: fixes n :: "nat"   and t\<^sub>1 :: "Term_Class.term"   and t\<^sub>2 :: "Term_Class.term" shows "wellformed' n (t\<^sub>1 $ t\<^sub>2) = (wellformed' n t\<^sub>1 \<and> wellformed' n t\<^sub>2)"
inject_2: fixes x2 :: "ty"   and x1 :: "ty"   and y2 :: "ty"   and y1 :: "ty" shows "(x2\<rightarrow>x1 = y2\<rightarrow>y1) = (x2 = y2 \<and> x1 = y1)"
sub_inserted1: fixes Y :: "'c set"   and a :: "'c"   and X :: "'c set" assumes "Y \<subseteq> insert a X"   and "\<not> Y \<subseteq> X" shows "a \<notin> X \<and> a \<in> Y"
case_1: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b" shows "(case Err of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = f1"
charpoly_iarrays_def: fixes A :: "'c iarray iarray" shows "charpoly_iarrays A = det_iarrays_rings (mat_iarray (monom (1::'c) (Suc 0)) (nrows_iarray A) - mat2matofpoly_iarrays A)"
igSwapIPresIGWls_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSwapIPresIGWls MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> igWls MOD s (igSwap MOD zs z1 z2 X)"
memberdecl_acc_modi_def: fixes m :: "memberdecl" shows "accmodi m = (case m of fdecl f \<Rightarrow> accmodi f | mdecl m \<Rightarrow> accmodi m)"
cases: fixes P :: "'a prog"   and a1 :: "char list"   and a2 :: "char list" assumes "subcls1p P a1 a2" obtains C :: "char list" and D :: "char list" and rest :: "fdecl list \<times> 'a mdecl list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(D, rest)\<rfloor>" and "C \<noteq> Object"
pmf_set_integral_def: fixes p :: "'b pmf"   and f :: "'b \<Rightarrow> real"   and A :: "'b set" shows "pmf_set_integral p f A = measure_pmf.expectation p (\<lambda>x. indicat_real A x * f x)"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> F"   and "y \<in> F" shows "x = y"
put_cont_def: fixes put :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f"   and m :: "('d, 'e) contT" shows "put_cont put s m = ContT (\<lambda>k. put s (run_cont m k))"
uminus_zmultiset_def: shows "uminus \<equiv> map_fun rep_zmultiset abs_zmultiset (\<lambda>(Mp, Mn). (Mn, Mp))"
last_no_split: fixes xs :: "'b list"   and ys :: "'b list"   and y :: "'b" shows "(xs \<le>p ys \<cdot> [y]) = (xs = ys \<cdot> [y] \<or> xs \<le>p ys)"
decomposed: shows "mset (bh @ bt) = I + J \<and> mset (ah @ at) = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> R)"
can_syncI: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and m :: "'heap"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and x' :: "'addr option \<times> 'addr frame list"   and m' :: "'heap"   and LT :: "('addr + 'thread_id + 'thread_id) set" assumes "execd_mthr.r_syntax P t x m ta x' m'"   and "LT = collect_waits ta" shows "execd_mthr.can_sync P t x m LT"
in_rangeC: fixes f :: "'a \<Rightarrow> 'b"   and A :: "('a \<Rightarrow> 'b) set"   and x :: "'a" assumes "f \<in> A" shows "f x \<in> rangeC A"
sees_methods_decl_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "'a prog"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "P \<turnstile> C sees_methods Mm" shows "\<exists>Mm' Mm\<^sub>2. P \<turnstile> C' sees_methods Mm' \<and> Mm' = Mm ++ Mm\<^sub>2 \<and> (\<forall>M m D. Mm\<^sub>2 M = \<lfloor>(m, D)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C)"
ring_id_eval: fixes a :: "'a" assumes "a \<in> carrier R" shows "ring_id a = a"
open_euclidean_space_def: fixes U :: "'a euclidean_space set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
reachable_daI: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w" shows "v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w"
dims_subtensor: fixes A :: "'b tensor"   and i :: "nat" assumes "dims A \<noteq> []"   and "i < hd (dims A)" shows "dims (subtensor A i) = tl (dims A)"
int_F: fixes n :: "nat" shows "integrable M (\<lambda>x. - u n x / real n)"
0: shows "rawpsubst (subst \<phi> s y) (zip (map Var ws) (map snd txs)) = subst (rawpsubst \<phi> (zip (map Var ws) (map snd txs))) s y"
le_cases: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
aux: shows "wf_matchF_aux \<sigma> V n R I r aux (Monitor.progress \<sigma> P (formula.MatchF I r) j) 0"
weakenBisimWeakCoinductAux: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and X :: "('e \<times> ('d, 'e, 'f) psi \<times> ('d, 'e, 'f) psi) set" assumes "weak substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> weak.weakenStatImp substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P X Q \<and> weak.weakenSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P X Q \<and> (\<forall>\<Psi>'. (SCompose' \<Psi> \<Psi>', P, Q) \<in> X) \<and> (\<Psi>, Q, P) \<in> X" shows "(\<Psi>, P, Q) \<in> weak.weakenBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
strongly_strategyproof_profile_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and Ri' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "social_decision_scheme agents alts sds" shows "social_decision_scheme.strongly_strategyproof_profile sds R i Ri' = SD (R i) (sds (R(i := Ri'))) (sds R)"
qstar_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<squnion> x \<cdot> y \<le> y" shows "qstar x \<cdot> z \<le> y"
return_heap_def: fixes h :: "'a" shows "return_heap h = Prog (\<lambda>_. h \<turnstile> return ())"
ideduct_mono: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and M' :: "('fun, 'var) Term.term set" assumes "M \<turnstile> t"   and "M \<subseteq> M'" shows "M' \<turnstile> t"
impl_intro: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "P x \<Longrightarrow> Q x" shows "(P \<longrightarrow> Q) x"
\<mu>\<nu>: shows "C.VV.arr \<mu>\<nu>"
neighborhoods_eq: shows "neighborhoods (identity S x) = {U. is_open U \<and> identity S x \<in> U}"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) \<chi>.A.map (J_C.Map d) \<chi>"
assms: shows "vbrelation (r \<union>\<^sub>\<circ> s)"
intros: fixes s :: "('i, 'a) state" assumes "\<And>x\<^sub>i. \<lbrakk>\<not> \<U> s; Some x\<^sub>i = min_lvar_not_in_bounds s; \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i\<rbrakk> \<Longrightarrow> check_dom (check' Positive x\<^sub>i s)"   and "\<And>x\<^sub>i. \<lbrakk>\<not> \<U> s; Some x\<^sub>i = min_lvar_not_in_bounds s; \<not> \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i\<rbrakk> \<Longrightarrow> check_dom (check' Negative x\<^sub>i s)" shows "check_dom s"
distr_lift_IntPairVal: fixes M :: "val measure"   and \<delta> :: "val \<Rightarrow> ennreal"   and f :: "int \<Rightarrow> int \<Rightarrow> int"   and g :: "(int \<times> int \<Rightarrow> ennreal) \<Rightarrow> int \<Rightarrow> ennreal"   and f' :: "real \<Rightarrow> real \<Rightarrow> real" assumes "has_density M (stock_measure (PRODUCT INTEG INTEG)) \<delta>"   and "\<And>M \<delta>. has_density M (count_space UNIV) \<delta> \<Longrightarrow> has_density (distr M (count_space UNIV) (\<lambda>(x, y). f x y)) (count_space UNIV) (g \<delta>)" shows "has_density (distr M (stock_measure INTEG) (lift_RealIntVal2 f f')) (stock_measure INTEG) (g (\<lambda>x. \<delta> (IntPairVal x)) \<circ> extract_int)"
Map_axioms: shows "Map RBT_Set.empty RBT_Map.update RBT_Map.delete lookup M.invar"
ereal_of_enat_gt_zero_cancel_iff: fixes n :: "enat" shows "(0 < ereal_of_enat n) = (0 < n)"
rbt_is_empty: fixes t :: "('c \<times> 'd) tree" shows "rbt_is_empty t = (Tree2.inorder t = [])"
sublens_def: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" shows "X \<subseteq>\<^sub>L Y = (\<exists>Z. vwb_lens Z \<and> X = Z ;\<^sub>L Y)"
tagged_division_of_finite: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" assumes "s tagged_division_of i" shows "finite s"
select_convs_5: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "next \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = next"
coinduct: fixes X :: "('h, 'i, 'j, 'k) converter \<Rightarrow> bool"   and x :: "('h, 'i, 'j, 'k) converter"   and \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>conv. x = conv \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> colossless_gpv \<I>' (run_converter conv x)) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> results_gpv \<I>' (run_converter conv x) \<longrightarrow> X xb \<or> colossless_converter \<I> \<I>' xb)" shows "colossless_converter \<I> \<I>' x"
ins_eq: fixes succs :: "'c \<Rightarrow> 'c list"   and is_node :: "'c \<Rightarrow> bool"   and invariant :: "'d \<Rightarrow> bool"   and ins :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empt :: "'d"   and x :: "'c"   and y :: "'c"   and S :: "'d" assumes "DFS succs is_node invariant ins memb empt"   and "is_node x"   and "is_node y"   and "invariant S"   and "\<not> memb y S" shows "memb x (ins y S) = (x = y \<or> memb x S)"
sylvester_mat_dim_2: fixes p :: "'c poly"   and q :: "'c poly" shows "dim_col (sylvester_mat p q) = degree p + degree q"
conservative_axioms_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "conservative_axioms M T \<equiv> \<forall>A. A \<in> sets M \<longrightarrow> 0 < emeasure M A \<longrightarrow> (\<exists>n>0. 0 < emeasure M ((T ^^ n) -` A \<inter> A))"
bij_is_inj: fixes f :: "'b \<Rightarrow> 'c" assumes "bij f" shows "inj f"
interpret_slp_of_fa: fixes fa :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M' :: "(floatarith, nat) mapping"   and slp' :: "floatarith list"   and xs :: "real list" assumes "slp_of_fa fa M slp = (M', slp')"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs" shows "subterms fa \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"
simps_2: fixes x :: "'b \<times> 'c"   and xs :: "('b \<times> 'c) list" shows "map_of_alist (x # xs) =  map_of_alist xs { fst x \<rightarrow> snd x }"
assms: shows "set (add_effects_of op) \<inter> set (delete_effects_of op) = {}"
polypoly_def: fixes bs :: "'b list"   and p :: "poly" shows "polypoly bs p = map (Ipoly bs) (coefficients p)"
assms_4: shows "g : a \<mapsto>\<^bsub>dg_prod J \<AA>\<^esub> b"
assms_3: shows "atom ` domA \<Delta> \<sharp>* S"
not_eventuallyD: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "\<not> eventually P F" shows "\<exists>x. \<not> P x"
powsum1: 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 x :: "'b"   and n :: "nat" assumes "class.dioid_one_zero plus times one zero less_eq less" shows "dioid_one_zero.powsum plus times one zero x n 0 = power.power one times x n"
uncurry_simp: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "uncurry f (a, b) = f a b"
univ_asTerm_alphaGood: fixes f :: "('e, 'f, 'varSort, 'var, 'g) qTerm \<Rightarrow> 'h"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "f respectsP alphaGood"   and "qGood X" shows "Equiv_Relation2.univ f (asTerm X) = f X"
right_neutral: fixes a :: "'a \<Rightarrow> bool" shows "(a \<and>* \<box>) = a"
termFSwSbMorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "termFSwSbMorph h hA MOD \<equiv> termFSwMorph h hA MOD \<and> presSubstAll h hA MOD"
stake_sdrop: fixes n :: "nat"   and s :: "'c stream" shows "stake n s @- sdrop n s = s"
generator_closed: fixes G :: "('d, 'e) cyclic_group_scheme" assumes "Cyclic_Group.cyclic_group G" shows "\<^bold>g\<^bsub>G\<^esub> \<in> carrier G"
conv_dist_sum: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv" shows "comm_monoid_add.sum sup bot (\<lambda>k. conv (f k)) {k. True} = conv (comm_monoid_add.sum sup bot f {k. True})"
powr_one_eq_one: fixes a :: "'d" shows "(1::'d) powr a = (1::'d)"
V_constotal_1_def: shows "V_constotal_1 \<equiv> {f. \<exists>j p. f = [j] \<odot> p \<and> 2 \<le> j \<and> p \<in> \<R>\<^sub>0\<^sub>1 \<and> \<phi> j = f}"
rtrancl_unfold: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
simps_11: fixes b :: "nat"   and y :: "nat"   and \<omega> :: "Formula.agg_type \<times> event_data"   and b' :: "nat"   and f :: "Formula.trm"   and \<phi> :: "Formula.formula" shows "Formula.fvi b (formula.Agg y \<omega> b' f \<phi>) = Formula.fvi (b + b') \<phi> \<union> Formula.fvi_trm (b + b') f \<union> (if b \<le> y then {y - b} else {})"
pd'_def: shows "pd' = perp_foot (Rep_hyp2 c') l'"
len_replace_ge_same: fixes ys :: "'c list"   and xs :: "'c list"   and x :: "'c" assumes "1 \<le> |ys|" shows "|xs| \<le> |replace x ys xs|"
prems_1: shows "(s2', t2') |\<in>| possible_steps (finsert x e2) s2 r2 l i"
closed_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "closed A f = (\<forall>x\<in>A. \<forall>y\<in>A. x \<squnion>\<^bsub>f\<^esub> y \<in> A)"
psubst_eql: fixes t1 :: "'trm"   and t2 :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "t1 \<in> atrm"   and "t2 \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "distinct (map snd txs)" shows "psubst (eql t1 t2) txs = eql (psubstT t1 txs) (psubstT t2 txs)"
range_constant: fixes x :: "'b" shows "range (\<lambda>_. x) = \<eta> x"
insertGuardedAssertion: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "guarded P" shows "insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F \<langle>\<epsilon>, \<Psi> \<otimes> \<one>\<rangle>"
h_in_linear_span: fixes H :: "'a set"   and h :: "'a" assumes "H \<subseteq> carrier M"   and "h \<in> H" shows "h \<in> linear_span R M (carrier R) H"
rGamma_series_minus_of_nat: fixes k :: "nat" shows "\<forall>\<^sub>F n in sequentially. rGamma_series (- of_nat k) n = (0::'a)"
invertible_mat_smult_mat: fixes A :: "'a mat"   and k :: "'a" assumes "invertible_mat A"   and "k dvd (1::'a)" shows "invertible_mat (k \<cdot>\<^sub>m A)"
simps_1: fixes k :: "'a" shows "delete_aux k [] = []"
d3_conv: fixes x :: "'a"   and y :: "'a" shows "d (x \<sqinter> y) \<le> d (x \<sqinter> d y)"
var_tys_Some_eq: fixes G :: "prog"   and oi :: "obj_tag"   and r :: "oref"   and n :: "vn"   and T :: "ty" shows "(var_tys G oi r n = Some T) = (case r of Inl a \<Rightarrow> case oi of CInst C \<Rightarrow> \<exists>nt. n = Inl nt \<and> fields_table G C (\<lambda>n f. \<not> static f) nt = Some T | Arr t k \<Rightarrow> \<exists>i. n = Inr i \<and> i in_bounds k \<and> t = T | Inr C \<Rightarrow> \<exists>nt. n = Inl nt \<and> fields_table G C (\<lambda>fn f. declclassf fn = C \<and> static f) nt = Some T)"
anonymous_neutral_indifference: fixes P :: "'i \<Rightarrow> ('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "profile A Is P"   and "hasw [x, y] A"   and "card {i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y} = card {i \<in> Is. y \<^bsub>(P i)\<^esub>\<prec> x}" shows "x \<^bsub>(scf P)\<^esub>\<approx> y"
C_eq_RAD_aux: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> []" shows "C (list2FWpolicy p) x = C (list2FWpolicy (removeAllDuplicates p)) x"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case One of One \<Rightarrow> f1 | Two \<Rightarrow> f2 | Three \<Rightarrow> f3) = f1"
countable_datatype: fixes Rep :: "'a \<Rightarrow> ('b, unit) Old_Datatype.node set"   and Abs :: "('b, unit) Old_Datatype.node set \<Rightarrow> 'a"   and rep_set :: "('b, unit) Old_Datatype.node set \<Rightarrow> bool" assumes "type_definition Rep Abs (Collect rep_set)"   and "\<And>x. rep_set x \<Longrightarrow> finite_item x" shows "OFCLASS('a, countable_class)"
Sup_image_sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and Y :: "'b set"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "Y \<noteq> {}" shows "Sup (sup x ` Y) = sup x (Sup Y)"
that: fixes c :: "('a, 'b) chart" assumes "c \<in> atlas"   and "x \<in> domain c" shows "thesis"
Call_lemma: fixes P :: "'b prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b option"   and D :: "String.literal"   and C' :: "String.literal"   and wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T \<and> P \<turnstile> C' \<preceq>\<^sup>* D' \<and> is_type P T' \<and> (\<forall>T\<in>set Ts'. is_type P T) \<and> (m' \<noteq> None \<longrightarrow> wf_md P D' (M, Ts', T', the m'))"
assms_2: shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<sigma> = {}"
connectedin_intermediate_closure_of: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "connectedin X S"   and "S \<subseteq> T"   and "T \<subseteq> X closure_of S" shows "connectedin X T"
components_are_equivalences: fixes a :: "'c" assumes "C.obj a" shows "C.equivalence_map (\<epsilon>\<^sub>0 a)"
simps_1: fixes p1 :: "'a"   and p2 :: "'a"   and v :: "bool" shows "isReceiverOf p1 <p2, inM v> = (p1 = p2)"
sgn_abs: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (sgn a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
sat_test: fixes b :: "'d bool_expr" shows "sat_test ifex_of b = (\<exists>s. val_bool_expr b s)"
comp_cong: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c"   and f' :: "'d \<Rightarrow> 'a"   and g' :: "'e \<Rightarrow> 'd"   and x' :: "'e" assumes "f (g x) = f' (g' x')" shows "(f \<circ> g) x = (f' \<circ> g') x'"
lossless_R2_14_OT: fixes m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool"   and m4 :: "bool"   and c0 :: "bool"   and c1 :: "bool" shows "lossless_spmf (R2_14_OT (m1, m2, m3, m4) (c0, c1))"
\<chi>_def: shows "\<chi> = rawpsubst \<phi> (zip (map Var us) (map snd txs))"
Cons_1: fixes qs :: "'b list"   and tab :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b" assumes "length ps = length qs" shows "tab(ps [\<mapsto>] qs, x \<mapsto> y) = tab(ps @ [x] [\<mapsto>] qs @ [y])"
p_star_contl: fixes X :: "'a list set"   and Y :: "'a list set" shows "p_prod X (p_star Y) = (\<Union>n. p_prod X (path_dioid.power Y n))"
F: shows "0 < offs_num (length ns) xs index key mi ma k"
rcs_tool2: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and a :: "'e"   and x :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "x \<in> H \<bullet>\<^bsub>G\<^esub> a" shows "\<exists>h\<in>H. h \<cdot>\<^bsub>G\<^esub> a = x"
extr_insert_chan_eq: fixes bad :: "agent set"   and IK :: "msg set"   and X :: "chan"   and CH :: "chan set" shows "extr bad IK (insert X CH) = extr bad IK {X} \<union> extr bad IK CH"
assms_5: shows "proj2_incident (apply_cltn2 q C) m"
subfield_axioms_def: fixes K :: "'e set"   and R :: "('e, 'f) ring_scheme" shows "subfield_axioms K R \<equiv> Units (R\<lparr>carrier := K\<rparr>) = K - {\<zero>\<^bsub>R\<^esub>}"