hcju/isabelleps huggingface dataset:
ProblemIkTpartG_GEModel_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "('b, 'c) epsym \<Rightarrow> univ list \<Rightarrow> bool" shows "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP \<equiv> ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl \<and> CM.Model (ProblemIkTpartG.GE_wtFsym wtFsym resOf protCl) (ProblemIkTpartG.GE_wtPsym wtPsym prot protFw) (ProblemIkTpartG.GE_arOf arOf) (ProblemIkTpartG.GE_resOf resOf) (ProblemIkTpartG.GE_parOf parOf) (ProblemIkTpartG.gPB wtFsym arOf resOf \<Phi> prot protFw protCl) eintT eintF eintP"
nth_append_take_is_nth_conv: fixes i :: "nat"   and j :: "nat"   and xs :: "'c list"   and ys :: "'c list" assumes "i < j"   and "j \<le> length xs" shows "(take j xs @ ys) ! i = xs ! i"
test_simps2_1: fixes y :: "'b list" shows "test None y = (case y of [] \<Rightarrow> 1::'a | x # xa \<Rightarrow> 2::'a)"
i_set0_mult_def: fixes k :: "nat" shows "i_set0_mult k \<equiv> insert {} (i_set_mult k)"
GFP_idem: fixes f :: "'a \<Rightarrow> 'a" assumes "f \<in> carrier L \<rightarrow> carrier L"   and "Mono\<^bsub>L\<^esub> f"   and "Idem\<^bsub>L\<^esub> f" shows "GFP\<^bsub>L\<^esub> f .=\<^bsub>L\<^esub> f \<top>\<^bsub>L\<^esub>"
Un_Int_eq_2: fixes S :: "'b set"   and T :: "'b set" shows "(S \<union> T) \<inter> T = T"
comp_fun_commute_product_fold: fixes B :: "'b set" assumes "finite B" shows "comp_fun_commute (\<lambda>x z. Finite_Set.fold (\<lambda>y. insert (x, y)) z B)"
preal_imp_pos: fixes A :: "rat set"   and r :: "rat" assumes "Dedekind_Real.cut A"   and "r \<in> A" shows "0 < r"
less_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f < g) = (f \<le> g \<and> \<not> g \<le> f)"
alpha_refl: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm" assumes "qGood X" shows "X #= X"
simps: fixes p :: "'a extended"   and i :: "'a"   and u :: "'a extended"   and uu :: "'c option" shows "delete (InsertMessage p i u uu) = InsertMessage p i u None"
assms: shows "Some (a1, a2) = div_int_poly_step q b (Some (c1, c2))"
simps_1: fixes \<zeta> :: "('s, 'v) hd"   and i :: "nat" shows "coef (Hd \<zeta>) i = coef_hd \<zeta> i"
assms_1: shows "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q"
hyps: shows "(of_bool b + (2::'a) * a) div (2::'a) = a"
then_return_eq: fixes p :: "'a + 'c"   and q :: "'a + 'b"   and f :: "'b" shows "(p \<bind> (\<lambda>_. q) = Inr f) = (isOK p \<and> q = Inr f)"
edges2: fixes E :: "'b set set"   and e :: "'b set" assumes "Graph_E E"   and "e \<in> E" shows "card e = 2"
ed_ou_witness_ed_ou: fixes X :: "ed_ou_witness set" shows "ed_ou X"
col__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Col A B C" shows "Coplanar A B C D"
inject_3: fixes x3 :: "char list"   and x1 :: "NominalSequents.var"   and x2 :: "form"   and y3 :: "char list"   and y1 :: "NominalSequents.var"   and y2 :: "form" shows "(x3 \<nabla> [x1].x2 = y3 \<nabla> [y1].y2) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
mk_eqcl'_Some: fixes i :: "nat"   and xs :: "nat option list"   and ii :: "nat"   and j :: "nat"   and l :: "nat"   and T :: "bool list list" assumes "i < length xs"   and "xs ! i \<noteq> None" shows "mk_eqcl' xs ii j l T ! i = xs ! i"
inv_simp: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.inv f = (local.inv (fst f), local.inv (snd f))"
Cons_1: fixes qs :: "'b list"   and tab :: "'a \<Rightarrow> 'b option" assumes "length ps = length qs" shows "tab(ps [\<mapsto>] qs, xs [\<mapsto>] ys) = tab(ps @ xs [\<mapsto>] qs @ ys)"
fmdrop_comm: fixes a :: "'c"   and b :: "'c"   and m :: "('c, 'd) fmap" shows "fmdrop a (fmdrop b m) = fmdrop b (fmdrop a m)"
get_min: fixes t :: "'b tree" assumes "heap t"   and "t \<noteq> \<langle>\<rangle>" shows "get_min t = Min_mset (mset_tree t)"
fixp_lowerbound: fixes f :: "'b \<Rightarrow> 'b"   and z :: "'b" assumes "monotone leq2 leq2 f"   and "leq2 (f z) z" shows "leq2 (c2.fixp f) z"
\<Gamma>_sound: fixes CC :: "'a literal multiset multiset"   and D :: "'a literal multiset"   and E :: "'a literal multiset"   and I :: "'a set" assumes "Infer CC D E \<in> \<Gamma>"   and "I \<Turnstile>m CC"   and "I \<Turnstile> D" shows "I \<Turnstile> E"
ltrm_cfs_above_deg: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "deg R f < n" shows "monom P (f (deg R f)) (deg R f) n = \<zero>"
igSwapAbsIPresIGWlsAbsSTR_imp_igSwapAbsIPresIGWlsAbs: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" assumes "igSwapAbsIPresIGWlsAbsSTR MOD" shows "igSwapAbsIPresIGWlsAbs MOD"
blin: fixes x :: "'d" shows "bounded_linear ((THE f'. \<forall>x. ((\<lambda>x. 0::'e) has_derivative f' x) (at x)) x)"
subspaceI: fixes S :: "'b set" assumes "(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> c *\<^sub>C x \<in> S" shows "csubspace S"
Upair_inject: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(Upair a b = Upair c d) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
wellformed_items_\<J>_induct_u: fixes k :: "nat"   and u :: "nat" assumes "wellformed_items (\<J> k u)" shows "wellformed_items (\<J> k (Suc u))"
Digamma_real_mono: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "x \<le> y" shows "Digamma x \<le> Digamma y"
inject_2: fixes x21 :: "floatarith"   and x22 :: "floatarith"   and x23 :: "form"   and y21 :: "floatarith"   and y22 :: "floatarith"   and y23 :: "form" shows "(Assign x21 x22 x23 = Assign y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
negligible_inverse_powerI: fixes k :: "real" assumes "1 < \<bar>k\<bar>" shows "Negligible.negligible (\<lambda>\<eta>. 1 / k ^ \<eta>)"
sfilter_supto: fixes d :: "int"   and n :: "int"   and m :: "int" assumes "0 \<le> d" shows "sfilter\<cdot>(\<Lambda> x. le\<cdot>(MkI\<cdot>n - x)\<cdot>(MkI\<cdot>d))\<cdot>(supto\<cdot>(MkI\<cdot>m)\<cdot>(MkI\<cdot>n)) = supto\<cdot>(MkI\<cdot>(if m \<le> n - d then n - d else m))\<cdot>(MkI\<cdot>n)"
domain: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Domainp (pcr_fract T) = (\<lambda>x. \<exists>y. rel_prod T T x y \<and> fractrel y y)"
add_left: fixes a :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c"   and a' :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c"   and b :: "'d" shows "(a + a') *\<^sub>V b = a *\<^sub>V b + a' *\<^sub>V b"
inject_2: fixes x21 :: "e list"   and x22 :: "nat"   and x23 :: "e list"   and x24 :: "Lholed"   and x25 :: "e list"   and y21 :: "e list"   and y22 :: "nat"   and y23 :: "e list"   and y24 :: "Lholed"   and y25 :: "e list" shows "(LRec x21 x22 x23 x24 x25 = LRec y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
inline1_parametric': fixes S :: "'d \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'k \<Rightarrow> bool"   and C' :: "'g \<Rightarrow> 'l \<Rightarrow> bool"   and R' :: "'h \<Rightarrow> 'm \<Rightarrow> bool"   and A :: "'n \<Rightarrow> 'o \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun C (rel_gpv'' (rel_prod R S) C' R'))) (rel_fun (rel_gpv'' A C R) (rel_fun S (rel_spmf (rel_sum (rel_prod A S) (rel_prod C' (rel_prod (rel_fun R' (rel_gpv'' (rel_prod R S) C' R')) (rel_fun R (rel_gpv'' A C R)))))))) inline1 inline1"
tell_option_def: fixes tell :: "'i \<Rightarrow> 'h \<Rightarrow> 'h"   and w :: "'i"   and m :: "('g, 'h) optionT" shows "tell_option tell w m = OptionT (tell w (run_option m))"
s1_1_def: shows "s1_1 = (\<lambda>n x. comp_by_index n (index_of_c_pair_n x))"
to_hnr_prod_def: shows "to_hnr_prod \<equiv> (\<times>\<^sub>a)"
Dgt: shows "{x \<in> S. 0 < \<bar>det (matrix (g' x))\<bar> * f (g x)} \<in> sets lebesgue"
add_pCons: fixes a :: "'b"   and p :: "'b poly"   and b :: "'b"   and q :: "'b poly" shows "pCons a p + pCons b q = pCons (a + b) (p + q)"
vfsequenceI: fixes xs :: "V" assumes "vsv xs"   and "\<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>" shows "vfsequence xs"
id: fixes i :: "nat" assumes "i < n" shows "v1 ! i = v2 ! i"
simps_3: fixes x :: "'a"   and va :: "('a, 'c) rbt"   and vb :: "'a"   and vc :: "'c"   and vd :: "('a, 'c) rbt"   and y :: "'a"   and s :: "'c"   and b :: "('a, 'c) rbt" shows "rbt_del_from_left x (Branch R va vb vc vd) y s b = Branch R (rbt_del x (Branch R va vb vc vd)) y s b"
rsemimodule3: fixes s :: "'b"   and a :: "'a"   and b :: "'a" shows "\<lambda> (s, a ; b) = \<lambda> (\<lambda> (s, a), b)"
cancel_comm_monoid_add_ow: shows "class.cancel_comm_monoid_add = cancel_comm_monoid_add_ow UNIV"
Sup_3'': fixes \<xi> :: "'a" assumes "\<parallel>\<xi>\<parallel> < r" shows "\<parallel>f *\<^sub>v (x - \<xi>)\<parallel> \<in> (\<lambda>\<xi>. \<parallel>f *\<^sub>v (x + \<xi>)\<parallel>) ` ball (0::'a) r"
locally_full: fixes f :: "'a"   and f' :: "'a"   and \<nu> :: "'b" assumes "C.ide f"   and "C.ide f'"   and "src\<^sub>C f = src\<^sub>C f'"   and "trg\<^sub>C f = trg\<^sub>C f'"   and "\<guillemotleft>\<nu> : F f \<Rightarrow>\<^sub>D F f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C f'\<guillemotright> \<and> F \<mu> = \<nu>"
bounded_linear_Blinfun_apply: fixes f :: "'e \<Rightarrow> 'f" assumes "bounded_linear f" shows "blinfun_apply (Blinfun f) = f"
D: shows "\<langle>n, Auth_PriKey n\<rangle> \<in> spied s"
distinct_names: shows "all_distinct (Node Tip b False (Node Tip c False Tip))"
positive_part_lemma: fixes C :: "'at Literal set" shows "positive_part C = validated_part UNIV C"
connectivity: shows "\<^bold>\<forall>x xa. G x xa \<or> G xa x \<or> x \<^bold>= xa"
axioms_2: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e" assumes "diff k charts1 charts2 f" shows "diff_axioms k charts1 charts2 f"
if_redT_NewHeapElem_typed: fixes s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and ad :: "'addr"   and CTn :: "htype" assumes "h.mthr.if.redT_syntax1 s t ta s'"   and "NormalAction (NewHeapElem ad CTn) \<in> set \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>"   and "typeof_addr ad \<noteq> None" shows "typeof_addr ad = \<lfloor>CTn\<rfloor>"
r_inv: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "x \<otimes>\<^bsub>DirProds G I\<^esub> inv\<^bsub>DirProds G I\<^esub> x = \<one>\<^bsub>DirProds G I\<^esub>"
assms_4: shows "((\<lambda>x. exp (c x)) expands_to E) basis"
invPU: shows "invariant (\<lambda>g. P g \<and> U g) f"
addr_loc_type_cases: fixes P :: "'a prog"   and a :: "'addr"   and a1 :: "addr_loc"   and a2 :: "ty" assumes "P \<turnstile> a@a1 : a2" obtains U :: "htype" and F :: "String.literal" and T :: "ty" and fm :: "fmod" and D :: "String.literal" where "a1 = CField D F" and "a2 = T__" and "typeof_addr a = \<lfloor>U\<rfloor>" and "P \<turnstile> class_type_of U has F:T__ (fm) in D"   | T :: "ty" and n' :: "nat" and n :: "nat" where "a1 = ACell n" and "a2 = T" and "typeof_addr a = \<lfloor>Array_type T n'\<rfloor>" and "n < n'"
bit_yx: fixes x :: "'a"   and y :: "'a"   and init :: "'a list"   and qs :: "'a list"   and r :: "'a list" assumes "x \<noteq> y"   and "init \<in> {[x, y], [y, x]}"   and "qs \<in> lang (Star (Times (Atom y) (Atom x)))" shows "T\<^sub>p_on_rand' BIT (type1 init x y) (qs @ r) = 75 / 10\<^sup>2 * real (length qs) + T\<^sub>p_on_rand' BIT (type1 init x y) r \<and> Partial_Cost_Model.config'_rand BIT (type1 init x y) qs = type1 init x y"
ex_pos: shows "{0..} \<subseteq> existence_ivl0 x"
prod_fv_supp: shows "prod_fv supp supp = supp"
sum_dim_vec_equals_sum_dim_poly: fixes A :: "rat vec"   and X :: "nat \<Rightarrow> rat" shows "(\<Sum>a = 0..<dim_vec A. Abstract_Linear_Poly.coeff (vec_to_lpoly A) a * X a) = (\<Sum>a = 0..<dim_poly (vec_to_lpoly A). Abstract_Linear_Poly.coeff (vec_to_lpoly A) a * X a)"
increasing_def: fixes M :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> 'c" shows "increasing M \<mu> = (\<forall>x\<in>M. \<forall>y\<in>M. x \<subseteq> y \<longrightarrow> \<mu> x \<le> \<mu> y)"
match_prereq_def: fixes i :: "of_match_field"   and s :: "of_match_field set"   and p :: "(32, 'b) simple_packet_ext_scheme" shows "match_prereq i s p = (if prerequisites i s then Some (match_no_prereq i p) else None)"
while_rel1_def: shows "while_rel1 = measure (\<lambda>x. trivCond (x = [])) <*lex*> measure (\<lambda>x. trivCond (x = (0::'d))) <*lex*> measure (\<lambda>x. trivCond (\<not> x))"
assms_3: shows "all_fulfill (in_all_resolvents_upon S E2) R"
card_DF: fixes k :: "nat"   and i :: "nat" shows "card (DF k i) = Suc k"
small_Un: fixes X :: "'c set"   and Y :: "'c set" assumes "small X"   and "small Y" shows "small (X \<union> Y)"
2: shows "f ` {0..<length xs} = {0..<length ys}"
higher_deriv_cong_ev: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and n :: "nat" assumes "\<forall>\<^sub>F x in nhds x. f x = g x"   and "x = y" shows "(deriv ^^ n) f x = (deriv ^^ n) g y"
closed: fixes A :: "'b sat set" assumes "finite A"   and "A \<noteq> {}"   and "\<And>x y. min x y \<in> {x, y}" shows "Inf A \<in> A"
hyps: shows "bound (of_nat n * v) \<le> n * bound v"
abc_abc_neq: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"
1: shows "is_greatest_postfixpoint (\<lambda>x. \<nu> f x) (\<nu> (\<lambda>x. \<nu> f x))"
rel_sel_conj_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A"   and "\<gamma>\<^sub>T\<^sub>R K (\<phi> .\<and>. \<psi>) \<inter> (K .\<bottom>. \<phi>) \<noteq> {}" shows "\<gamma>\<^sub>T\<^sub>R K \<phi> \<subseteq> \<gamma>\<^sub>T\<^sub>R K (\<phi> .\<and>. \<psi>)"
ring_functions_def: fixes R :: "('a, 'b) ring_scheme" shows "ring_functions R \<equiv> ring R"
ih_T\<^sub>2: fixes \<Gamma> :: "binding list" assumes "\<turnstile> \<Gamma> ok"   and "T\<^sub>2 closed_in \<Gamma>" shows "\<Gamma>\<turnstile>T\<^sub>2<:T\<^sub>2"
Sup_downset_id: shows "Inf \<circ> ord.downset (\<lambda>x y. y \<le> x) = id"
the_adj_chamber_neq: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "the_adj_chamber C z \<noteq> C"
word_less_eq_iff_unsigned: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'c word"   and b :: "'c word" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "(a \<le> b) = less_eq (semiring_1.unsigned one plus zero a) (semiring_1.unsigned one plus zero b)"
sup_commute: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "sup_prod x y = sup_prod y x"
Vring_prime_maximal: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "prime_ideal (Vr K v) I"   and "I \<noteq> {\<zero>\<^bsub>Vr K v\<^esub>}" shows "maximal_ideal (Vr K v) I"
assms_3: shows "are_all_operators_non_interfering (a # ops)"
wlp_equiv: fixes c :: "com"   and c' :: "com" assumes "c \<sim> c'" shows "local.wlp c = local.wlp c'"
arr_iff_has_trg: fixes \<mu> :: "'a" shows "arr \<mu> = (targets \<mu> \<noteq> {})"
ocircline_set_circline_set_x_axis: shows "ocircline_set o_x_axis = circline_set x_axis"
CISK_proof_obligations_satisfied: shows "Controllable_Interruptible_Separation_Kernel rstep routput_f \<up> s0 rcurrent rcswitch rkinvolved rifp rvpeq rAS_set rinvariant rprecondition raborting rwaiting rset_error_code"
hyps: shows "\<Sum>\<^sub># (mset (replicate n m1)) \<subseteq># \<Sum>\<^sub># (mset (replicate n m2))"
disjoint_unionD1: fixes A :: "'b set set"   and B :: "'b set set" assumes "disjoint (A \<union> B)" shows "disjoint A"
prj0_in_hom': fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "\<guillemotleft>\<p>\<^sub>0[f, g] : local.dom \<p>\<^sub>0[f, g] \<rightarrow> local.dom g\<guillemotright>"
nat_mult_not_infty: fixes c :: "eint"   and n :: "int" assumes "c \<noteq> \<infinity>" shows "eint n * c \<noteq> \<infinity>"
simps_3: fixes rel :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list"   and y :: "'d"   and ys :: "'d list" shows "merge_wrt rel (x # xs) (y # ys) = (if x = y then y # merge_wrt rel xs ys else if rel x y then x # merge_wrt rel xs (y # ys) else y # merge_wrt rel (x # xs) ys)"
simps_2: fixes s :: "'d stream"   and n :: "nat" shows "s !! Suc n = stl s !! n"
mem_diff: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc"   and "\<forall>x\<in>mds' AsmNoWrite. mem' x = mem'' x" shows "\<langle>c', mds', mem''\<rangle> \<in> loc_reach lc"
A: shows "p\<^sub>1 \<in> set (p\<^sub>2 # ps)"
sympI: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "\<And>a b. r a b \<Longrightarrow> r b a" shows "symp r"
tranclD4: fixes S :: "'a"   and T :: "'a"   and R :: "('a \<times> 'a) set" assumes "(S, T) \<in> R\<^sup>+"   and "(S, T) \<notin> R" shows "\<exists>U. (S, U) \<in> R\<^sup>+ \<and> (U, T) \<in> R"
three_ins_1: fixes i :: "'a"   and t :: "'a"   and e :: "'a" shows "i \<in> set [i, t, e]"
exec_block_Normal_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and bdy :: "('e, 'd, 'f) com"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate"   and p :: "'d" assumes "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> \<Rightarrow> t" obtains t' :: "'e" where "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t'__" and "\<Gamma>\<turnstile> \<langle>c s t'__,Normal (return s t'__)\<rangle> \<Rightarrow> t"   | t' :: "'e" where "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Abrupt t'" and "t = Abrupt (return s t')"   | f :: "'f" where "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Fault f" and "t = Fault f"   | "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Stuck" and "t = Stuck"   | "\<Gamma> p = None" and "t = Stuck"
simps_1: fixes r :: "real"   and p :: "real" shows "ereal r * ereal p = ereal (r * p)"
face_cycle_succ_no_arc: fixes a :: "'b" assumes "a \<notin> arcs G" shows "GM.face_cycle_succ a = a"
split_paired_Ball_Sigma: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and P :: "'c \<times> 'd \<Rightarrow> bool" shows "(\<forall>z\<in>Sigma A B. P z) = (\<forall>x\<in>A. \<forall>y\<in>B x. P (x, y))"
mapExtendEmpty: fixes ps :: "('a, 'b) sequent list" shows "map (extend ( \<Empt> \<Rightarrow>* \<Empt>)) ps = ps"
dord_pm_alt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.dord_pm ord = (\<lambda>x y. deg_pm x < deg_pm y \<or> deg_pm x = deg_pm y \<and> ord x y)"
weakStepSimulation_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q \<equiv> (\<forall>Q' a x. Q \<longmapsto> a<\<nu>x> \<prec> Q' \<longrightarrow> x \<sharp> P \<longrightarrow> (\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>Q' \<alpha>. Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
ccpo: fixes leq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lub :: "'b set \<Rightarrow> 'b" assumes "partial_function_definitions leq lub" shows "class.ccpo (fun_lub lub) (fun_ord leq) (mk_less (fun_ord leq))"
extendLevel: fixes m :: "nat" shows "\<exists>belowNext Next. (extendLevel ^^ m) (atoms, depth1) = (belowNext, Next) \<and> i.R belowNext = {\<phi>. max_depth \<phi> < 1 + m} \<and> i.R Next = {\<phi>. max_depth \<phi> = 1 + m}"
div_unit_factor: fixes a :: "'a" shows "a div unit_factor a = normalize a"
mono_def: fixes f :: "'b \<Rightarrow> 'c" shows "mono f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
not_zle_0_negative: fixes n :: "nat" shows "\<not> 0 \<le> - int (Suc n)"
perm_finite: fixes A :: "'a list" shows "finite {B. B <~~> A}"
fold_insert_idem2: fixes A :: "'a set"   and z :: "'a"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold sup z (insert x A) = Finite_Set.fold sup (sup x z) A"
abs_update_map_of: fixes l :: "('a \<times> 'b) list list"   and k :: "'a"   and v :: "'b" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l" shows "map_of (concat (abs_update k v l)) = map_of (concat l)(k \<mapsto> v)"
drinks_end: fixes r :: "nat \<Rightarrow>f value option"   and a :: "String.literal"   and b :: "value list" shows "possible_steps drinks 2 r a b = {||}"
Int: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and a :: "'b set"   and b :: "'b set" assumes "semiring_of_sets \<Omega> M"   and "a \<in> M"   and "b \<in> M" shows "a \<inter> b \<in> M"
siftDown_in_tree: fixes t :: "'c Tree" assumes "t \<noteq> E" shows "in_tree (val (siftDown t)) t"
ide_eval_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "ide \<lbrace>t\<rbrace>"
countable_ordLeq: fixes A :: "'c set"   and B :: "'d set" assumes "|A| \<le>o |B|"   and "countable B" shows "countable A"
if_then_else_distributive: fixes a :: "bool"   and f :: "'m \<Rightarrow> 'l"   and b :: "'m"   and c :: "'m" shows "(if a then f b else f c) = f (if a then b else c)"
l_get_owner_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_get_owner_document_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr \<equiv> known_ptr = ShadowRootClass.known_ptr"
simps: fixes var :: "nat"   and p :: "real mpoly"   and xs :: "real list" shows "convert_poly var p xs = (if MPoly_Type.degree p var < 3 then let (A, B, C) = get_coeffs var p in Some (insertion (nth_default 0 xs) A, insertion (nth_default 0 xs) B, insertion (nth_default 0 xs) C) else None)"
in_unforwarded_non_volatile_reads_non_volatile_Read\<^sub>s\<^sub>b: fixes a :: "nat"   and sb :: "'b memref list"   and W :: "nat set" assumes "a \<in> unforwarded_non_volatile_reads sb W" shows "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb"
hyps: fixes j :: "nat" shows "(htuple i = htuple j) = (i = j)"
assms_2: fixes x :: "real" assumes "x \<in> A" shows "f x \<in> {- 1<..<1}"
snd_eqD: fixes x :: "'b"   and y :: "'a"   and a :: "'a" assumes "snd (x, y) = a" shows "y = a"