hcju/isabelleps huggingface dataset:
llist_all2_all_lnthI: fixes xs :: "'c llist"   and ys :: "'d llist"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "llength xs = llength ys"   and "\<And>n. enat n < llength xs \<Longrightarrow> P (lnth xs n) (lnth ys n)" shows "llist_all2 P xs ys"
inext_nth_singleton: fixes a :: "nat"   and n :: "nat" shows "{a} \<rightarrow> n = a"
gauss_int_norm_of_numeral: fixes n' :: "num" shows "gauss_int_norm (numeral n') = numeral (Num.sqr n')"
intro: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "2 \<le> a"   and "2 \<le> b"   and "2 \<le> c"   and "x \<^sup>@ a \<cdot> y \<^sup>@ b = z \<^sup>@ c" shows "LS x a y b z c"
intros_2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and zer :: "'b"   and suc :: "'b \<Rightarrow> 'b"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tms :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'a" assumes "Syntax_Arith var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num zer suc pls tms"   and "x \<in> var" shows "Var x \<in> Syntax_Arith.atrm var Var num suc pls tms"
fds_nth_shift: fixes c :: "'a"   and f :: "'a fds"   and n :: "nat" shows "fds_nth (fds_shift c f) n = fds_nth f n * nat_power n c"
enn2real_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "\<And>i. i \<in> I \<Longrightarrow> f i < \<top>" shows "enn2real (sum f I) = sum (enn2real \<circ> f) I"
measurable_return_prob_space: fixes N :: "'b measure" shows "return N \<in> N \<rightarrow>\<^sub>M prob_algebra N"
funion_absorb2: fixes B :: "'b fset"   and A :: "'b fset" assumes "B |\<subseteq>| A" shows "A |\<union>| B = A"
simps: fixes as :: "('b \<times> 'c atom) set"   and bi :: "((nat \<Rightarrow> 'c option) \<times> (nat \<Rightarrow> 'c option)) \<times> (nat \<Rightarrow> 'b) \<times> (nat \<Rightarrow> 'b)" shows "as \<Turnstile>\<^sub>i bi = (\<forall>I v. (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as \<longrightarrow> (I, v) \<Turnstile>\<^sub>i\<^sub>b bi)"
fac_body'_def: shows "fac_body' \<equiv> \<Lambda> r n. unbox\<cdot>(If box\<cdot>n =\<^sub>B 0 then 1 else unbox\<cdot>(box\<cdot>n - 1) >>= r >>= (\<Lambda> b. box\<cdot>n * box\<cdot>b))"
fls_lr_inverse_mult_ring1_1: fixes x :: "'a"   and f :: "'a fls"   and y :: "'a"   and g :: "'a fls" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * x = (1::'a)"   and "y * fls_nth g (fls_subdegree g) = (1::'a)"   and "fls_nth g (fls_subdegree g) * y = (1::'a)" shows "fls_left_inverse (f * g) (y * x) = fls_left_inverse g y * fls_left_inverse f x"
pop_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M" shows "a \<plusminus> b = b \<plusminus> a"
non_loop_tail: fixes f :: "'s \<Rightarrow> 'b"   and u :: "'b"   and X :: "'s cfg set"   and r :: "ennreal"   and cfg :: "'s cfg"   and x :: "'s" assumes "\<forall>l. \<forall>cfg\<in>\<Union> (cfg_on ` f -` {u}) \<inter> X. emeasure (measure_pmf (action cfg)) (f -` {u}) \<le> r"   and "r < 1"   and "cfg \<in> cfg_on x"   and "almost_everywhere (MC.T cfg) (pred_stream (\<lambda>x. x \<in> X))" shows "AE \<omega> in MC.T cfg. \<not> ev (alw (\<lambda>xs. shd xs = u)) (smap (f \<circ> state) \<omega>)"
conj_mono: fixes c\<^sub>0 :: "'a::refinement_lattice"   and d\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice" assumes "(c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>0::'a::refinement_lattice)"   and "(c\<^sub>1::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'a::refinement_lattice)" shows "(c\<^sub>0::'a::refinement_lattice) \<iinter> (c\<^sub>1::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>0::'a::refinement_lattice) \<iinter> (d\<^sub>1::'a::refinement_lattice)"
simps: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and s :: "nat set"   and tab :: "'b option list set"   and st :: "nat set"   and tup :: "'b option list" assumes "getIJ getIJ" shows "New_max.semiJoin (s, tab) (st, tup) = (s, Set.filter (New_max.isSameIntersection tup (s \<inter> st)) tab)"
ide_def: fixes a :: "'j" shows "J.ide a \<equiv> J a a \<noteq> J.null \<and> (\<forall>f. (J f a \<noteq> J.null \<longrightarrow> J f a = f) \<and> (J a f \<noteq> J.null \<longrightarrow> J a f = f))"
assms_6: shows "s\<^sub>2' r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r''))])"
kronecker_factorization_main_complete: fixes p :: "int poly" assumes "kronecker_factorization_main df dp bnd p = None"   and "2 \<le> degree p" shows "\<nexists>q. 1 \<le> degree q \<and> degree q \<le> bnd \<and> q dvd p"
eq2: shows "connect_obsf (absorb \<A> (obsf_converter (sim_outer |\<^sub>= 1\<^sub>C))) (obsf_resource middle) = connect_obsf \<A> (obsf_resource (sim_outer |\<^sub>= 1\<^sub>C \<rhd> middle))"
QTrans: shows "Q \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>y> \<prec> Q'"
zero_ratfps_def: shows "0 \<equiv> Abs_ratfps 0"
no_outstanding_write_to_read_only_memory: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "no_write_to_read_only_memory \<S> sb"
valid_indexE: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" obtains "isa__ = []" and "dsa__ = []"   | "is" :: "nat list" and ds :: "nat list" and i :: "nat" and d :: "nat" where "isa__ = i # is" and "dsa__ = d # ds" and "is \<lhd> ds" and "i < d"
field_cont_on_gamma4_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (g2 x, x)} (\<lambda>a. F a \<bullet> j)"
mult_less_mono2: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "0 < k" shows "k * i < k * j"
hd_def: fixes xs :: "'x" shows "local.hd xs = list.hd (fst (list_of_oalist xs))"
lipschitz: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and L :: "real"   and t :: "'b" assumes "global_lipschitz T X f L"   and "t \<in> T" shows "L-lipschitz_on X (f t)"
AX_def: fixes f :: "'b set" shows "AX f \<equiv> {s. {f0. s \<rightarrow>\<^sub>i f0} \<subseteq> f}"
Throw\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,s\<^sub>1\<rangle>"
Act_tau_eq_iff: fixes x1 :: "('a, 'b, 'act) formula"   and \<alpha> :: "'act"   and x2 :: "('a, 'b, 'act) formula" shows "(Act \<tau> x1 = Act \<alpha> x2) = (\<alpha> = \<tau> \<and> x2 = x1)"
lattice_stable: fixes F :: "'b program" shows "lattice {X. F \<in> stable X}"
cet_exp: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e : REAL" shows "\<Gamma> \<turnstile>\<^sub>c exp\<^sub>c e : REAL"
comb_seq_0: fixes \<omega> :: "nat \<Rightarrow> 'b"   and \<omega>' :: "nat \<Rightarrow> 'b" shows "comb_seq 0 \<omega> \<omega>' = \<omega>'"
Fvars_instInpP: fixes \<phi> :: "'fmla"   and \<tau> :: "'fmla"   and n :: "nat" assumes "\<phi> \<in> fmla"   and "\<tau> \<in> ptrm n"   and "Variable (Suc (Suc n)) \<notin> Fvars \<phi>" shows "Fvars (instInpP \<phi> n \<tau>) = Fvars \<phi> - {inp} \<union> Variable ` {Suc 0..n}"
induct: fixes P :: "bool list \<Rightarrow> bool list \<Rightarrow> bool"   and a0 :: "bool list"   and a1 :: "bool list" assumes "\<And>bs. P [] bs"   and "\<And>v va. P (v # va) []"   and "\<And>a as b bs. P as bs \<Longrightarrow> P (a # as) (b # bs)" shows "P a0 a1"
A: fixes x :: "V" assumes "x \<in> A" shows "Card x"
solve_equation: fixes H :: "(Vertex \<Rightarrow> Vertex) set"   and x :: "Vertex \<Rightarrow> Vertex"   and y :: "Vertex \<Rightarrow> Vertex" assumes "subgroup H tetrahedral_group"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<otimes>\<^bsub>tetrahedral_group\<^esub> x"
WTrtFAccNT: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and F :: "String.literal"   and D :: "String.literal"   and T :: "ty" assumes "P,E,h \<turnstile> e : NT" shows "P,E,h \<turnstile> e\<bullet>F{D} : T"
cf_set_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and e :: "nat \<times> nat"   and x :: "int" shows "cf_set cf e x \<equiv> assert (e \<in> E \<union> E\<inverse>) \<bind> (\<lambda>_. return (cf(e := x)))"
enum_all_ChildAct_def: fixes P :: "ChildAct \<Rightarrow> bool" shows "enum_class.enum_all P = (P SayIKnow \<and> P SayNothing)"
has_locks_Suc_has_lock: fixes l :: "('d \<times> nat) option"   and t :: "'d"   and n :: "nat" assumes "has_locks l t = Suc n" shows "has_lock l t"
paste_sub_Un: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" shows "P +* Q \<subseteq> P \<union> Q"
path_weight_elem_trans: fixes s :: "'sum"   and l1 :: "'loc"   and l2 :: "'loc"   and s' :: "'sum"   and l3 :: "'loc" assumes "s \<in>\<^sub>A path_summary l1 l2"   and "s' \<in>\<^sub>A path_summary l2 l3" shows "\<exists>u. u \<in>\<^sub>A path_summary l1 l3 \<and> u \<le> followed_by s s'"
L_flat_Prf: fixes r :: "'b rexp" shows "lang r = {flat v |v. \<turnstile> v : r}"
polysub_same_0: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "p -\<^sub>p p = 0\<^sub>p"
uniform_distrI_borel: fixes X :: "'b \<Rightarrow> real"   and M :: "'b measure"   and A :: "real set"   and r :: "real" assumes "X \<in> borel_measurable M"   and "emeasure lborel A = ennreal r"   and "0 < r"   and "A \<in> sets borel"   and "\<And>a. emeasure M {x \<in> space M. X x \<le> a} = emeasure lborel (A \<inter> {..a}) / ennreal r" shows "distributed M lborel X (\<lambda>x. ennreal (indicat_real A x / content A))"
append_exec_def: fixes e :: "'c \<times> 'd list"   and e' :: "'e \<times> 'd list" shows "append_exec e e' \<equiv> (fst e, append (snd e') (snd e))"
Suc_2: shows "S |\<subseteq>| initials G"
eta_expand_eqvt: fixes \<pi> :: "perm"   and n :: "nat"   and e :: "exp" shows "\<pi> \<bullet> eta_expand n e = eta_expand (\<pi> \<bullet> n) (\<pi> \<bullet> e)"
"1_1": shows "(S ===> A ===> rel_spmf (rel_prod B (rel_sum (rel_resource A B) S))) f g"
assms: fixes ps :: "((('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and gs :: "(('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" assumes "ps \<noteq> []" shows "sel gs bs ps data \<noteq> [] \<and> set (sel gs bs ps data) \<subseteq> set ps"
simps: fixes a1 :: "lam"   and a2 :: "lam" shows " a1 \<longrightarrow>\<^sub>1 a2 = ((\<exists>M. a1 = M \<and> a2 = M) \<or> (\<exists>t1 t2 s1 s2. a1 = App t1 s1 \<and> a2 = App t2 s2 \<and>  t1 \<longrightarrow>\<^sub>1 t2 \<and>  s1 \<longrightarrow>\<^sub>1 s2) \<or> (\<exists>s1 s2 a. a1 = Lam [a].s1 \<and> a2 = Lam [a].s2 \<and>  s1 \<longrightarrow>\<^sub>1 s2) \<or> (\<exists>a s1 s2 t1 t2. a1 = App (Lam [a].t1) s1 \<and> a2 = t2[a::=s2] \<and> a \<sharp> (s1, s2) \<and>  s1 \<longrightarrow>\<^sub>1 s2 \<and>  t1 \<longrightarrow>\<^sub>1 t2))"
is_valid_operator_sas_plus_then_5: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (sas_plus_operator.precondition_of op). \<forall>(v', a')\<in>set (sas_plus_operator.precondition_of op). v \<noteq> v' \<or> a = a'"
pref_imp_trans: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool"   and H :: "'a \<Rightarrow> bool" assumes "|~ F \<longrightarrow> G"   and "|~ G \<longrightarrow> H" shows "|~ F \<longrightarrow> H"
hfunction_ins: fixes f :: "hf"   and x :: "hf"   and y :: "hf" assumes "hfunction f"   and "x \<^bold>\<notin> hdomain f" shows "hfunction (f \<triangleleft> \<langle>x, y\<rangle>)"
Stream_snth: fixes x :: "'a"   and s :: "'a stream"   and n :: "nat" shows "(x ## s) !! n = (case n of 0 \<Rightarrow> x | Suc n \<Rightarrow> s !! n)"
Prf_flat_lang: fixes v :: "'b val"   and r :: "'b rexp" assumes "\<turnstile> v : r" shows "flat v \<in> lang r"
vfsequence_vcons_ex: assumes "xs \<noteq> []\<^sub>\<circ>" obtains xs' :: "V" and x :: "V" where "xs = xs' #\<^sub>\<circ> x" and "vfsequence xs'"
solves_odeI: fixes y :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "real set"   and X :: "'c set" assumes "(y has_vderiv_on (\<lambda>t. f t (y t))) T"   and "\<And>t. t \<in> T \<Longrightarrow> y t \<in> X" shows "(y solves_ode f) T X"
map_SSup_map_ds_inv: shows "map_SSup \<circ> map_ds = id"
NSDERIV_NSLIM_iff: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a" shows "NSDERIV f x :> D = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'a\<rightarrow>\<^sub>N\<^sub>S D"
zip_with_Pair: fixes xs :: "'b list"   and ys :: "'c list" shows "zip_with Pair xs ys = zip xs ys"
less_top_Value_Top: fixes x :: "'a" shows "top.Value x < Top"
intro: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<BB>"   and "digraph \<alpha> \<CC>"   and "subdigraph_axioms \<BB> \<CC>" shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
Max_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq A = Sup A"
vsv_vinsert_match_appI: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "vinsert \<langle>a, b\<rangle> r\<lparr>a\<rparr> = b"
liftMM_comm_ctxt: fixes E :: "ctxt"   and n :: "nat" shows "liftM_ctxt (liftM_ctxt E n) 0 = liftM_ctxt (liftM_ctxt E 0) (n + 1)"
emeasure_map_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf"   and A :: "'c set" shows "emeasure (measure_spmf (map_spmf f p)) A = emeasure (measure_spmf p) (f -` A)"
inf_le1: fixes x :: "'a upred"   and y :: "'a upred" shows "x \<sqsubseteq> (x \<and> y)"
Try: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"
bit_seq_of_nat_set_def: shows "bit_seq_of_nat_set \<equiv> map_fun id Sequence.seq.Abs_seq (\<lambda>A i. i \<in> A)"
is_UNIV_unfold_1: fixes A :: "'b set" shows "(A = UNIV) = is_UNIV A"
of_weak_ranking_weak_ranking_aux: shows "of_weak_ranking (weak_ranking_aux carrier) = le"
src_trg: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "src (trg \<mu>) = trg \<mu>"
intros_10: fixes \<phi> :: "'a ltln"   and \<chi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<turnstile>\<^sub>s \<chi>" shows "\<phi> \<turnstile>\<^sub>s \<psi> U\<^sub>n \<chi>"
RS1n_nMT: fixes p :: "('b, 'c) Combinators list" assumes "p \<noteq> []" shows "removeShadowRules1_alternative p \<noteq> []"
prems: shows "del x \<langle>l, y, r\<rangle> = Some t'"
f''_ge0_imp_convex: fixes C :: "real set"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and f'' :: "real \<Rightarrow> real" assumes "convex C"   and "\<And>x. x \<in> C \<Longrightarrow> (f has_real_derivative f' x) (at x)"   and "\<And>x. x \<in> C \<Longrightarrow> (f' has_real_derivative f'' x) (at x)"   and "\<And>x. x \<in> C \<Longrightarrow> 0 \<le> f'' x" shows "convex_on C f"
fps_is_left_unit_iff_zeroth_is_left_unit: fixes f :: "'a fps" shows "(\<exists>g. 1 = f * g) = (\<exists>k. (1::'a) = fps_nth f 0 * k)"
gso_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "fs.gs.gso i \<in> Rn"
poly_compat2: fixes p1 :: "('c monom \<times> 'a) list"   and p2 :: "('c monom \<times> 'a) list"   and p3 :: "('c monom \<times> 'a) list" assumes "p1 >p p2"   and "p2 \<ge>p p3" shows "p1 >p p3"
minus_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
S_def: shows "S = {q \<in> qs. \<mu> U\<^sub>r \<eta> \<in> old q \<longrightarrow> \<eta> \<in> old q}"
left_right_simps_1: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and l :: "'b"   and r :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.left unapp (app l r) = l"
length_upto_enum: fixes a :: "'b word"   and b :: "'b word" shows "length [a .e. b] = Suc (unat b) - unat a"
poly_add_monoms: fixes p1 :: "('c monom \<times> 'd) list"   and p2 :: "('c monom \<times> 'd) list" shows "poly_monoms (poly_add p1 p2) \<subseteq> poly_monoms p1 \<union> poly_monoms p2"
pr_auth_data_items: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and b :: "rat" assumes "(evs, S, A, U) \<in> protocol" shows "Pri_AgrK (priAK n) \<notin> items (A \<union> spies evs) \<and> (IntMapK (S (Card n, n, run)) = Some b \<longrightarrow> Pri_AgrK b \<notin> items (A \<union> spies evs))"
accD_pos: fixes x :: "'b"   and y :: "'b" assumes "(x, y) \<in> K2.acc" shows "\<exists>n. 0 < K2.p x y n"
simps_17: fixes v :: "\<I>"   and va :: "Formula.formula Regex.regex" shows "remove_neg (formula.MatchP v va) = formula.MatchP v va"
sorted_sorted_list_of_set: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sorted_list_of_set less_eq A)"
le_setdist_iff: fixes d :: "real"   and S :: "'b set"   and T :: "'b set" shows "(d \<le> setdist S T) = ((\<forall>x\<in>S. \<forall>y\<in>T. d \<le> dist x y) \<and> (S = {} \<or> T = {} \<longrightarrow> d \<le> 0))"
refl: fixes g :: "'c"   and succs :: "'c \<Rightarrow> 'c list" shows "g [succs]\<rightarrow>* g"
f4: assumes "rk {R, a} = 1" shows "rk {P, Q, R, A', A, B, C, B', C'} \<le> 3"
homotopic_into_contractible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "continuous_on S g"   and "g ` S \<subseteq> T"   and "contractible T" shows "homotopic_with_canon (\<lambda>h. True) S T f g"
inf_sub_left: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "\<infinity>\<^sub>h -\<^sub>h z = \<infinity>\<^sub>h"
mat_det_inv: fixes A :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A \<noteq> 0" shows "mat_det (mat_inv A) = 1 / mat_det A"
minus_numeral_div_numeral: fixes m :: "num"   and n :: "num" shows "- numeral m div numeral n = - Divides.adjust_div (divmod m n)"
one_mult_of: fixes R :: "('b, 'c) ring_scheme" shows "\<one>\<^bsub>mult_of R\<^esub> = \<one>\<^bsub>R\<^esub>"
lt__le: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Lt C D" shows "A B Le C D"
pseudo_proj_True_Suc_prefix: fixes n :: "nat"   and w :: "bool stream" shows "pseudo_proj_True (Suc n) w = w !! 0 ## pseudo_proj_True n (stl w)"
finite_capacity: fixes x :: "'v" assumes "x \<noteq> sink \<Delta>" shows "d_IN (capacity \<Delta>) x \<noteq> \<top> \<or> d_OUT (capacity \<Delta>) x \<noteq> \<top>"
carrier_def: fixes charts :: "('c, 'd) chart set" shows "manifold.carrier charts = \<Union> (domain ` charts)"
inj_map: fixes f :: "'a \<Rightarrow> 'b" shows "inj (map f) = inj f"
filtrated_prob_space_axioms_def: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure" shows "filtrated_prob_space_axioms M F \<equiv> Filtration.filtration M F"
assms: fixes x :: "'a"   and y1 :: "'a"   and y2 :: "'a" assumes "r\<^sup>*\<^sup>* x y1"   and "r x y2" shows "\<exists>z. r\<^sup>*\<^sup>* y1 z \<and> r\<^sup>*\<^sup>* y2 z"
subsume_fin: fixes u\<^sub>1 :: "'action list"   and w :: "'action stream"   and v\<^sub>1 :: "'action list" assumes "u\<^sub>1 \<preceq>\<^sub>F\<^sub>I w"   and "v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w" obtains w\<^sub>1 :: "'action list" where "u\<^sub>1 \<preceq>\<^sub>F w\<^sub>1" and "v\<^sub>1 \<preceq>\<^sub>F w\<^sub>1"
sup_absorb1: fixes y :: "'c multiset"   and x :: "'c multiset" assumes "y \<subseteq># x" shows "x \<union># y = x"
Development_existence: fixes M :: "lam" shows "\<exists>M'.  M \<longrightarrow>\<^sub>d M'"
delete_var_1_vimage: fixes V :: "nat set" shows "case_nat 0 (\<lambda>x. Suc (Suc x)) -` shift_var_set (shift_var_set V) = shift_var_set V"
lim_def: fixes X :: "nat \<Rightarrow> 'c" shows "lim X = (THE L. X \<longlonglongrightarrow> L)"
circline_set_moebius_circline_E: fixes M :: "moebius"   and z :: "complex_homo"   and H :: "circline" assumes "moebius_pt M z \<in> circline_set (moebius_circline M H)" shows "z \<in> circline_set H"
lg_safe_4: fixes x :: "nat" assumes "0 < x" shows "lg (x + x) = 1 + lg x"
exhaust: fixes y :: "chanRef" obtains x :: "varRef" where "y = ChanRef x"
elims_3: fixes x :: "'a option"   and xa :: "'a option" assumes "\<not> less_eq_option x xa" obtains uv_ :: "'a" where "x = Some uv_" and "xa = None"   | a :: "'a" and b :: "'a" where "x = Some a" and "xa = Some b" and "\<not> less_eq0 a b"
zip_map1: fixes f :: "'i \<Rightarrow> 'g"   and xs :: "'i list"   and ys :: "'h list" shows "zip (map f xs) ys = map2 (\<lambda>x. Pair (f x)) xs ys"
b41: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "b ((4 * real m + 1) / 2 ^ Suc n) = leftcut (a ((2 * real m + 1) / 2 ^ n)) (b ((2 * real m + 1) / 2 ^ n)) (c ((2 * real m + 1) / 2 ^ n))"
inf_absorb1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x \<sqinter> y = x"
dom_map_add: fixes m :: "'d \<Rightarrow> 'e option"   and n :: "'d \<Rightarrow> 'e option" shows "dom (m ++ n) = dom n \<union> dom m"