hcju/isabelleps huggingface dataset:
not_zero_a_A: shows "\<not> is_zero_row a A"
that: fixes m :: "nat" assumes "t \<^sup>@ m \<le>s u"   and "u <s t \<cdot> t \<^sup>@ m" shows "thesis"
intervalI: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "(a, x) \<in> r"   and "(x, b) \<in> r" shows "x \<in> interval r a b"
in_idealE_homomorphism_finite: fixes h :: "'b \<Rightarrow> 'b"   and B :: "'b set"   and p :: "'b" assumes "\<And>x y. h (x + y) = h x + h y"   and "\<And>x y. h (x * y) = h x * h y"   and "\<And>x. h (h x) = h x"   and "finite B"   and "B \<subseteq> range h"   and "p \<in> range h"   and "p \<in> ideal B" obtains q :: "'b \<Rightarrow> 'b" where "\<And>b. q b \<in> range h" and "p = (\<Sum>b\<in>B. q b * b)"
lfp_exp_lowerbound: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "t P \<tturnstile> P"   and "sound P" shows "lfp_exp t \<tturnstile> P"
as_proj_eq_as: fixes as :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) list"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set" assumes "no_effectless_act as"   and "as \<in> valid_plans PROB"   and "prob_dom PROB \<subseteq> vs" shows "as_proj as vs = as"
instance_helper: fixes a :: "'a fls"   and b :: "'a fls"   and c :: "'a fls" assumes "a \<noteq> b"   and "a \<noteq> c"   and "dist a c < dist a b" shows "fls_subdegree (a - b) < fls_subdegree (a - c)"
ereal_infty_less_eq2_1: fixes a :: "ereal"   and b :: "ereal" assumes "a \<le> b"   and "a = \<infinity>" shows "b = \<infinity>"
enter_room: fixes s :: "state"   and k :: "key"   and k' :: "key"   and g :: "guest"   and r :: "room" assumes "s \<in> reach"   and "(k, k') \<in> state.cards s g"   and "state.roomk s r \<in> {k, k'}" shows "s\<lparr>isin := (state.isin s)(r := state.isin s r \<union> {g}), roomk := (state.roomk s)(r := k'), safe := (state.safe s)(r := state.owns s r = \<lfloor>g\<rfloor> \<and> state.isin s r = {} \<and> k' = state.currk s r \<or> state.safe s r)\<rparr> \<in> reach"
empty: fixes g :: "'c \<Rightarrow> 'b" shows "sum g {} = (0::'b)"
sets_UNIV: fixes A :: "'c set" shows "A \<in> sets (count_space UNIV)"
curry_K: fixes c :: "'c" shows "curry (\<lambda>x. c) = (\<lambda>x y. c)"
sc: shows "ta_seq_consist P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"
DF_def: fixes A :: "'b set" shows "DF A \<equiv> \<mu> x. \<sqinter>xa\<in>A \<rightarrow>  x"
succ_pred_no_2: fixes w :: "num" shows "word_pred (numeral w) = numeral w - 1"
rep_eq: fixes x :: "'a mpoly" shows "coeffs x = Poly_Mapping.range (mapping_of x)"
deg_nzero_nzero: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_cring R"   and "deg R p \<noteq> 0" shows "p \<noteq> \<zero>\<^bsub>UP R\<^esub>"
subset_trans: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "order.greater_eq B A"   and "order.greater_eq C B" shows "order.greater_eq C A"
guarantees_equiv: fixes F :: "'b program"   and X :: "'b program set"   and Y :: "'b program set" shows "(F \<in> X guarantees Y) = (\<forall>H. H \<in> X \<longrightarrow> F component_of H \<longrightarrow> H \<in> Y)"
add_le_cancel_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus c a) (plus c b) = less_eq a b"
rStatEq: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"
SeqRedsThrow: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>e;; e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree"   and a1 :: "'b"   and a2 :: "('b \<times> color) tree" assumes "\<And>t1 a t2 b t3 c t4. P (R (R t1 a t2) b t3) c t4"   and "\<And>a t2 b t3 c t4. P (R \<langle>\<rangle> a (R t2 b t3)) c t4"   and "\<And>v vc vb a t2 b t3 c t4. P (R (B v vc vb) a (R t2 b t3)) c t4"   and "\<And>a t2. P \<langle>\<rangle> a t2"   and "\<And>v vb a t2. P (B \<langle>\<rangle> v vb) a t2"   and "\<And>va a t2. P \<langle>\<langle>\<rangle>, va, \<langle>\<rangle>\<rangle> a t2"   and "\<And>va v ve vd a t2. P \<langle>\<langle>\<rangle>, va, B v ve vd\<rangle> a t2"   and "\<And>vc vf ve v vb a t2. P (B (B vc vf ve) v vb) a t2"   and "\<And>vc vf ve va a t2. P \<langle>B vc vf ve, va, \<langle>\<rangle>\<rangle> a t2"   and "\<And>vc vf ve va v vh vg a t2. P \<langle>B vc vf ve, va, B v vh vg\<rangle> a t2"   and "\<And>v vc vb a t2. P (B v vc vb) a t2" shows "P a0 a1 a2"
irreducible_imp_squarefree: fixes p :: "'b" assumes "irreducible p" shows "squarefree p"
Hahn_Banach: fixes E :: "'c set"   and F :: "'c set"   and p :: "'c \<Rightarrow> real"   and f :: "'c \<Rightarrow> real" assumes "vectorspace E"   and "Subspace.subspace F E"   and "seminorm E p"   and "linearform F f"   and "\<forall>x\<in>F. f x \<le> p x" shows "\<exists>h. linearform E h \<and> (\<forall>x\<in>F. h x = f x) \<and> (\<forall>x\<in>E. h x \<le> p x)"
Q: shows "{x \<in> space M. Q x} \<in> events"
hom: shows "VectorSpaceHom (\<sharp>\<cdot>) V (\<sharp>\<star>) T"
rel_id: shows "\<langle>v_rel, w_rel\<rangle>g.rel = br g.\<alpha> g.invar"
IH: shows "\<Prod>\<^sub># (image_mset f M) = (\<Prod>x\<in>set_mset M. f x ^ count M x)"
simps_2: fixes \<eta> :: "nat"   and vkey :: "'vkey"   and skey :: "'sigkey"   and log :: "('message \<times> 'signature) list"   and m :: "'message" shows "sign_oracle \<eta> (Some (vkey, skey, log)) m = (if valid_message \<eta> m then sign \<eta> skey m \<bind> (\<lambda>sig. return_spmf (sig, Some (vkey, skey, (m, sig) # log))) else return_pmf None)"
fmrel_def: shows "fmrel \<equiv> \<lambda>R. BNF_Def.vimage2p fmlookup fmlookup (rel_map R)"
ANDLEFT1_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set" shows "pi \<bullet> ANDLEFT1 (A AND B) X = ANDLEFT1 (A AND B) (pi \<bullet> X)"
vsv_axioms: shows "vsv (\<FF>\<lparr>ArrMap\<rparr>)"
unfinished_dnodes_def: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and s :: "'a"   and S :: "'a set" shows "Dijkstra.unfinished_dnodes w s S \<equiv> Dijkstra.all_dnodes w s - S"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "int option \<Rightarrow> int option \<Rightarrow> 'a"   and ivl :: "ivl" shows "P (case ivl of I x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. ivl = I x1 x2 \<longrightarrow> P (f x1 x2))"
DSourcesA5_L0: shows "DSources level0 sA5 = {sA4}"
runit_char_2: fixes a :: "'a" assumes "ide a" shows "C.tensor (C.runit a) C.unity = C.tensor a \<iota> \<cdot> C.assoc a C.unity C.unity"
ord_isom_Tw_segment: fixes E :: "('e, 'f) Order_scheme"   and a :: "'a" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))"   and "a \<in> carrier D" shows "ord_isom (Iod D (segment D a)) (Iod E (segment E ((Tw\<^bsub>D,E\<^esub>) a))) (restrict (Tw\<^bsub>D,E\<^esub>) (segment D a))"
cod_in_codomains: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "partial_magma C"   and "partial_magma.codomains C f \<noteq> {}" shows "partial_magma.cod C f \<in> partial_magma.codomains C f"
sym_preserve_diff: fixes A :: "'p"   and B :: "'p"   and M :: "'p"   and A' :: "'p"   and B' :: "'p" assumes "A \<noteq> B"   and "M Midpoint A A'"   and "M Midpoint B B'" shows "A' \<noteq> B'"
Rep_inject: fixes x :: "'a word"   and y :: "'a word" shows "(to_bl x = to_bl y) = (x = y)"
SelectIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E SelectIteratorCall src k its its_ty body : \<tau>" obtains \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>, \<sigma>, \<rho>)" and "length its \<le> Suc 0" and "\<rho> \<le> Boolean[?]"
is_aligned_less_sz: fixes a :: "'b word"   and sz :: "nat" assumes "is_aligned a sz"   and "a \<noteq> 0" shows "\<not> a < 2 ^ sz"
distinct_1: fixes x1 :: "('e, 'f) atm"   and x2 :: "('e, 'f) atm" shows "Pos x1 \<noteq> Neg x2"
no_loops: fixes a :: "'b \<times> 'b" assumes "a \<in> parcs (list_digraph G_list)" shows "fst a \<noteq> snd a"
case_prodD: fixes f :: "'ap \<Rightarrow> 'aq \<Rightarrow> bool"   and a :: "'ap"   and b :: "'aq" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
assms_9: shows "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
asym: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subset># b"   and "b \<subset># a" shows "False"
ack_less_ack_Suc2: fixes i :: "nat"   and j :: "nat" shows "ack i j < ack i (Suc j)"
lemma1: fixes F :: "'fmla set"   and \<psi> :: "'fmla" assumes "nprv (insert (subst \<tau> t out) F) \<psi>"   and "subst \<tau> t out \<in> fmla"   and "\<psi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F \<psi>"
finite_Int: fixes F :: "'h set"   and G :: "'h set" assumes "finite F \<or> finite G" shows "finite (F \<inter> G)"
compat: fixes y :: "nat"   and x :: "nat"   and z :: "nat" assumes "y \<le> x"   and "z < y" shows "z < x"
rev_nonempty_induct: fixes xs :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "\<And>x. P [x]"   and "\<And>x xs. \<lbrakk>xs \<noteq> []; P xs\<rbrakk> \<Longrightarrow> P (xs @ [x])" shows "P xs"
fail\<^sub>R_def: shows "fail\<^sub>R = {uu_. \<exists>r \<nu> s q q'. uu_ = (r, \<nu>, s) \<and> r q \<noteq> None \<and> q' = \<delta> q \<nu> \<and> q' \<notin> F \<and> sink q'}"
eq1: shows "f \<one>\<^bsub>R\<^esub> = g \<one>\<^bsub>R\<^esub>"
ordering_0: fixes A :: "ordinal set" shows "OrdinalVeblen.ordering A 0 = (LEAST z. z \<in> A)"
dgrad_p_setD: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'t" assumes "p \<in> dgrad_p_set d m"   and "v \<in> keys p" shows "d (pp_of_term v) \<le> m"
ndvd: shows "\<not> q ^ Suc (Suc i) dvd (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
bounded_iff: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "(a \<le> min b c) = (a \<le> b \<and> a \<le> c)"
indmhom_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'f" shows "f\<^sup>\<flat>\<^bsub>R M, N\<^esub> = (\<lambda>X\<in>set_mr_cos M (ker\<^bsub>M,N\<^esub> f). f (SOME x. x \<in> X))"
multiplicative_dirichlet_prod: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f"   and "multiplicative_function g" shows "multiplicative_function (f \<star> g)"
tab_simps_4: shows "local.dom (\<rho> \<cdot> \<phi>) = g'"
comp_arr_dom: fixes f :: "'c"   and a :: "'c" assumes "C.arr f"   and "C.dom f = a" shows "C f a = f"
assms_2: shows "instance_of\<^sub>t\<^sub>s ts\<^sub>2 ts\<^sub>3"
simps_1: shows "Tree2.bst \<langle>\<rangle> = True"
doms_2: shows "subst_domain (rm_vars (subst_domain \<theta> - X) \<theta>) \<subseteq> subst_domain \<theta>"
oprod_minus_Id: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Total r"   and "\<not> r \<subseteq> Id"   and "Total r'"   and "\<not> r' \<subseteq> Id" shows "r *o r' - Id \<subseteq> (r - Id) *o (r' - Id)"
wfzf_is_option_of: shows "wfzf is_option_of"
is_closed_subst_bvs1'_closeds: fixes t :: "Core.term"   and us :: "Core.term list"   and k :: "nat" assumes "\<not> is_dependent t"   and "\<forall>x\<in>set us. is_closed x" shows "\<not> is_dependent (subst_bvs1' t (Suc k) us)"
power_eq_if: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "'c"   and m :: "nat" shows "power.power one times p m = (if m = 0 then one else times p (power.power one times p (m - 1)))"
a: shows "\<nexists>fs. set fs \<subseteq> carrier (Ring_Divisibility.mult_of R) \<and> wfactors (Ring_Divisibility.mult_of R) fs a"
nAct: shows "\<nexists>i. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
iterator_to_ordered_iterator_correct: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sort_fun :: "'c list \<Rightarrow> 'c list"   and it :: "('c list \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'c list \<Rightarrow> 'c list) \<Rightarrow> 'c list \<Rightarrow> 'c list"   and S0 :: "'c set" assumes "\<And>l. sorted_wrt R (sort_fun l) \<and> mset (sort_fun l) = mset l"   and "set_iterator it S0" shows "set_iterator_genord (iterator_to_ordered_iterator sort_fun it) S0 R"
unordered_pair_element_equality: fixes p :: "'b"   and q :: "'b"   and r :: "'b"   and s :: "'b" assumes "{p, q} = {r, s}"   and "p = r" shows "q = s"
nvi_1: fixes i :: "nat" assumes "i < length us" shows "normal_vectors ws ! i \<in> carrier_vec n"
indRelRTPO_to_TRel_4: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q" shows "\<forall>TP TQ. TP \<in>T P \<and> TQ \<in>T Q \<longrightarrow> (TP, TQ) \<in> TRel\<^sup>+"
funs_term_subterms_eq_1: fixes t :: "('c, 'd) Term.term" shows "\<Union> (funs_term ` subterms t) = funs_term t"
F_4: fixes i :: "'i"   and k :: "nat" shows "emeasure (M i) (F i k) \<noteq> \<infinity>"
wp_weaken_pre: fixes Q :: "'a \<Rightarrow> real"   and a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "Q \<tturnstile> wp a R"   and "P \<tturnstile> Q" shows "P \<tturnstile> wp a R"
rtranclp_tranclp_absorb: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "R\<^sup>*\<^sup>*\<^sup>+\<^sup>+ = R\<^sup>*\<^sup>*"
closed_subsets_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<aa> :: "'c set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.closed_subsets R addition multiplication zero unit \<aa> \<equiv> {I. pr_ideal R I addition multiplication zero unit \<and> \<aa> \<subseteq> I}"
fact_dvd_pochhammer: fixes k :: "nat"   and a :: "int" shows "fact k dvd pochhammer a k"
diagonal_to_Smith_aux_PQ_JNF_dim: fixes P' :: "'b mat"   and m :: "nat"   and A' :: "'b mat"   and n :: "nat"   and Q' :: "'b mat"   and xs :: "nat list"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" assumes "P' \<in> carrier_mat m m \<and> A' \<in> carrier_mat m n \<and> Q' \<in> carrier_mat n n"   and "diagonal_to_Smith_aux_PQ_JNF xs bezout (P', A', Q') = (P, A, Q)" shows "P \<in> carrier_mat m m \<and> A \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
dens': fixes M :: "int measure"   and \<delta> :: "int \<Rightarrow> ennreal" assumes "has_density M (count_space UNIV) \<delta>" shows "has_density (distr M (count_space UNIV) f) (count_space UNIV) (g \<delta>)"
map_pmf_def: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'e pmf" shows "map_pmf f M = M \<bind> (\<lambda>x. return_pmf (f x))"
commute: fixes a :: "'f multiset"   and b :: "'f multiset" shows "a \<union># b = b \<union># a"
game_hash_set_def: fixes W :: "'\<beta> set" shows "local.hash.game_hash_set W = seed_gen \<bind> (\<lambda>seed. return_spmf (\<not> inj_on (h seed) W))"
4: shows "prv (neg (cnj (PP \<langle>\<phi>\<rangle>) (PP (encF N \<langle>\<phi>\<rangle>))))"
SBHRMWReadOnly: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "\<not> cond (\<theta>(t \<mapsto> m a))" shows "(RMW a t (D, f) cond ret A L R W # is, \<theta>, [], m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>(t \<mapsto> m a), [], m, False, \<O>, Map.empty, \<S>)"
extend_dec_env_def: fixes new_env :: "v sem_env"   and env :: "v sem_env" shows "extend_dec_env new_env env = make_sem_env (nsAppend (sem_env.v new_env) (sem_env.v env)) (nsAppend (c new_env) (c env))"
FL_distinguished_bounded_support: fixes L :: "'b \<times> 'c fs_set \<times> 'c \<Rightarrow> 'c fs_set"   and satisfies :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and transition :: "'d \<Rightarrow> ('b, 'd) residual \<Rightarrow> bool"   and effect_apply :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c fs_set"   and x :: "('a, 'e, 'b, 'c) FL_Formula.formula"   and P :: "'d"   and Q :: "'d" assumes "indexed_effect_nominal_ts TYPE('a) L satisfies transition effect_apply"   and "effect_nominal_ts.is_FL_formula L F x"   and "indexed_effect_nominal_ts.is_distinguishing_formula satisfies transition effect_apply x P Q" obtains y :: "('a, 'e, 'b, 'c) FL_Formula.formula" where "effect_nominal_ts.is_FL_formula L F y" and "supp y \<subseteq> supp (F, P)" and "indexed_effect_nominal_ts.is_distinguishing_formula satisfies transition effect_apply y P Q"
\<mu>\<nu>\<tau>: shows "VVV.ide \<mu>\<nu>\<tau>"
rep_eq: fixes x :: "('c, 'd) fps" shows "($) (x\<^sup>\<star>) = star_fps_rep (($) x)"
elim: assumes "s1\<Colon>\<preceq>(G, L)"   and "normal s1 \<Longrightarrow> G,L,snd s1\<turnstile>t\<succ>v\<Colon>\<preceq>T"   and "error_free s0 = error_free s1" shows "P"
scaleR_left_le_one_le: fixes x :: "'b"   and a :: "real" assumes "(0::'b) \<le> x"   and "a \<le> 1" shows "a *\<^sub>R x \<le> x"
gAbsPresGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gAbsPresGWls MOD \<equiv> \<forall>xs s x X' X. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> gWlsAbs MOD (xs, s) (gAbs MOD xs x X' X)"
Scan_\<J>_subset_\<J>: fixes k :: "nat"   and u :: "nat" shows "Scan (\<T> k (Suc u)) k (\<J> k u) \<subseteq> \<J> k (Suc u)"
entails_def: fixes N1 :: "'a literal multiset set"   and N2 :: "'a literal multiset set" shows "(N1 \<TTurnstile>\<G>e N2) = (\<forall>q\<in>UNIV. \<Union> (\<G>_F ` N1) \<TTurnstile>e \<Union> (\<G>_F ` N2))"
sorted_nth_mono: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "sorted xs"   and "i \<le> j"   and "j < length xs" shows "xs ! i \<le> xs ! j"
abs_eq: fixes x :: "dflt_size word" shows "bit (Abs_uint x) = bit x"
rtranclp_add_poly_p_polynomial_of_mset_full: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and r' :: "(char list multiset \<times> int) multiset" assumes "add_poly_p\<^sup>*\<^sup>* (p, q, {#}) ({#}, {#}, r')" shows "polynomial_of_mset r' = polynomial_of_mset p + polynomial_of_mset q"
fps_unit_dvd: fixes f :: "'b fps"   and g :: "'b fps" assumes "f $ 0 \<noteq> (0::'b)" shows "f dvd g"
mod_lemma: fixes c :: "nat"   and r :: "nat"   and b :: "nat"   and q :: "nat" assumes "0 < c"   and "r < b" shows "b * (q mod c) + r < b * c"
elims_2: fixes x :: "real \<times> real" assumes "invar x" obtains n :: "real" and l :: "real" where "x = (n, l)" and "l0 \<le> l \<and> (l0 \<le> l / c \<longrightarrow> f1 * l \<le> n) \<and> n \<le> f2 * l"
list_wset_autoref_empty: fixes R :: "('a \<times> 'b) set" shows "([], {}) \<in> \<langle>R\<rangle>list_wset_rel"
mset_pos_empty: shows "mset_pos {#}\<^sub>z = {#}"
equiv_trans_refl: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "equiv_trans t t"
mono_SUP: fixes f :: "'b \<times> 'c \<Rightarrow> 'd"   and A :: "'e \<Rightarrow> 'b \<times> 'c"   and I :: "'e set" assumes "prod.mono f" shows "(\<Squnion>x\<in>I. f (A x)) \<le> f (Sup_prod (A ` I))"
nths_assoc_three_A: fixes A :: "nat set"   and B :: "nat set"   and C :: "nat set"   and xs :: "'a list" assumes "A \<inter> B = {}"   and "(A \<union> B) \<inter> C = {}" shows "nths (nths xs (ind_in_set (A \<union> B \<union> C) ` (A \<union> B))) (ind_in_set (A \<union> B) ` A) = nths xs (ind_in_set (A \<union> B \<union> C) ` A)"
simps_1: fixes Ks :: "('a, 'c) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option" shows "subsumes_list [] Ks \<sigma> = True"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsA at" shows "\<pi>1 x = \<pi>2 x"
worecSL_def: fixes S :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" shows "worecSL S L \<equiv> local.worec (mergeSL S L)"
assms_5: shows "\<forall>k\<in>set vs. 0 < k \<longrightarrow> k \<le> n \<and> (\<exists>c. v c = k)"
simps_5: fixes edges' :: "'c edge set \<Rightarrow> 'c edge set"   and root :: "'c"   and edges :: "'c edge set"   and more :: "'d" shows "edges_update edges' \<lparr>root = root, edges = edges, \<dots> = more\<rparr> = \<lparr>root = root, edges = edges' edges, \<dots> = more\<rparr>"
simps_2: fixes x :: "'b"   and xs :: "['b]"   and ys :: "['b]" shows "(x : xs) ++ ys = x : xs ++ ys"
iT_Plus_neg_finite_iff: fixes I :: "nat set"   and k :: "nat" shows "finite (I \<oplus>- k) = finite I"
continuous_on_dist: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. dist (f x) (g x))"
space_C: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} (\<lambda>_. M))" shows "space (C n m x) = space (Pi\<^sub>M {0..<n + m} (\<lambda>_. M))"
scale_minus_left: fixes a :: "'c"   and x :: "'c" shows "- a * x = - (a * x)"
closed_plane: fixes i :: "'c"   and c :: "real" shows "closed (plane i c)"
a_host_shadow_root_rel_finite: fixes h :: "(_) heap" shows "finite (host_shadow_root_rel h)"
induct: fixes x1 :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and x2 :: "('e \<Rightarrow> real) set set"   and x3 :: "'d"   and x4 :: "('e \<Rightarrow> real) set"   and x5 :: "'d"   and x6 :: "('e \<Rightarrow> real) set"   and P :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint) \<Rightarrow> ('e \<Rightarrow> real) set set \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> real) set \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> real) set \<Rightarrow> bool" assumes "x1,x2 \<turnstile> \<langle>x3, x4\<rangle> \<leadsto>* \<langle>x5, x6\<rangle>"   and "\<And>A \<R> l R. P A \<R> l R l R"   and "\<And>A \<R> l R l' R' l'' R''. \<lbrakk>A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle>; P A \<R> l R l' R'; A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto> \<langle>l'', R''\<rangle>\<rbrakk> \<Longrightarrow> P A \<R> l R l'' R''" shows "P x1 x2 x3 x4 x5 x6"