hcju/isabelleps huggingface dataset:
join_ivl_def: fixes i1 :: "ivl"   and i2 :: "ivl" shows "i1 \<squnion> i2 = (if is_empty i1 then i2 else if is_empty i2 then i1 else case (i1, i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (min_option False l1 l2) (max_option True h1 h2))"
case_sum_inject: fixes f1 :: "'b \<Rightarrow> 'd"   and f2 :: "'c \<Rightarrow> 'd"   and g1 :: "'b \<Rightarrow> 'd"   and g2 :: "'c \<Rightarrow> 'd" assumes "case_sum f1 f2 = case_sum g1 g2" obtains "f1 = g1" and "f2 = g2"
walk_in_V: fixes xs :: "'a list" assumes "walk xs" shows "set xs \<subseteq> V"
greatest_le: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and A :: "'c set"   and a :: "'c" assumes "greatest L x A"   and "a \<in> A" shows "a \<sqsubseteq>\<^bsub>L\<^esub> x"
zmset_subset_eqI: fixes A :: "'b zmultiset"   and B :: "'b zmultiset" assumes "\<And>a. zcount A a \<le> zcount B a" shows "A \<subseteq>#\<^sub>z B"
objT_def: fixes P :: "ast_problem" shows "ast_problem.objT P \<equiv> map_of (objects P) ++ ast_domain.constT (domain P)"
decl: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c :: "('a, 'c, 'd) Stmt"   and mu :: "'a ModeUpd"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and E :: "('a, 'c, 'd) Stmt list" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((c, sifum_lang_no_dma.update_modes mu mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "(((sifum_lang_no_dma.cxt_to_stmt E (c@[mu]), mds), mem), (sifum_lang_no_dma.cxt_to_stmt E c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
disjoint: shows "A \<Gamma> \<inter> B \<Gamma> = {}"
mbisim_thrD1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x1"   and ln :: "'l \<Rightarrow>f nat" assumes "s1 \<approx>m s2"   and "thr s1 t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>x'. thr s2 t = \<lfloor>(x', ln)\<rfloor> \<and> t \<turnstile> (x, shr s1) \<approx> (x', shr s2) \<and> (wset s1 t = None \<or> x \<approx>w x')"
Nonce_Tag: fixes X :: "nonce_t"   and X' :: "tag" shows "Nonce X \<noteq> Tag X'"
simps_9: fixes i :: "int"   and a :: "MIR.num" shows "bound0 (Dvd i a) = numbound0 a"
order_iff_strict: fixes a :: "'b list"   and b :: "'b list" shows "(a \<le>p b) = (a <p b \<or> a = b)"
ir_If_True_False: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>2' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and c\<^sub>1' :: "com" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> bval b s \<and> \<not> bval b' s') c\<^sub>1 c\<^sub>2' Q" shows "ir_hoare P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (IF b' THEN c\<^sub>1' ELSE c\<^sub>2') Q"
leadsto_def: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "P \<leadsto> Q \<equiv> \<box>(P \<longrightarrow> \<diamond>Q)"
times_V_def: fixes x :: "V" shows "(*) x \<equiv> transrec (\<lambda>f y. SUP u\<in>elts y. lift (f u) x)"
iter_def: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_or_infinite_iteration.iter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) \<equiv> lfp (\<lambda>x::'b::refinement_lattice. nil \<sqinter> seq c x)"
eq: fixes \<omega> :: "'a stream" assumes "P \<omega>" shows "Q1 \<omega> = Q2 \<omega>"
wf_trmI: fixes T :: "('fun, 'b) Term.term list"   and f :: "'fun" assumes "\<And>t. t \<in> set T \<Longrightarrow> wf\<^sub>t\<^sub>r\<^sub>m t"   and "length T = arity f" shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"
basis_max_indep: fixes X :: "'a set"   and x :: "'a" assumes "basis X"   and "x \<in> carrier - X" shows "\<not> indep (insert x X)"
b: shows "v = column (LEAST n. A $ b $ n \<noteq> (0::'a)) A"
PPs_insertE: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and x :: "'b"   and X :: "'b set" assumes "t \<in> .[insert x X]" obtains e :: "nat" and tx :: "'b \<Rightarrow>\<^sub>0 nat" where "tx \<in> .[X]" and "t = monomial e x + tx"
Cor_2_4_1_multi: fixes CT :: "nat \<Rightarrow> classDef option"   and e :: "exp"   and e' :: "exp"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and C :: "nat" assumes "CT \<turnstile> e \<rightarrow>* e'"   and "CT OK"   and "CT;\<Gamma> \<turnstile> e : C" shows "\<exists>C'. CT;\<Gamma> \<turnstile> e' : C' \<and> CT \<turnstile> C' <: C"
coprime_imp_no_common_roots: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" assumes "coprime p q" shows "\<not> (poly p x = (0::'a) \<and> poly q x = (0::'a))"
composition_closed: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "monoid M composition unit"   and "a \<in> M"   and "b \<in> M" shows "composition a b \<in> M"
add_idem: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.join_semilattice plus less_eq less" shows "plus x x = x"
supteq_not_supt: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<unrhd> t"   and "(s, t) \<notin> {\<rhd>}" shows "s = t"
simps_1: fixes T1 :: "'c trm"   and T2 :: "'c trm" shows "nvA (Eq T1 T2) = nvT T1 \<union> nvT T2"
h01: shows "\<forall>t\<in>{0..1}. h (t, 1) = h (t, 0)"
ide_compE: fixes g :: "(bool, unit) arr"   and f :: "(bool, unit) arr" assumes "ide (COMP g f)" obtains "seq g f" and "seq f g" and "COMP g f = local.dom f" and "COMP g f = cod g"
preserves_cod: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "D.cod (cmp f) = FoH\<^sub>C.map (C.VV.cod f)"
deriv: shows "\<forall>x. (f has_derivative F x) (at x)"
Fract: fixes a :: "'a"   and b :: "'a" assumes "q = Fract a b"   and "b \<noteq> (0::'a)"   and "a \<noteq> (0::'a)" shows "thesis"
par_strict_symmetry: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "C D ParStrict A B"
roofed_greaterI: fixes x :: "'a"   and S :: "'a set"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set" assumes "x \<in> S" shows "x \<in> roofed_gen G B S"
distrib: fixes g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Prod>x\<in>A. g x * h x) = prod g A * prod h A"
ta_tb_wf_1: shows "wf\<^sub>t\<^sub>r\<^sub>m ta"
stream_pred_snth: fixes P :: "'b \<Rightarrow> bool"   and w :: "'b stream" shows "pred_stream P w = (\<forall>i. P (w !! i))"
v11_app_vconverse_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>(\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>a\<rparr>\<rparr> = a"
Assign: fixes D :: "nat set"   and \<theta> :: "nat \<Rightarrow> nat option"   and volatile :: "bool"   and a :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and e :: "expr"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and t :: "nat" assumes "D \<subseteq> dom \<theta>" shows "\<theta>\<turnstile> (Assign volatile (Tmp (D, a)) e A L R W, t) \<rightarrow>\<^sub>s ((Skip, t + used_tmps e), issue_expr t e @ [Write volatile (a \<theta>) (eval_expr t e) (A \<theta>) (L \<theta>) (R \<theta>) (W \<theta>)])"
l_one: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<one> \<otimes> x = x"
P1UR: shows "P1 * UR \<in> carrier_mat n m"
incl: assumes "[] \<in> A" shows "lQuot c (star A) \<subseteq> lQuot c A @@ star A"
distinct_map_of_rev: fixes xs :: "('c \<times> 'd) list" assumes "distinct (map fst xs)" shows "map_of (rev xs) = map_of xs"
InputR_def: fixes x2 :: "name"   and x1 :: "name" shows "x2<x1> \<equiv> Abs_freeRes (InputR_Rep x2 x1)"
UniN: fixes H :: "('a, 'b) fm set"   and p :: "('a, 'b) fm" assumes "Hintikka H"   and "(\<^bold>\<not> \<^bold>\<forall> p) \<in> H" shows "\<exists>a. (\<^bold>\<not> p\<langle>\<^bold>\<dagger> a []/0\<rangle>) \<in> H"
eventually_le_at_bot: fixes c :: "'b" shows "\<forall>\<^sub>F x in at_bot. x \<le> c"
assms_7: shows "sum f1 B1 = sum f2 B2"
less_1_simp: fixes n :: "'b word"   and m :: "'b word" shows "(n - 1 < m) = (n \<le> m \<and> n \<noteq> 0)"
surj_on_push_forward_inclusion: fixes p :: "'a" assumes "p \<in> sub.carrier" shows "tangent_space p \<subseteq> inclusion.push_forward ` sub.tangent_space p"
fin: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> U" shows "\<exists>i n\<^sub>0. \<psi> i = f \<and> (\<forall>n<n\<^sub>0. s (f \<triangleright> n) \<down>= 0) \<and> (\<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= Suc i)"
prv_pls_imp_cong: fixes t1 :: "'trm"   and t1' :: "'trm"   and t2 :: "'trm"   and t2' :: "'trm" assumes "t1 \<in> atrm"   and "t1' \<in> atrm"   and "t2 \<in> atrm"   and "t2' \<in> atrm" shows "prv (imp (eql t1 t1') (imp (eql t2 t2') (eql (pls t1 t2) (pls t1' t2'))))"
1: shows "row (vec_vec_Tensor v w # vec_mat_Tensor v N) i = vec_vec_Tensor v w ! i # row (vec_mat_Tensor v N) i"
recurse_traverse: fixes f :: "'e \<Rightarrow> ('d, 'c) state"   and g :: "'c \<Rightarrow> ('d, 'e) state" assumes "f \<bullet> g = pure" shows "recurse f \<bullet> traverse g = pure"
fmset_wfactors_mult: fixes cs :: "'a list"   and as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "fmset G cs = fmset G as + fmset G bs"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G"   and "set cs \<subseteq> carrier G"   and "wfactors G as a"   and "wfactors G bs b"   and "wfactors G cs c" shows "c \<sim> a \<otimes> b"
Conj_def: fixes xset :: "('d, 'e, 'f) formula set['d]" shows "Conj xset = Abs_formula (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
abs_eq: fixes xa :: "'a"   and x :: "nat" shows "Polynomial.monom xa x = Abs_poly (\<lambda>n. if x = n then xa else (0::'a))"
ab_left_minus: fixes a :: "'a" shows "- a + a = (0::'a)"
eq_up_to_refl: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" shows "\<pi> =\<^bsub>k\<^esub> \<pi>"
closed_sum_closure_right: fixes A :: "'d set"   and B :: "'d set" shows "A +\<^sub>M closure B = A +\<^sub>M B"
inject: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat"   and y21 :: "'b"   and y22 :: "'b tree"   and y23 :: "'b tree"   and y24 :: "nat" shows "(MKT x21 x22 x23 x24 = MKT y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
bounded_subset_cball: fixes S :: "'c set" shows "bounded S = (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
insertion_fun_def: fixes f :: "nat \<Rightarrow> 'a"   and p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'a" shows "insertion_fun f p = (\<Sum>m. p m * (\<Prod>v. f v ^ lookup m v))"
lexordp_eq_simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "ord.lexordp_eq less xs [] = (xs = [])"
GSMP_disjoint_term: fixes l :: "'b"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and l' :: "'b"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec" shows "t \<notin> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) \<or> t \<notin> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) \<or> t \<in> Sec \<or> {} \<turnstile>\<^sub>c t"
abs_eq_iff: fixes x :: "('e, 'f, 'g) Tree"   and y :: "('e, 'f, 'g) Tree" shows "(abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> y) = (x =\<^sub>\<alpha> y)"
ClassD: fixes b :: "'a"   and a :: "'a" assumes "b \<in> additive.Class a"   and "a \<in> R" shows "(b, a) \<in> Ring_Congruence"
mult_cancel_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times a c = times b c) = (c = zero \<or> a = b)"
pair_complete_iff_finite_complete: shows "pair_complete A (\<lambda>x y. y \<sqsubseteq> x) = finite_complete A (\<lambda>x y. y \<sqsubseteq> x)"
prems: shows "typ_of1 Ts (t1 $ t2) = Some T"
some_geodesic_is_geodesic_segment_1: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.geodesic_space dist uniformity open" shows "geodesic_segment_between {x--y} x y"
uniformly_convergent_eq_cauchy: fixes P :: "'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" shows "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e) = (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e)"
maxsimpD_simplex: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimp X x" shows "x \<in> X"
derive_define_flip: shows "{#b \<inter> bl. b \<in># remove1_mset bl \<B>#} = \<B>\<^sup>D"
simps_1: shows "jkbpC\<^bsub>0\<^esub> = {tInit s |s. s \<in> set envInit}"
satC_cong: fixes c :: "('fsym, 'psym) lit list"   and \<xi>1 :: "Preliminaries.var \<Rightarrow> 'univ"   and \<xi>2 :: "Preliminaries.var \<Rightarrow> 'univ" assumes "\<And>x. x \<in> varsC c \<Longrightarrow> \<xi>1 x = \<xi>2 x" shows "satC \<xi>1 c = satC \<xi>2 c"
ceiling_of_int: fixes z :: "int" shows "\<lceil>of_int z\<rceil> = z"
scale_def: fixes r :: "rat"   and lp :: "(nat, rat) fmap" shows "scale r lp = (if r = 0 then fmempty else fmmap ((*) r) lp)"
While: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and c :: "('e, 'd, 'f) com"   and i :: "'e set"   and A :: "'e set"   and b :: "'e set"   and Q :: "'e set"   and r :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c i, A"   and "i \<inter> b \<subseteq> pre P"   and "i \<inter> - b \<subseteq> Q"   and "r \<subseteq> i" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnWhile r i P) (While b c) Q, A"
inv_mult: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> y) = inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y"
ennreal_enn2real_if: fixes r :: "ennreal" shows "ennreal (enn2real r) = (if r = \<top> then 0 else r)"
dictator_def: fixes F :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> (alt \<times> alt) set"   and i :: "indi" shows "dictator F i \<equiv> \<forall>P\<in>Prof. F P = P i"
S5n_def: fixes M :: "('d, 'e, 'f) KripkeStructure" shows "S5n M \<equiv> \<forall>a. equiv (worlds M) (relations M a)"
mat_of_rows_carrier_3: fixes n :: "nat"   and vs :: "'b vec list" shows "dim_col (mat_of_rows n vs) = n"
vimage2p_rel_fun: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_fun (BNF_Def.vimage2p f g R) R f g"
in_X: fixes x :: "'b" assumes "v x \<le> n" shows "x \<in> clk_set A"
Arity_zero_top: fixes x :: "Arity" shows "x \<sqsubseteq> 0"
terminating_iff: fixes x :: "'a" shows "terminating x = (backward_terminating x \<and> forward_terminating x)"
length_distincts: fixes xss :: "'b list list" assumes "distincts xss" shows "length xss = card (sset xss)"
poly_lim_inf: fixes p :: "real poly" shows "\<forall>\<^sub>F x in at_top. sgn (poly p x) = poly_inf p"
crossproduct_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and w :: "'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(plus (times w y) (times x z) = plus (times w z) (times x y)) = (w = x \<or> y = z)"
le_inf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "traces.le_inf ind w\<^sub>1 w\<^sub>2"   and "traces.le_inf ind w\<^sub>2 w\<^sub>3" shows "traces.le_inf ind w\<^sub>1 w\<^sub>3"
error_reads: fixes e :: "'b"   and h :: "'a"   and h' :: "'a" shows "reads {} (error e) h h'"
valid_cfg_action: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and cfg :: "'a cfg"   and s :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "s \<in> set_pmf (action cfg)" shows "s \<in> S"
inject_24: fixes x311 :: "t"   and x312 :: "cvtop"   and x313 :: "t"   and x314 :: "sx option"   and y311 :: "t"   and y312 :: "cvtop"   and y313 :: "t"   and y314 :: "sx option" shows "(Cvtop x311 x312 x313 x314 = Cvtop y311 y312 y313 y314) = (x311 = y311 \<and> x312 = y312 \<and> x313 = y313 \<and> x314 = y314)"
nonneg_exp: fixes f :: "'a fds" assumes "nonneg_dirichlet_series f" shows "nonneg_dirichlet_series (fds_exp f)"
ext_language_Collapse_dfa: fixes u :: "'a list" shows "(u \<in> Collapse.language) = (u \<in> language)"
maximum_def: fixes i :: "nat_int" assumes "i \<noteq> bot" shows "maximum i = Max (Rep_nat_int i)"
simps_1: fixes uu :: "'b infty" shows "(Infty < uu) = False"
rep_residual_def: shows "rep_residual \<equiv> quot_type.rep Rep_residual"
leg0_simps_3: shows "trg (tab\<^sub>0 (r \<star> s)) = src (r \<star> s)"
safe_delayed_def: fixes cfg :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" shows "safe_delayed cfg \<equiv> let (ts, m, \<S>) = cfg in \<forall>i<length ts. let (p, is, \<theta>, x, \<D>, \<O>, \<R>) = ts ! i in map owned ts,map released ts,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
multiple_is_design: fixes n :: "nat" shows "design \<V>' (target.multiple_blocks n)"
set_shadow_root_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_shadow_root_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h = ShadowRootClass.type_wf h'"
subsetFn_def: fixes X :: "'b"   and Y :: "'a set" shows "subsetFn X Y \<equiv> \<lambda>x. if x \<in> Y then x else SOME y. y \<in> Y"
simps_2: fixes c :: "'form"   and i :: "nat"   and "is" :: "nat list" shows "inst (c, i # is) = iSubst (tree_at (it' c) is)"
psubstInp_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option"   and inp :: "'b \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option" shows "inp %[rho] \<equiv> lift (psubst rho) inp"
sublens_implies_sublens': fixes Y :: "'d \<Longrightarrow> 'e"   and X :: "'f \<Longrightarrow> 'e" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "X \<subseteq>\<^sub>L' Y"
regular_event_cannot_induce_snapshot: fixes c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration" assumes "\<not> ps c p \<noteq> None"   and "c \<turnstile> ev \<mapsto> c'" shows "regular_event ev \<longrightarrow> \<not> ps c' p \<noteq> None"
ascP_f_Cons: fixes f :: "'c list \<Rightarrow> 'c list"   and x :: "'c"   and xs :: "'c list" assumes "ascP f" shows "f (x # xs) = f [] @ x # xs"
normalize_mult_unit_left: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" shows "normalize (a * b) = normalize b"
\<epsilon>o_in_hom: fixes x :: "'b" assumes "B.ide x" shows "\<guillemotleft>F'G'.\<epsilon>o x : F' (G' x) \<rightarrow>\<^sub>B x\<guillemotright>"
simps_1: fixes x1 :: "'b"   and x2 :: "'c"   and y1 :: "'b"   and y2 :: "'c" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
poly_mapping_rangeE: fixes c :: "'c"   and p :: "'d \<Rightarrow>\<^sub>0 'c" assumes "c \<in> Poly_Mapping.range p" obtains k :: "'d" where "k \<in> keys p" and "c = lookup p k"
fst_snd_flip: fixes xy :: "'a \<times> 'b" shows "fst xy = (snd \<circ>\<circ> case_prod) (\<lambda>x y. (y, x)) xy"
pdiv_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and q :: "'c list" assumes "ring R" shows "ring.pdiv R p q = (if q = [] then [] else fst (ring.long_division R p q))"
trcl_pair_induct: fixes xc1 :: "'f"   and xc2 :: "'g"   and xb :: "'h list"   and xa1 :: "'f"   and xa2 :: "'g"   and t :: "(('f \<times> 'g) \<times> 'h \<times> 'f \<times> 'g) set"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h list \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "((xc1, xc2), xb, xa1, xa2) \<in> trcl t"   and "\<And>a b. P a b [] a b"   and "\<And>a b aa ab ba w ac bb. \<lbrakk>((a, b), aa, ab, ba) \<in> t; ((ab, ba), w, ac, bb) \<in> trcl t; P ab ba w ac bb\<rbrakk> \<Longrightarrow> P a b (aa # w) ac bb" shows "P xc1 xc2 xb xa1 xa2"
HS_If_alg: fixes G :: "Var \<Rightarrow> L"   and b :: "BExpr"   and p :: "L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L"   and d :: "IMP" assumes "(G, b, p) \<in> HS_B"   and "(p, G, c, H) \<in> HS"   and "(p, G, d, H) \<in> HS" shows "(p, G, Iff b c d, H) \<in> HS"
lucas_lehmer_distrib_right: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat"   and z :: "nat \<times> nat" assumes "1 < m" shows "lucas_lehmer_mult m (lucas_lehmer_add m x y) z = lucas_lehmer_add m (lucas_lehmer_mult m x z) (lucas_lehmer_mult m y z)"
def_hashmap_size_uint32_def: shows "def_hashmap_size = (\<lambda>_. 16)"
fixp_induct: fixes P :: "('c literal set set \<Rightarrow> 'c literal set set option) \<Rightarrow> bool" assumes "option.admissible P"   and "P Map.empty"   and "\<And>S. P S \<Longrightarrow> P (\<lambda>Sa. let R = res Sa \<union> Sa in if R = Sa then Some Sa else S R)" shows "P Res"
iso_trans: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and I :: "('f, 'g) monoid_scheme" assumes "G \<cong> H"   and "H \<cong> I" shows "G \<cong> I"
subtensor_combine_induct: fixes P :: "'b tensor \<Rightarrow> bool"   and A :: "'b tensor" assumes "\<And>A. dims A = [] \<Longrightarrow> P A"   and "\<And>As ds. \<lbrakk>\<And>A. A \<in> set As \<Longrightarrow> P A; \<And>A. A \<in> set As \<Longrightarrow> dims A = ds\<rbrakk> \<Longrightarrow> P (subtensor_combine ds As)" shows "P A"
lunit_ide_simp: fixes f :: "'a" assumes "ide f" shows "Cop.lunit f = \<r>[f]"
M_def: shows "M = (\<lambda>n. mat_extension dims vars1 (testN n))"
smult_minus_1: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and v :: "'d" assumes "Module.module R M"   and "v \<in> carrier M" shows "\<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> v = \<ominus>\<^bsub>M\<^esub> v"
monad_right_unit: fixes m :: "'e\<cdot>'f" shows "m \<bind> return = m"