hcju/isabelleps huggingface dataset:
ftv_Ctxt_subst: fixes X :: "W.tvar"   and \<Gamma> :: "(W.var \<times> tyS) list"   and Y :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "X \<in> set (ftv \<Gamma>)"   and "Y \<in> set (ftv (lookup \<theta> X))" shows "Y \<in> set (ftv (\<theta><\<Gamma>>))"
rInput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list"   and C :: "'d"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (M\<lparr>\<lambda>*xvec N\<rparr>.P) K (N[xvec::=Tvec]) (P[xvec::=Tvec])"
Collect_conv_if: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "{x. x = a \<and> P x} = (if P a then {a} else {})"
vg_def: shows "vg = (\<lambda>w. inverse (g w))"
Least_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and k :: "'b" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
T_splay_simps_9: fixes b :: "'a"   and x :: "'a"   and c :: "'a"   and AB :: "'a tree"   and C :: "'a tree"   and D :: "'a tree" assumes "b < x"   and "c < x" shows "T_splay x \<langle>AB, b, \<langle>C, c, D\<rangle>\<rangle> = (if D = \<langle>\<rangle> then 1 else T_splay x D + 1)"
N_of_norm_2: fixes f :: "'c" shows "Norm N_of_norm f = norm f"
mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> O[F](g1)"   and "f2 \<in> O[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)"
CollectI: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "P a" shows "a \<in> {x. P x}"
obtain_shortest_path: fixes s :: "'v"   and u :: "'v" obtains p :: "enat list" where "path s p u" and "\<delta> s u = sum_list p"
wp_loop_mono: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and u :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "le_trans t u"   and "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "\<And>P. sound P \<Longrightarrow> sound (u P)" shows "le_trans (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) (wp (body ;; Embed u \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
mk_eqcl_set_snd: fixes i :: "nat"   and zs :: "nat list"   and j :: "nat"   and xs :: "nat option list"   and T :: "bool list list" assumes "i \<notin> set zs"   and "i < j" shows "i \<notin> set (snd (mk_eqcl xs zs j T))"
push_forward_def: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real" shows "d23.push_forward X = restrict0 d23.dest.diff_fun_space (\<lambda>ga. X (restrict0 d12.dest.carrier (ga \<circ> g)))"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> 'univ"   and xa :: "('fsym, 'psym) lit" assumes "\<not> satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>__" and "xa = Pos a__" and "\<not> satA \<xi>__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> satA \<xi> a"
is_parser_def: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list" shows "is_parser p = (\<forall>s r x. p s = Inr (x, r) \<longrightarrow> length r \<le> length s)"
circ_mult_increasing: fixes x :: "'a" shows "x \<le> x * x\<^sup>\<circ>"
map\<^sub>0_def: fixes a :: "'c" shows "F.map\<^sub>0 a \<equiv> src\<^sub>D (F a)"
weak_preservation_of_barbs_and_closures_3: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_weakly_preserves_barbs Rel CWB" shows "rel_weakly_preserves_barbs (Rel\<^sup>*) CWB"
group_iso_rel_def: fixes G :: "'d group"   and H :: "'d group" shows "group_iso_rel G H = (\<exists>\<phi>. \<phi> \<in> Group.iso (Rep_group G) (Rep_group H))"
fmdrop_fmupd_same: fixes x :: "'e"   and y :: "'f"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmdrop x (m(x \<mapsto>\<^sub>f y)) = fmdrop x m"
list_e_eq_lel_1: fixes e :: "'b"   and l1 :: "'b list"   and e' :: "'b"   and l2 :: "'b list" shows "([e] = l1 @ e' # l2) = (l1 = [] \<and> e' = e \<and> l2 = [])"
homeomorphism_def: fixes s :: "'a set"   and t :: "'c set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'a" shows "homeomorphism s t f g = ((\<forall>x\<in>s. g (f x) = x) \<and> f ` s = t \<and> continuous_on s f \<and> (\<forall>y\<in>t. f (g y) = y) \<and> g ` t = s \<and> continuous_on t g)"
S_transform_formula_def: shows "S_transform_formula \<equiv> (Rep_formula ---> id) S_transform_Tree\<^sub>\<alpha>"
assms_1: shows "h \<turnstile> f \<rightarrow>\<^sub>r x"
subset_refl: fixes A :: "'q set" shows "A \<subseteq> A"
nprv_eql_transE012: fixes t1 :: "'trm"   and t2 :: "'trm"   and F :: "'fmla set"   and t3 :: "'trm"   and \<psi> :: "'fmla" assumes "eql t1 t2 \<in> F"   and "eql t2 t3 \<in> F"   and "\<psi> = eql t1 t3 \<or> \<psi> \<in> F"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "t3 \<in> trm" shows "nprv F \<psi>"
assms_1: shows "q2 \<in> io_targets M q1 io"
scc_of_def: fixes u :: "'a" shows "H.scc_of u \<equiv> {v. u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v \<and> v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> u}"
comp_arr_mkPoint: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and t :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "t \<in> Dom f" shows "local.comp f (mkPoint (local.dom f) t) = mkPoint (cod f) (Fun f t)"
heap_merge: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree" assumes "Leftist_Heap.linorder_class.heap l"   and "Leftist_Heap.linorder_class.heap r" shows "Leftist_Heap.linorder_class.heap (merge l r)"
check_FSwImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "FSwImorph check check (errMOD MOD) MOD"
assms_3: shows "summable (\<lambda>i. LINT x|M. norm (f i x))"
bl_bin_bk_oc: fixes xs :: "cell list" shows "bl_bin (xs @ [Bk, Oc]) = bl_bin xs + 2 * 2 ^ length xs"
mem_carrier: fixes x :: "'a" assumes "x \<in> B" shows "x \<in> carrier G"
ldrop_LCons: fixes n :: "enat"   and x :: "'c"   and xs :: "'c llist" shows "ldrop n (LCons x xs) = (case n of 0 \<Rightarrow> LCons x xs | eSuc n' \<Rightarrow> ldrop n' xs)"
Quotient_rel: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and r :: "'b"   and s :: "'b" assumes "Quotient R Abs Rep T" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
assms_3: assumes "ip \<noteq> ip'"   and "ip \<in> kD rt" shows "P rt ip ip'"
merkle_interfaceI: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "merge_on UNIV h bo m" shows "merkle_interface h bo m"
internal_invariant_translation: fixes Sa :: "('a, 'b) spec"   and Pa :: "'a set"   and Qa :: "'b set"   and pi :: "'c \<Rightarrow> 'b"   and Q :: "'c set"   and S :: "('d, 'c) spec"   and P :: "'d set" assumes "reach Sa \<subseteq> Pa"   and "Pa \<subseteq> obs Sa -` Qa"   and "pi -` Qa \<subseteq> Q"   and "obs S -` Q \<subseteq> P"   and "implements pi Sa S" shows "reach S \<subseteq> P"
even_power: 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (power.power one times a n) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<and> 0 < n)"
eq_sym: fixes a :: "cmsg"   and b :: "cmsg" shows "eq a b = eq b a"
k_2: shows "D \<le> 5 * delta * real (k + 1) + 15 / 2 * delta + C / 2"
carr_dsumM_mem_eq: fixes I :: "'h set"   and M :: "'h \<Rightarrow> ('i, 'a, 'j) Module_scheme"   and x :: "'h \<Rightarrow> 'i"   and y :: "'h \<Rightarrow> 'i" assumes "\<forall>i\<in>I. R module M i"   and "x \<in> carr_dsumag I M"   and "y \<in> carr_dsumag I M"   and "\<forall>j\<in>I. x j = y j" shows "x = y"
interact: fixes l :: "nat"   and U :: "nat list set" assumes "Form l U"   and "0 < l" obtains ka :: "nat" and kb :: "nat" and xs :: "nat list" and ys :: "nat list" and zs :: "nat list" where "l = ka + kb - 1" and "U = {xs, ys}" and "Form_Body ka kb xs ys zs" and "0 < kb" and "kb \<le> ka" and "ka \<le> Suc kb"
c_neq_large: shows "\<forall>\<^sub>F n in sequentially. c (n + 1) \<noteq> c n"
suffix_def: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'd" shows "suffix k x \<equiv> \<lambda>n. x (k + n)"
sorted_mid_iff: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "sorted_less (xs @ y # ys) = (sorted_less (xs @ [y]) \<and> sorted_less (y # ys))"
trancl_into_trancl: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
Goedel_Form_Classical_HBL1_rev_prv_axioms: shows "Goedel_Form_Classical_HBL1_rev_prv var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi fls prv bprv enc S P"
h_def: shows "h \<equiv> \<lambda>r. if Resumption.resumption.is_Done r then f (Resumption.resumption.result r) else g (Resumption.resumption.output r) (Resumption.resumption.resume r) r"
prems_8: shows "consts (name $$ ts) |\<subseteq>| fmdom \<Gamma> |\<union>| C"
in_map_sndD: fixes a :: "'b"   and b :: "'c"   and f :: "'d \<Rightarrow> 'c"   and xs :: "('b \<times> 'd) list" assumes "(a, b) \<in> set (map_snd f xs)" shows "\<exists>b'. (a, b') \<in> set xs"
MClkRelayArg_def: fixes marg :: "memOp" shows "MClkRelayArg marg = memcall marg"
s_var: fixes a :: "name" shows "Var a \<rightarrow> Var a"
evalHeap_cong': fixes heap :: "('d \<times> 'e) list"   and eval1 :: "'e \<Rightarrow> 'f"   and eval2 :: "'e \<Rightarrow> 'f" assumes "\<And>x. x \<in> domA heap \<Longrightarrow> eval1 (the (map_of heap x)) = eval2 (the (map_of heap x))" shows "evalHeap heap eval1 = evalHeap heap eval2"
flt_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "flt a b = (fcompare a b = Lt)"
star_n_inverse: fixes X :: "nat \<Rightarrow> 'a" shows "inverse (star_n X) = star_n (\<lambda>n. inverse (X n))"
inf_l_def: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
interval: fixes x0 :: "'c" shows "is_interval (existence_ivl0 x0)"
mk_globally_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and x :: "'a ltln" shows "w \<Turnstile>\<^sub>n mk_globally x = w \<Turnstile>\<^sub>n G\<^sub>n x"
simps_10: fixes f1 :: "'d"   and f2 :: "'d"   and f3 :: "'e \<Rightarrow> 'e ifex \<Rightarrow> 'e ifex \<Rightarrow> 'd"   and x31 :: "'e"   and x32 :: "'e ifex"   and x33 :: "'e ifex" shows "(case IF x31 x32 x33 of Trueif \<Rightarrow> f1 | Falseif \<Rightarrow> f2 | IF x xa xb \<Rightarrow> f3 x xa xb) = f3 x31 x32 x33"
lift_opts_length: fixes ty_opts :: "'a option list"   and tys :: "'a list" assumes "lift_opts ty_opts = Some tys" shows "length ty_opts = length tys"
return_returns_heap: fixes h :: "'l"   and x :: "'n"   and h' :: "'l" shows "h \<turnstile> return x \<rightarrow>\<^sub>h h' = (h = h')"
split_ivls_at_halfspace_def: fixes sctn :: "'b sctn"   and XS :: "'b set" shows "split_ivls_at_halfspace sctn XS = sets_of_coll XS \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>X. split_ivl_at_halfspace sctn X \<bind> (\<lambda>(A, B). RETURN (filter_empty_ivls (mk_coll A \<union> mk_coll B)))) (\<lambda>X X'. RETURN (X' \<union> X)))"
bigo_plus_absorb_lemma2: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(g)" shows "O(g) \<subseteq> f +o O(g)"
dim_ty_sem_DimTimes_def: fixes x :: "('e \<cdot> 'f) itself" shows "dim_ty_sem x = QD('e) \<cdot> QD('f)"
finfun_comp_const: fixes g :: "'c \<Rightarrow> 'b"   and c :: "'c" shows "g \<circ>$ (K$ c) = (K$ g c)"
similar_mat_witD_5: fixes n :: "nat"   and A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "B \<in> carrier_mat n n"
Nml_HcompD_3: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "Nml t"
wordinterval_CIDR_split1_distinct: fixes r :: "'b wordinterval"   and s :: "'b prefix_match"   and u :: "'b wordinterval" assumes "wordinterval_CIDR_split1 r = (Some s, u)" shows "wordinterval_empty (wordinterval_intersection (prefix_to_wordinterval s) u)"
Pi_pmf_map: fixes A :: "'b set"   and f :: "'d \<Rightarrow> 'c"   and dflt :: "'d"   and dflt' :: "'c"   and g :: "'b \<Rightarrow> 'd pmf" assumes "finite A"   and "f dflt = dflt'" shows "Pi_pmf A dflt' (\<lambda>x. map_pmf f (g x)) = map_pmf ((\<circ>) f) (Pi_pmf A dflt g)"
border_trans: fixes t :: "'b list"   and x :: "'b list"   and w :: "'b list" assumes "t \<le>b x"   and "x \<le>b w" shows "t \<le>b w"
gen_dfs_le_gen_dfsT: shows "gen_dfs \<le> gen_dfsT"
noStuckI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate" assumes "\<And>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<Longrightarrow> t \<noteq> Stuck" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Stuck}"
term_to_literal_cancel: fixes l :: "fterm literal" shows "term_to_literal (literal_to_term l) = l"
simps: fixes a :: "('a, 'b, 'c) formula" shows "fsafe a = ((\<exists>t1 t2. a = Geq t1 t2 \<and> dsafe t1 \<and> dsafe t2) \<or> (\<exists>args p. a = $\<phi> p args \<and> (\<forall>x. dsafe (args x))) \<or> (\<exists>p. a = ! p \<and> fsafe p) \<or> (\<exists>p q. a = (p && q) \<and> fsafe p \<and> fsafe q) \<or> (\<exists>p x. a = Exists x p \<and> fsafe p) \<or> (\<exists>aa p. a = (\<langle> aa \<rangle> p) \<and> hpsafe aa \<and> fsafe p) \<or> (\<exists>f C. a = InContext C f \<and> fsafe f))"
predictable_is_dsp: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "predict_stoch_proc F X N" shows "X \<in> stoch_procs M N"
psubst_eqv: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "psubst (eqv \<phi>1 \<phi>2) txs = eqv (psubst \<phi>1 txs) (psubst \<phi>2 txs)"
alternating_def: fixes cfg :: "('a \<times> 'b) cfg" shows "alternating cfg = almost_everywhere (MDP.MC.T cfg) (infs {cfg. \<forall>cfg'\<in>set_pmf (K_cfg cfg). fst (state cfg') = fst (state cfg)})"
Eps_Opt_eq_Some: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>x'. \<lbrakk>P x; P x'\<rbrakk> \<Longrightarrow> x' = x" shows "(Eps_Opt P = Some x) = P x"
lexext_trans_strong: fixes zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "lexext gt zs ys"   and "lexext gt ys xs" shows "lexext gt zs xs"
isom_unique: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'e" assumes "Worder D"   and "Worder E"   and "ord_isom D E f"   and "ord_isom D E g" shows "f = g"
h1: shows "\<forall>i. var \<notin> vars (if i mod 2 = 0 then Const 1 else Const (- 1))"
simplex_like_def: fixes P :: "'b set" shows "order.simplex_like P \<equiv> finite P \<and> (\<exists>f A. OrderingSetIso dual_order.greater_eq dual_order.greater dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A)"
contf: shows "continuous_on (cball a r \<inter> T) f"
Q: shows "((\<forall>nm\<in>incoming (fst (n, ns)). nm < name (fst (n, ns))) \<and> 0 < name (fst (n, ns)) \<and> (\<forall>q\<in>snd (n, ns). name q < name (fst (n, ns)) \<and> (\<forall>nm\<in>incoming q. nm < name (fst (n, ns))))) \<and> expand_name_ident (snd (n, ns))"
invc_baliR: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invc l"   and "invc2 r" shows "invc (baliR l a r)"
subsuming_states_def': fixes s :: "nat \<Rightarrow> nat option"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and ps :: "nat \<Rightarrow> nat option" shows "(s \<in> ast_problem.subsuming_states prob ps) = (s \<in> ast_problem.valid_states prob \<and> ps \<subseteq>\<^sub>m s)"
rank_conjugate_le: fixes A :: "'b mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n d" shows "vec_space.rank n (conjugate A) \<le> vec_space.rank n A"
blinfun_apply_blinfun_compose: fixes a :: "'c \<Rightarrow>\<^sub>L 'a"   and b :: "'b \<Rightarrow>\<^sub>L 'c"   and c :: "'b" shows "(a o\<^sub>L b) *\<^sub>v c = a *\<^sub>v b *\<^sub>v c"
wf_trmI: fixes T :: "('fun, 'a) 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)"
nat_pow_0: fixes x :: "'a" shows "[0] \<cdot> x = \<zero>"
psubstAll_preserves_wlsAll: fixes P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsPar P" shows "(wls s X \<longrightarrow> (\<forall>rho\<in>envsOfS P. wls s X #[rho])) \<and> (wlsAbs (xs, s') A \<longrightarrow> (\<forall>rho\<in>envsOfS P. wlsAbs (xs, s') A $[rho]))"
connected_with_inside: fixes S :: "'a set" assumes "closed S"   and "connected S" shows "connected (S \<union> inside S)"
dgrad_set_le_def: fixes d :: "'e \<Rightarrow> nat"   and S :: "'e set"   and T :: "'e set" shows "dgrad_set_le d S T = (\<forall>s\<in>S. \<exists>t\<in>T. d s \<le> d t)"
lcf_closed: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)" shows "f (deg Zp f) \<in> carrier Zp"
decompose_prod_root_unity_main: fixes p :: "complex poly"   and ks :: "nat list"   and f :: "complex poly"   and k :: "nat"   and ks' :: "nat list"   and g :: "complex poly" assumes "p = prod_root_unity ks * f"   and "decompose_prod_root_unity_main p k = (ks', g)"   and "\<And>x. cmod x = 1 \<Longrightarrow> poly f x \<noteq> 0"   and "\<And>k'. k < k' \<Longrightarrow> \<not> root_unity k' dvd p" shows "p = prod_root_unity ks' * f \<and> f = g \<and> set ks = set ks'"
ide_char': fixes a :: "'a" shows "S.ide a = (S.arr a \<and> (S.dom a = a \<or> S.cod a = a))"
ret_no_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and k :: "nat" assumes "\<pi> n = local.return" shows "\<not> n cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
B: shows "Esigma 12285 = 12285 + 14595"
ec: shows "spr_simAbs ec = SPRdet.sim_equiv_class a t"
induct: fixes x1 :: "char list \<Rightarrow> ty"   and x2 :: "com"   and P :: "(char list \<Rightarrow> ty) \<Rightarrow> com \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>\<Gamma>. P \<Gamma> SKIP"   and "\<And>\<Gamma> a x. \<Gamma> \<turnstile> a : \<Gamma> x \<Longrightarrow> P \<Gamma> (x ::= a)"   and "\<And>\<Gamma> c1 c2. \<lbrakk>\<Gamma> \<turnstile> c1; P \<Gamma> c1; \<Gamma> \<turnstile> c2; P \<Gamma> c2\<rbrakk> \<Longrightarrow> P \<Gamma> (c1;; c2)"   and "\<And>\<Gamma> b c1 c2. \<lbrakk>\<Gamma> \<turnstile> b; \<Gamma> \<turnstile> c1; P \<Gamma> c1; \<Gamma> \<turnstile> c2; P \<Gamma> c2\<rbrakk> \<Longrightarrow> P \<Gamma> (IF b THEN c1 ELSE c2)"   and "\<And>\<Gamma> b c. \<lbrakk>\<Gamma> \<turnstile> b; \<Gamma> \<turnstile> c; P \<Gamma> c\<rbrakk> \<Longrightarrow> P \<Gamma> (WHILE b DO c)" shows "P x1 x2"
pred_True: shows "pred_gpv (\<lambda>_. True) (\<lambda>_. True) = (\<lambda>_. True)"
less_supI2: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less x b" shows "less x (sup a b)"
h3: shows "distinct (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))"
sequence': fixes P :: "'b \<Rightarrow> bool"   and M :: "'b \<Rightarrow> ('c \<times> 'b) option"   and Q :: "'b \<Rightarrow> bool"   and M' :: "'b \<Rightarrow> ('d \<times> 'b) option"   and R :: "'d \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<lbrace>P\<rbrace> M \<lbrace>\<lambda>_. Q\<rbrace>"   and "\<lbrace>Q\<rbrace> M' \<lbrace>R\<rbrace>" shows "\<lbrace>P\<rbrace> M ;- M' \<lbrace>R\<rbrace>"
m: fixes x :: "'V" assumes "x \<in> C" shows "{(m x, m x)} = getRel (S_Const x) (map_graph_fn G' h)"
comp_ide_self: fixes a :: "'a arrow_of_spans_data" assumes "V.ide a" shows "a \<bullet> a = a"
inverse_arrowsI: fixes g :: "'a"   and f :: "'a" assumes "C.ide (g \<cdot>\<^sub>C f)"   and "C.ide (f \<cdot>\<^sub>C g)" shows "C.inverse_arrows f g"
runit_naturality: fixes \<mu> :: "'b" assumes "arr \<mu>" shows "\<mu> \<cdot> \<r>[local.dom \<mu>] = \<r>[cod \<mu>] \<cdot> (\<mu> \<star> src \<mu>)"
fps_one_mult_2: fixes f :: "'a fps" shows "f * 1 = f"
no_vs_change_snapshot_s_vs_is_valid_bound_i: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list"   and vs :: "'a set"   and s :: "('a, bool) fmap"   and as' :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "vars_change as vs s = []"   and "no_effectless_act as"   and "sat_precond_as s as"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as) = fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as')"   and "subseq as' as"   and "length as' \<le> problem_plan_bound (snapshot PROB (fmrestrict_set vs s))" shows "fmrestrict_set (fmdom' (exec_plan s as) - prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as) = fmrestrict_set (fmdom' (exec_plan s as) - prob_dom (snapshot PROB (fmrestrict_set vs s))) s \<and> fmrestrict_set (fmdom' (exec_plan s as') - prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as') = fmrestrict_set (fmdom' (exec_plan s as') - prob_dom (snapshot PROB (fmrestrict_set vs s))) s"
sup_state_conv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and ST :: "ty list"   and LT :: "ty err list"   and ST' :: "ty list"   and LT' :: "ty err list" shows "P \<turnstile> (ST, LT) \<le>\<^sub>i (ST', LT') = (P \<turnstile> ST [\<le>] ST' \<and> P \<turnstile> LT [\<le>\<^sub>\<top>] LT')"
synth_analz_monotone: fixes c :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "c \<in> synth (analz G)"   and "G \<subseteq> H" shows "c \<in> synth (analz H)"
that: fixes b2 :: "'a set" assumes "b = b2 \<inter> bl"   and "b2 \<in># remove1_mset bl \<B>" shows "thesis"
fa_nonzero: shows "f \<bullet> a \<noteq> \<zero>"
getIJ_def: 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" shows "getIJ getIJ \<equiv> \<forall>V Q_pos Q_neg I J. 2 \<le> card V \<longrightarrow> getIJ Q_pos Q_neg V = (I, J) \<longrightarrow> 1 \<le> card I \<and> 1 \<le> card J \<and> V = I \<union> J \<and> I \<inter> J = {}"