hcju/isabelleps huggingface dataset:
wt_Inj_elim: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and t :: "Term.term"   and U :: "tys" assumes "E,dt\<Turnstile>t\<Colon>U" shows "case t of In1 (Inl e) \<Rightarrow> \<exists>T. U = Inl T | In1 (Inr s) \<Rightarrow> U = Inl (PrimT Void) | In2 e \<Rightarrow> \<exists>T. U = Inl T | In3 e \<Rightarrow> \<exists>T. U = Inr T"
linear_span_sub: fixes A :: "'b set"   and H :: "'a set" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "linear_span R M A H \<subseteq> carrier M"
rb_spp_aux_Nil: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and ss :: "'t list"   and z :: "nat" shows "rb_spp_aux ((bs, ss, []), z) = ((bs, ss, []), z)"
balanced_optimal: fixes kdt :: "'b kdt"   and kdt' :: "'c kdt" assumes "balanced kdt"   and "KD_Tree.size_kdt kdt \<le> KD_Tree.size_kdt kdt'" shows "height kdt \<le> height kdt'"
filter_filter: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
assms_5: shows "\<forall>x\<in>set eq. case x of (d, e, f) \<Rightarrow> d * ((- aa - sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a))\<^sup>2 + e * (- aa - sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a) + f = 0"
T_append2: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and s :: "'c list"   and qs' :: "'c list"   and as' :: "(nat \<times> nat list) list" assumes "length qs = length as" shows "Move_to_Front.T s (qs @ qs') (as @ as') = Move_to_Front.T s qs as + Move_to_Front.T (Partial_Cost_Model.steps s qs as) qs' as'"
simps_4: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)"
gfp_f: shows "gfp f = (INF i. (f ^^ i) (f top))"
\<psi>_eq: shows "\<psi> = (\<lambda>\<mu>'. if \<guillemotleft>\<mu>' : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright> then \<mu>' \<cdot> \<r>[local.dom \<mu>'] \<cdot> \<l>[local.dom \<mu>' \<star> trg e\<^sub>0] \<cdot> (\<epsilon>\<^sub>1 \<star> local.dom \<mu>' \<star> \<epsilon>\<^sub>0) \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>[local.dom \<mu>', e\<^sub>0, d\<^sub>0]) \<cdot> \<a>\<^sup>-\<^sup>1[e\<^sub>1, d\<^sub>1, (local.dom \<mu>' \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> (e\<^sub>1 \<star> \<a>[d\<^sub>1, local.dom \<mu>' \<star> e\<^sub>0, d\<^sub>0]) else null)"
length_akiyama_tanigawa_step_aux: fixes m :: "nat"   and xs :: "real list" shows "length (akiyama_tanigawa_step_aux m xs) = length xs - 1"
simps_31: fixes \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (SKIP, s) = Some (SKIP, s)"
filtersD2: fixes F :: "'b set set"   and T :: "'b set set" assumes "F \<in> Filters\<^bsub>T\<^esub>" shows "\<Union> F \<subseteq> carrier\<^bsub>T\<^esub>"
eval_type_sound: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and accC :: "qtname"   and L :: "lenv"   and T :: "tys"   and A :: "assigned" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>T"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"   and "wf_prog G"   and "s0\<Colon>\<preceq>(G, L)" shows "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"
X_minus_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R" shows "X_poly_minus R a \<in> carrier (UP R)"
sum_divide_distrib: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and r :: "'c" shows "sum f A / r = (\<Sum>n\<in>A. f n / r)"
flat_ord_compat_1: fixes x :: "'b"   and y :: "'b" assumes "flat_le x y" shows "x \<le> y"
lepoll_refl: fixes A :: "'b set" shows "A \<lesssim> A"
assms_2: shows "unit_counit_adjunction C D F G \<eta> \<epsilon>'"
err_le: shows "\<bar>- e1 - e1' - e2 + inverse (aform_val e X) - alpha * aform_val e X - zeta\<bar> \<le> snd (inverse_aform' p X)"
infinite: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "infinite A" shows "prod g A = (1::'c)"
complex_cnj_3: fixes x :: "complex"   and y :: "complex" shows "cnj (x + y) = cnj x + cnj y"
assms_3: shows "\<forall>(k2, \<gamma>2)\<in>one_chain2. \<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
less_imp_le: fixes D :: "('e, 'f) Order_scheme"   and a :: "'e"   and b :: "'e" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b" shows "a \<preceq>\<^bsub>D\<^esub> b"
fds_truncate_add_strong: fixes m :: "nat"   and f :: "'a fds"   and g :: "'a fds" shows "fds_truncate m (f + g) = fds_truncate m f + fds_truncate m g"
poly_exp_add: fixes p :: "'a list"   and n :: "nat"   and d :: "nat"   and x :: "'a" shows "poly (p %^ (n + d)) x = poly (p %^ n *** p %^ d) x"
Collect_mem_eq: fixes A :: "'n set" shows "{x. x \<in> A} = A"
weak_conv_imp_bdd_ae_continuous_conv: fixes f :: "real \<Rightarrow> 'b"   and B :: "real" assumes "emeasure M {x. \<not> isCont f x} = 0"   and "\<And>x. norm (f x) \<le> B"   and "f \<in> borel_measurable borel" shows "(\<lambda>n. \<mu>.expectation n f) \<longlonglongrightarrow> M.expectation f"
subsp: fixes a :: "'b" assumes "a \<in> space M" shows "subprob_space (K a)"
intro: fixes P :: "'c \<Rightarrow> bool" assumes "simple_syntactic_and P"   and "subst_syntactic_and_axioms P" shows "subst_syntactic_and P"
VInterE1: fixes a :: "V"   and A :: "V"   and x :: "V" assumes "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A" obtains "x \<notin>\<^sub>\<circ> A"   | "a \<in>\<^sub>\<circ> x"
intros_1: fixes vs :: "e list"   and lholed :: "Lholed"   and es' :: "e list"   and es :: "e list" assumes "const_list vs"   and "lholed = LBase vs es'" shows "Lfilled 0 lholed es (vs @ es @ es')"
user_accessible_8: fixes p :: "'b \<times> word8" assumes "mmu_readable (get_acc_flag (snd p)) 8" shows "mmu_readable (get_acc_flag (snd p)) 10"
eval_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" obtains c :: "('Var, 'AExp, 'BExp) Stmt" and c'a :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E__ cb__" and "eval\<^sub>w_simple_abv (cb__, mem) (c'a__, mem')" and "c' = cxt_to_stmt E__ c'a__" and "mds' = mds"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "ca__ = c\<^sub>1'''__ ;; c\<^sub>2" and "c' = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1'''__, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem'\<rangle>"   | c :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c'a :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E (c@[mu])" and "c' = cxt_to_stmt E c'a" and "\<langle>c, update_modes mu mds, mem\<rangle> \<leadsto> \<langle>c'a, mds', mem'\<rangle>"   | b :: "'BExp" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" where "ca__ = Await b c\<^sub>1" and "ev\<^sub>B mem b" and "no_await c\<^sub>1" and "\<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto>\<^sup>+ \<langle>c', mds', mem'\<rangle>" and "is_final c'"
is_universal: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c" assumes "D.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> D.cones_map f \<pi> = \<chi>'"
infsetsum_reindex: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and f :: "'c \<Rightarrow> 'd" assumes "inj_on g A" shows "infsetsum f (g ` A) = (\<Sum>\<^sub>ax\<in>A. f (g x))"
prob0_jozsa_algo_1_is_const: assumes "prob0_fst_qubits n jozsa_algo = 1" shows "const 0 \<or> const 1"
set_card: fixes a :: "'s" assumes "S.ide a" shows "|S.set a| <o \<AA>"
differentiable_imp_continuous_on: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "f differentiable_on s" shows "continuous_on s f"
oghoare_sound_Parallel_Fault_case: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and x :: "'d"   and c' :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and P :: "('d, 'e, 'f) ann"   and f :: "'f"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and Q :: "'d set"   and A :: "'d set" assumes "\<Gamma> \<turnstile> (Parallel cs, Normal x) \<rightarrow>\<^sup>* (c', t)"   and "x \<in> pre P"   and "t = Fault f"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Parallel cs) Q, A"   and "final (c', t)" shows "f \<in> F"
leq_min_inf': fixes m :: "nat"   and n :: "nat" shows "m \<le> n \<longrightarrow> minimum (Abs_nat_int {m..n}) = m"
cond1_propagate: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and bs :: "'b list"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and b :: "'b" assumes "cond1 f bs y xs ys" shows "cond1 f (b # bs) y xs ys"
assms: shows "i1 \<notin> set (added_ids [Deliver (i, Add i2 e)] e)"
insertion_lowerPoly01: fixes p :: "'a mpoly"   and x :: "'a"   and L :: "'a list" assumes "0 \<notin> vars p" shows "insertion (nth_default (0::'a) (x # L)) p = insertion (nth_default (0::'a) L) (lowerPoly 0 1 p)"
distinct_toList: fixes xs :: "'c odlist" shows "distinct (toList xs)"
that: fixes r :: "real" assumes "0 < r"   and "ball \<xi> r \<subseteq> S"   and "open (f ` ball \<xi> r)"   and "inj_on f (ball \<xi> r)" shows "thesis"
accept\<^sub>G\<^sub>R_simp2: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and \<alpha> :: "(('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set set) set"   and w :: "nat \<Rightarrow> 'c" shows "accept\<^sub>G\<^sub>R (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>P\<in>\<alpha>. accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 P w)"
setcompr_eq_image: fixes f :: "'c \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> bool" shows "rel_ext (\<lambda>uu_. \<exists>x. uu_ = f x \<and> P x) = f ` rel_ext P"
rev: fixes xs :: "'a list" shows "prod_list (rev xs) = prod_list xs"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "C.seq g f"
eval_expression_list_no_jump: fixes Env :: "env"   and s0 :: "state"   and es :: "expr list"   and v :: "val list"   and s1 :: "state"   and j :: "jump"   and T :: "ty list" assumes "prg Env\<turnstile>s0 \<midarrow>es\<doteq>\<succ>v\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)"   and "Env,empty_dt\<Turnstile>es\<Colon>\<doteq>T"   and "wf_prog (prg Env)" shows "fst s1 \<noteq> Some (Jump j)"
nat_filtration_borel_measurable_integrable: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "integrable M f"
CD_on_range': fixes c :: "'a"   and ds :: "'b set"   and A :: "'a set" assumes "c \<in> Pdds'.CD_on ds A" shows "c \<in> A \<inter> (\<Union>da\<in>ds. Field (Pd_shuffle_to_top (insert d ds') Y da))"
shunting_var_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> - y \<le> z) = (x \<le> z \<squnion> - - y)"
assms: fixes \<ss>\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "P \<ss>\<^sub>0" shows "wp \<pi> (c \<ss>\<^sub>0) (Q \<ss>\<^sub>0) \<ss>\<^sub>0"
range: fixes k :: "natural"   and s :: "natural \<times> natural" assumes "0 < k" shows "fst (Random.range k s) < k"
is_down_def: shows "is_down \<equiv> map_fun (map_fun Word.Word Word.rep) id (\<lambda>_. LENGTH('b) \<le> LENGTH('a))"
del_list_sorted: fixes ps :: "('c \<times> 'd) list"   and a :: "'c"   and b :: "'d"   and qs :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted1 (ps @ (a, b) # qs)" shows "del_list x (ps @ (a, b) # qs) = (if x < a then del_list x ps @ (a, b) # qs else ps @ del_list x ((a, b) # qs))"
inferences_from_def: fixes \<Gamma> :: "'b inference set"   and CC :: "'b literal multiset set" shows "inference_system.inferences_from \<Gamma> CC = {\<gamma> \<in> \<Gamma>. infer_from CC \<gamma>}"
factorial_semiring_altI_aux: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "\<And>x. x \<noteq> zero \<Longrightarrow> finite {y. dvd.dvd times y x \<and> normalize y = y}"   and "\<And>x. comm_semiring_1.irreducible times one zero x \<Longrightarrow> comm_semiring_1.prime_elem times one zero x"   and "x \<noteq> zero" shows "\<exists>A. (\<forall>x. x \<in># A \<longrightarrow> comm_semiring_1.prime_elem times one zero x) \<and> normalize (comm_monoid_mult.prod_mset times one A) = normalize x"
p_zero: shows "delayed_safe_distance.other.p 0 = other.s \<delta>"
map2_elemE1: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "length xs = length ys"   and "x \<in> set xs" obtains y :: "'b" where "y \<in> set ys" and "f x y \<in> set (Term_Utils.map2 f xs ys)"
x0: fixes x :: "(real, 2) vec" assumes "x \<in> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w) ` cbox (- 1) 1" shows "x \<noteq> 0"
span_is_subset: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "span S \<subseteq> N"
coboundedI1: fixes a :: "'c"   and c :: "'c"   and b :: "'c" assumes "a \<le> c" shows "inf a b \<le> c"
real_powr_eq_powerI: fixes x :: "real"   and y :: "real"   and y' :: "nat" assumes "0 < x"   and "y = real y'" shows "x powr y = x ^ y'"
comb_planar_impl_def: fixes G :: "'b list \<times> ('b \<times> 'b) list" shows "comb_planar_impl G \<equiv> case G of (V, A) \<Rightarrow> let i = int (2 * card (sccs_verts_impl G)) - int (length (isolated_verts_impl G)) - int (length V) + int (length A) div 2 in \<exists>M\<in>set (all_maps_list G). (i - int (length (orbits_list_impl (lists_fc_succ M) A))) div 2 = 0"
num_def: shows "Nitpick.num \<equiv> fst \<circ> Nitpick.Rep_Frac"
OrdDom_shift: fixes s :: "hf"   and k :: "hf" assumes "OrdDom s"   and "Ord k" shows "OrdDom (shift s k)"
pfun_graph_inter: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "pfun_graph (f \<inter>\<^sub>p g) = pfun_graph f \<inter> pfun_graph g"
Suc: fixes dll :: "'a" shows "result (sigma_array_cost dmus dmusi dmusj dll l) = sigma_array dmus dmusi dmusj dll l \<and> cost (sigma_array_cost dmus dmusi dmusj dll l) \<le> 4 * l + 1"
LT9: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<leadsto> G) = (F \<leadsto> G)"
sublist_append_rightI: fixes xs :: "'b list"   and ss :: "'b list" shows "xs \<le>f xs \<cdot> ss"
simps_2: fixes x31 :: "'e"   and x32 :: "'f"   and y31 :: "'e"   and y32 :: "'f" shows "(Yield x31 x32 = Yield y31 y32) = (x31 = y31 \<and> x32 = y32)"
singleton_conv2: fixes a :: "'c" shows "{x. a = x} = {a}"
assms: fixes b :: "'a set" assumes "b \<in># B" shows "x \<in> b"
pre_between_def: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" shows "pre_between vs ram1 ram2 \<equiv> distinct vs \<and> ram1 \<in> set vs \<and> ram2 \<in> set vs \<and> ram1 \<noteq> ram2"
simps: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and X :: "'f set"   and gpv :: "('d, 'e, 'f) gpv" shows "pred_gpv' P Q X gpv = ((\<forall>x. x \<in> results_gpv (\<I>_uniform UNIV X) gpv \<longrightarrow> P x) \<and> (\<forall>x. x \<in> outs_gpv (\<I>_uniform UNIV X) gpv \<longrightarrow> Q x))"
refl_onPI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "\<And>a a'. r a a' \<Longrightarrow> a \<in> A \<and> a' \<in> A"   and "\<And>a. a \<in> A \<Longrightarrow> r a a" shows "refl_onP A r"
ideal_ring_multiple1: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and x :: "'c"   and r :: "'c" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "r \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> r \<in> I"
Mp_smult_1: fixes a :: "int"   and f :: "int poly" shows "Polynomial.smult (M a) f =m Polynomial.smult a f"
alw_discrCf_indis': fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream"   and t :: "'state" assumes "enabled cf \<omega>"   and "discrCf cf"   and "snd cf \<approx> t" shows "alw (holds (\<lambda>cf'. snd cf' \<approx> t)) \<omega>"
closed_except_const: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and name :: "name"   and S :: "name fset" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.closed_except frees (const name) S"
finite_gpv_induct: fixes \<I> :: "('d, 'e) \<I>"   and x :: "('f, 'd, 'e) gpv"   and P :: "('f, 'd, 'e) gpv \<Rightarrow> bool" assumes "finite_gpv \<I> x"   and "\<And>gpv. \<lbrakk>\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> finite_gpv \<I> (c input); \<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> P (c input)\<rbrakk> \<Longrightarrow> P gpv" shows "P x"
hcomplex_hRe_hIm_cancel_iff: fixes w :: "complex star"   and z :: "complex star" shows "(w = z) = (hRe w = hRe z \<and> hIm w = hIm z)"
univ_def: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set" shows "univ f X \<equiv> f (SOME x. x \<in> X)"
endo: shows "RModuleEnd (aezfun_setspan G) smult V T"
plus_perm_eq: fixes p :: "perm"   and q :: "perm" assumes "supp p \<inter> supp q = {}" shows "p + q = q + p"
cases: fixes a1 :: "cld"   and a2 :: "cl" assumes "superclass_name a1 a2" obtains dcl :: "char list" and cl :: "cl" and fds :: "fd list" and meth_defs :: "meth_def list" where "a1 = cld_def dcl cl fds meth_defs" and "a2 = cl"
p_2: shows "\<forall>i. coeff p i \<in> \<int>"
transcendental_cosh: fixes z :: "complex" assumes "algebraic z"   and "z \<noteq> 0" shows "\<not> algebraic (cosh z)"
callee_invariant_on_def: fixes callee :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf"   and I :: "'b \<Rightarrow> bool"   and \<I> :: "('d, 'e) \<I>" shows "callee_invariant_on callee I \<I> \<equiv> (\<forall>s x y s'. (y, s') \<in> set_spmf (callee s x) \<longrightarrow> I s \<longrightarrow> x \<in> outs_\<I> \<I> \<longrightarrow> I s') \<and> (\<forall>s. I s \<longrightarrow> \<I> \<turnstile>c callee s \<surd>)"
matrix_vector_mul_injective_on_rowspace: fixes A :: "((real, 'b) vec, 'a) vec"   and x :: "(real, 'b) vec"   and y :: "(real, 'b) vec" assumes "A *v x = A *v y"   and "x \<in> span (rows A)"   and "y \<in> span (rows A)" shows "x = y"
hyps_2: shows "(\<parallel>p\<parallel>\<^sup>2 = (0::'a)) = (p = 0)"
simps_4: fixes n1 :: "int"   and n2 :: "int" shows "(Num n1 < Num n2) = (n1 < n2)"
if_red_cong1: fixes e1 :: "exp"   and e1' :: "exp"   and e2 :: "exp"   and e3 :: "exp" assumes "e1 \<longrightarrow>* e1'" shows "EIf e1 e2 e3 \<longrightarrow>* EIf e1' e2 e3"
big_equal_iff_bigtheta: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(O[F](f) = O[F](g)) = (f \<in> \<Theta>[F](g))"
cons_cluster_wb: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and p :: "'b"   and C :: "'b set" assumes "stellar slices W"   and "p \<in> C"   and "with_w.is_cons_cluster (stellar.quorum_of slices W) W C" shows "p \<in> W"
Output: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P"
s2': shows "redT_upd s2 t ta2 x2' m2' s2'"
configuration_token_succeeds: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat"   and q :: "'d"   and n :: "nat"   and y :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "x \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n"   and "y \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n" shows "mojmir_def.token_succeeds \<delta> q\<^sub>0 w F x = mojmir_def.token_succeeds \<delta> q\<^sub>0 w F y"
module_hom_id: shows "Vector_Spaces.linear (*s) (*s) id"
columns_iarray_def: fixes A :: "'d iarray iarray" shows "columns_iarray A = {column_iarray i A |i. i \<in> {..<ncols_iarray A}}"
butlast_snoc: fixes xs :: "'a list"   and x :: "'a" shows "butlast (xs @ [x]) = xs"
Domainp_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Domainp (list_all2 R) = list_all (Domainp R)"
elims: fixes x :: "'b match_expr \<Rightarrow> 'b match_expr list"   and xa :: "'b rule list"   and y :: "'b rule list" assumes "normalize_rules x xa = y" obtains uu_ :: "'b match_expr \<Rightarrow> 'b match_expr list" where "x = uu_" and "xa = []" and "y = []"   | f :: "'b match_expr \<Rightarrow> 'b match_expr list" and m :: "'b match_expr" and a :: "action" and rs :: "'b rule list" where "x = f" and "xa = Rule m a # rs" and "y = map (\<lambda>m. Rule m a) (f m) @ normalize_rules f rs"
sel: fixes x2 :: "'b" shows "node.inner (Inner x2) = x2"
eq_inf_word: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a stream"   and w\<^sub>2 :: "'a stream"   and p :: "'b" assumes "transition_system_traces ex en ind"   and "traces.eq_inf ind w\<^sub>1 w\<^sub>2"   and "transition_system.run ex en w\<^sub>1 p" shows "transition_system.run ex en w\<^sub>2 p"
exec_gpv_try_gpv: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and s :: "('a, 'b) resource"   and f :: "'c \<times> ('a, 'b) resource \<Rightarrow> 'd"   and x :: "'c"   and z :: "'d" assumes "\<I> \<turnstile>g gpv \<surd>"   and "pfinite_gpv \<I> gpv"   and "lossless_resource \<I> s \<and> \<I> \<turnstile>res s \<surd>"   and "\<And>s. lossless_resource \<I> s \<and> \<I> \<turnstile>res s \<surd> \<Longrightarrow> f (x, s) = z"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; lossless_resource \<I> s \<and> \<I> \<turnstile>res s \<surd>\<rbrakk> \<Longrightarrow> lossless_spmf (run_resource s x)" shows "map_spmf f (exec_gpv run_resource (TRY gpv ELSE Done x) s) = TRY map_spmf f (exec_gpv run_resource gpv s) ELSE return_spmf z"