hcju/isabelleps huggingface dataset:
stab_s_def: fixes s :: "s"   and i :: "nat"   and j :: "nat" shows "stab_s s i j = (let stabinst = s.tab s ! i in if j < length stabinst then stabinst ! j else None)"
List_unique: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and as :: "ref list"   and bs :: "ref list" assumes "List p h as"   and "List p h bs" shows "as = bs"
par_rreq_rrep_nsqn_fresh_any_step: fixes i :: "nat" shows "opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. rreq_rrep_sn) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). anycast (msg_fresh \<sigma>) a)"
setop_def: shows "(\<diamondop>) \<equiv> if b then (\<inter>) else (-)"
abs_pos_floor: fixes b :: "int"   and Bnd :: "'b" assumes "\<bar>of_int b\<bar> \<le> Bnd" shows "b \<le> \<lfloor>Bnd\<rfloor>"
compet_def: fixes A :: "('c list \<Rightarrow> 'd) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> (nat \<times> nat list) \<times> 'd)"   and c :: "real"   and S :: "'c list set" shows "compet A c S = (\<forall>s\<in>S. \<exists>b\<ge>0. \<forall>rs. static s rs \<longrightarrow> real (Move_to_Front.T_on A s rs) \<le> c * real (Move_to_Front.T_opt s rs) + b)"
filter_mset_eq: fixes L :: "'b"   and A :: "'b multiset" shows "filter_mset ((=) L) A = replicate_mset (count A L) L"
is_lasso_prpl_pre_conv: fixes prpl :: "'Q list \<times> 'Q list" shows "is_lasso_prpl_pre prpl = (snd prpl \<noteq> [] \<and> is_lasso_pre (lasso_of_prpl prpl))"
sort_set_unique: fixes x :: "'a list"   and y :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "set x = set y"   and "sorted_wrt (<) (map f x)"   and "sorted_wrt (<) (map f y)" shows "x = y"
lcoset_el_reduce: fixes G :: "'a set"   and a :: "'a" assumes "Group G"   and "a \<in> G" shows "a +o G = G"
card_of_Un_ordLess_infinite: fixes C :: "'d set"   and A :: "'e set"   and B :: "'e set" assumes "infinite C"   and "|A| <o |C|"   and "|B| <o |C|" shows "|A \<union> B| <o |C|"
nonzero_inverse_inverse_eq: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "inverse (inverse a) = a"
that: fixes ms :: "'node list"   and m' :: "'node" assumes "g \<turnstile> m-ms\<rightarrow>m'"   and "var g v \<in> uses g m'"   and "\<forall>x\<in>set (tl ms). var g v \<notin> defs g x" shows "thesis"
i_one: fixes G :: "('e, 'f) Group_scheme" assumes "Group G" shows "\<rho>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>G\<^esub>"
simps_8: fixes V :: "'f"   and e :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (V:=e) A = \<D> e A"
fun_eq_def: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" shows "t1 \<sim> t2 \<equiv> t1 \<lesssim> t2 \<and> t2 \<lesssim> t1"
trelPS: shows "rel_preserves_barb_set TRel TWB {success}"
\<B>_def: shows "\<B> = {ball x (setdist {x} S / 2) |x. x \<in> U - S}"
sup_least: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and y :: "'b::type"   and x :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (y::'b::type) (x::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (z::'b::type) (x::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (y::'b::type) (z::'b::type)) (x::'b::type)"
fps_const_minus: fixes c :: "'a"   and d :: "'a" shows "fps_const c - fps_const d = fps_const (c - d)"
map_graph_homo_simp: fixes G :: "('d, 'e) labeled_graph"   and f :: "('e \<times> 'f) set" shows "graph_homomorphism G (map_graph f G) f = (univalent f \<and> vertices G = Domain f \<and> graph G)"
geodesic_segment_param_6: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and t :: "real" assumes "geodesic_segment_between G x y"   and "t \<in> {0..dist x y}" shows "dist x (geodesic_segment_param G x t) = t"
hyps: shows "support (postList xs) \<subseteq> lesvars xs"
hyps: fixes k :: "nat"   and thesis :: "bool" assumes "\<And>i j. \<lbrakk>i < j; j < length bs; v = ord_term_lin.max (term_of_pair (punit.lt (bs ! i), k + j)) (term_of_pair (punit.lt (bs ! j), k + i))\<rbrakk> \<Longrightarrow> thesis"   and "v \<in> set (Koszul_syz_sigs_aux bs k)" shows "thesis"
weakCongTransI: fixes P :: "ccs"   and P'' :: "ccs"   and \<alpha> :: "act"   and P''' :: "ccs"   and P' :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P''"   and "P'' \<longmapsto>\<alpha> \<prec> P'''"   and "P''' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<alpha> \<prec> P'"
greedy_path_ltl_ex: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a"   and P :: "'a llist" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v P" shows "P = LNil \<or> (\<exists>v. P = greedy_conforming_path p \<sigma> \<sigma>' v)"
RedSCallNonStatic: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> char list exp"   and D :: "char list"   and vs :: "val list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool" assumes "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),s,b\<rangle> \<rightarrow> \<langle>THROW IncompatibleClassChangeError,s,False\<rangle>"
split_mod: fixes P :: "nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" shows "P (m mod n) = ((n = 0 \<longrightarrow> P m) \<and> (n \<noteq> 0 \<longrightarrow> (\<forall>i j. j < n \<longrightarrow> m = n * i + j \<longrightarrow> P j)))"
tlookup_some: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and k :: "int"   and x :: "'b" shows "sortedTree h t \<and> tlookup h k t = Some x \<longrightarrow> x \<in> setOf t \<and> h x = k"
fact3: shows "spmf (connect \<A> (RES (lazy_channel_insec' \<oplus>\<^sub>O lazy_channel_send' \<oplus>\<^sub>O lazy_channel_recv') (False, Void, None, Map.empty))) True = spmf (connect \<A> (RES (local.lazy_channel_insec \<oplus>\<^sub>O lazy_channel_send \<oplus>\<^sub>O local.lazy_channel_recv) (Void, None, Map.empty))) True"
iso_unit_2: shows "\<guillemotleft>\<iota> : b \<star> b \<Rightarrow>\<^sub>S b\<guillemotright>"
\<delta>_def: shows "\<delta> \<equiv> max 0 (1 - \<alpha>)"
is_Less_iff: fixes a :: "atom" shows "is_Less a = (\<exists>i j. a = Less i j)"
alpha1_def: shows "alpha1 \<equiv> $pc1 = #a \<and> #0 < $sem \<and> pc1$ = #b \<and> sem$ = $sem - #1 \<and> Unchanged (x, y, pc2)"
scale_scale_spmf: fixes r :: "real"   and r' :: "real"   and p :: "'a spmf" shows "scale_spmf r (scale_spmf r' p) = scale_spmf (r * max 0 (min (inverse_class.inverse (weight_spmf p)) r')) p"
assms_2: shows "splits_at \<delta> i \<alpha> N \<beta>"
set_annul_result: fixes b :: "bool"   and s :: "'b sparc_state" shows "snd (set_annul b s) = False"
square_free_m_def: fixes f :: "int poly" shows "square_free_m f = (Mp f \<noteq> Mp 0 \<and> (\<forall>g. degree_m g \<noteq> 0 \<longrightarrow> \<not> g * g dvdm f))"
simps_2: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "FWNormalisationCore.sort (x # xs) l = FWNormalisationCore.insort x (FWNormalisationCore.sort xs l) l"
0: shows "Cring_Poly.compose R f g = Cring_Poly.compose R (Cring_Poly.truncate R f) g \<oplus>\<^bsub>P\<^esub> Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g"
hd_Suc_eq_Eps: fixes n :: "nat" shows "hd (F (Suc n)) = Eps (\<Phi> (F n))"
is_sup_related: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "is_sup x y y"
is_partial_magma: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "partial_magma (span_vertical_category.vcomp C)"
PMS_Crypt_sessionK_not_spied: fixes Na :: "nat"   and Nb :: "nat"   and PMS :: "nat"   and NA :: "nat"   and NB :: "nat"   and role :: "role"   and Y :: "msg"   and evs :: "event list" assumes "Crypt (sessionK ((Na, Nb, PRF (PMS, NA, NB)), role)) Y \<in> parts (knows Spy evs)"   and "evs \<in> tls" shows "Nonce PMS \<in> parts (knows Spy evs)"
distinct_1: fixes x11 :: "name"   and x12 :: "name" shows "x11[x12] \<noteq> \<tau>"
fls_integral_add: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_integral (f + g) = fls_integral f + fls_integral g"
A_B_step: fixes a :: "'a"   and b :: "'a"   and a' :: "'b" assumes "A a b"   and "sim1 a a'"   and "PA a"   and "PB a'" shows "\<exists>b'. B a' b' \<and> sim1 b b'"
assms: fixes r :: "real" shows "\<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)"
fAB: fixes x :: "'a" shows "f x = fA x + fB x"
problem_plan_bound_S_bound_2nd_step: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and vs :: "'a set"   and lss :: "('a, bool) fmap list"   and s :: "('a, bool) fmap"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "finite PROB"   and "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"   and "set lss = valid_states (prob_proj PROB vs)"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "\<exists>as'. exec_plan s as' = exec_plan s as \<and> subseq as' as \<and> length as' \<le> S vs lss PROB (fmrestrict_set vs s)"
keyfree_KeyE: fixes K :: "nat"   and P :: "bool" assumes "Key K \<in> keyfree" shows "P"
nhds_imp_cauchy_filter: fixes F :: "'a filter"   and x :: "'a" assumes "F \<le> nhds x" shows "cauchy_filter F"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.arr f"   and "S.dom f = a"   and "S.cod f = b" shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>"
AppendInv_def: shows "AppendInv \<equiv> \<lambda>s ind. \<forall>i<length (M s). ind \<le> i \<longrightarrow> i \<in> Reach (E s) \<longrightarrow> M s ! i = Black"
incidence_system_axioms: shows "incidence_system \<V> \<B>"
consistent_length_tcsigs_iff_exe_consistent_length_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "exe_ars_conds arss" shows "consistent_length_tcsigs (translate_ars arss) = exe_consistent_length_tcsigs arss"
functor_axioms: shows "functor \<F>C.CC.comp (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C"
Nash_Williams_WW: fixes M :: "nat set"   and h :: "nat list \<Rightarrow> nat"   and A :: "nat list set" assumes "infinite M"   and "h ` {l \<in> A. list.set l \<subseteq> M} \<subseteq> {..<2}"   and "thin A"   and "A \<subseteq> WW" obtains i :: "nat" and N :: "nat set" where "i < 2" and "infinite N" and "N \<subseteq> M" and "h ` {l \<in> A. list.set l \<subseteq> N} \<subseteq> {i}"
poincare_collinear3_poincare_lines_equal_general: fixes a :: "complex_homo"   and a1 :: "complex_homo"   and a2 :: "complex_homo" assumes "poincare_collinear {a, a1, a2}"   and "a \<noteq> a1"   and "a \<noteq> a2"   and "a \<noteq> inversion a1"   and "a \<noteq> inversion a2" shows "poincare_line a a1 = poincare_line a a2"
suminf_eq_zero_iff: fixes f :: "nat \<Rightarrow> 'c" assumes "summable f"   and "\<And>n. (0::'c) \<le> f n" shows "(suminf f = (0::'c)) = (\<forall>n. f n = (0::'c))"
indep_inI: fixes X :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>"   and "indep X" shows "indep_in \<E> X"
valid_shift_aux_2: fixes ts :: "nat" assumes "ts \<in> set (linearize tss)" shows "\<not> enat ts + right (args_ivl args) < enat nt"
Inf_par_distrib: fixes D :: "'a::refinement_lattice set"   and c :: "'a::refinement_lattice" assumes "(D::'a::refinement_lattice set) \<noteq> {}" shows "\<Sqinter> (D::'a::refinement_lattice set) \<parallel> (c::'a::refinement_lattice) = (\<Sqinter>d::'a::refinement_lattice\<in>D. d \<parallel> c)"
pos_stock_borel_adapted: shows "borel_adapt_stoch_proc F (prices Mkt pos_stock)"
tiTable_SampleT_i1_buffer: fixes ack :: "nat \<Rightarrow> GatewayStatus list"   and t :: "nat"   and req :: "nat \<Rightarrow> reqType list"   and d :: "nat"   and lose :: "nat \<Rightarrow> bool list"   and buffer :: "nat \<Rightarrow> ECall_Info list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and st :: "nat \<Rightarrow> GatewayStatus"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" assumes "ack t = [connection_ok]"   and "req (Suc t) = [send]"   and "\<forall>k\<le>Suc d. lose (t + k) = [False]"   and "buffer t = inf_last_ti dt t"   and "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st"   and "fin_inf_append [init_state] st (Suc t) = connection_ok" shows "i1 (Suc t) = inf_last_ti dt t"
r: shows "map_poly to_fract q = map_poly to_fract p * r"
fidx_eq: fixes x :: "'b"   and i :: "nat"   and s :: "'b fset" assumes "x |\<in>|\<^bsub>i\<^esub> s" shows "fidx s x = i"
multisetLeTrans: fixes r :: "('b \<times> 'b) set"   and x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" assumes "trans r"   and "multiset_le x y r"   and "multiset_le y z r" shows "multiset_le x z r"
MakeCat: fixes C :: "('d, 'e, 'f) Category_scheme" assumes "Category_axioms C" shows "Category (MakeCat C)"
rep_eq: shows "blinfun_apply no_real_inner.fstzero = (\<lambda>(x, y). (x, 0))"
assms_2: shows "pivot_fun (row_echelon (Macaulay_mat ps)) f (dim_col (row_echelon (Macaulay_mat ps)))"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c" assumes "natural_transformation A B F G \<tau>"   and "natural_isomorphism_axioms A B \<tau>" shows "natural_isomorphism A B F G \<tau>"
times_inf: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linorder_stone_relation_algebra_expansion one times uminus less_eq less bot top inf sup conv" shows "times x y = inf x y"
rel_G_pos_distr_cond_eq: fixes tytok :: "('t \<times> 'u \<times> 'v \<times> 'w \<times> 'x \<times> 'y \<times> 'z) itself" shows "rel_G_pos_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) tytok"
finite_Init_rhs: fixes CS :: "'b list set set"   and X :: "'b list set" assumes "finite CS" shows "finite (Init_rhs CS X)"
indets_homogenize_subset: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "indets (homogenize x p) \<subseteq> insert x (indets p)"
assms_2: shows "\<forall>i. (seq i, seq (Suc i)) \<in> R'"
inj: shows "\<forall>a b. PB (f a) \<and> PA b \<and> f a = f b \<longrightarrow> a = b"
p_in: shows "poly_of_pair p \<in> dgrad_sig_set' (length fs) dgrad"
noStuck_startD': fixes t :: "('d, 'e) xstate"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate" assumes "t \<noteq> Stuck"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "s \<noteq> Stuck"
min_avg: fixes T :: "nat \<Rightarrow> nat" shows "m * T (min\<^sub>k T m) \<le> sum T {1..m}"
inf_compl_bot_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (inf y (uminus x)) = bot"
cases: fixes a :: "'a list"   and A :: "'a set" assumes "a \<in> lists A" obtains "a = []"   | aa :: "'a" and l :: "'a list" where "a = aa # l" and "aa \<in> A" and "l \<in> lists A"
bounded_Int: fixes S :: "'b set"   and T :: "'b set" assumes "bounded S \<or> bounded T" shows "bounded (S \<inter> T)"
leq_trans: fixes leq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and lub :: "'g set \<Rightarrow> 'g"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "partial_function_definitions leq lub"   and "leq x y"   and "leq y z" shows "leq x z"
cblinfun_compose_add_left: fixes a :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and b :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and c :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "a + b o\<^sub>C\<^sub>L c = (a o\<^sub>C\<^sub>L c) + (b o\<^sub>C\<^sub>L c)"
Rep_inject: fixes x :: "'b word"   and y :: "'b word" shows "(unat x = unat y) = (x = y)"
Inr_in_sum_iff: fixes b :: "hf"   and A :: "hf"   and B :: "hf" shows "(HF.Inr b \<^bold>\<in> A + B) = (b \<^bold>\<in> B)"
simps_2: fixes h :: "'b heap"   and hs :: "'b heap list" shows "pass\<^sub>2 (h # hs) = merge h (pass\<^sub>2 hs)"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> b" shows "c \<sqsubseteq> a"
simps: fixes q :: "'a ltl_prop_equiv_quotient" shows "\<MM>.q\<^sub>\<R> q = (if q = q\<^sub>0\<^sub>M \<psi> then Some 0 else None)"
\<eta>: shows "\<guillemotleft>\<eta> : src\<^sub>C f \<Rightarrow>\<^sub>C g \<star>\<^sub>C f\<guillemotright> \<and> F \<eta> = \<Phi> (g, f) \<cdot>\<^sub>D \<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f))"
assms_1: shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
same_length_eq_append: fixes as :: "'b list"   and bs :: "'b list"   and cs :: "'b list"   and ds :: "'b list" assumes "length as = length bs"   and "as @ cs = bs @ ds" shows "as = bs"
RBT_set_def: shows "RBT_set = Collect \<circ> RBT_Set2.member"
nprv_eql_symE0: fixes t1 :: "'trm"   and t2 :: "'trm"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "eql t1 t2 \<in> F"   and "nprv (insert (eql t2 t1) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "nprv F \<psi>"
sub_emb: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "sub s t" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "f\<cdot>(f\<cdot>x\<cdot>y)\<cdot>z = f\<cdot>x\<cdot>(f\<cdot>y\<cdot>z)"
subst_uminus: fixes \<sigma> :: "'b \<Rightarrow> 'c"   and x :: "('a, 'c) uexpr" shows "\<sigma> \<dagger> - x = - (\<sigma> \<dagger> x)"
extremum_unique: fixes a :: "'k" shows "(\<top> \<le> a) = (a = \<top>)"
inv_one: shows "inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> \<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> = \<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub>"
eventually_ceiling_eq: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<notin> \<int>" shows "\<forall>\<^sub>F x in F. \<lceil>f x\<rceil> = \<lceil>l\<rceil>"
tllist_all2_is_TNilD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys" shows "is_TNil xs = is_TNil ys"
profile_rprD: fixes A :: "'a set"   and Is :: "'b set"   and P :: "'b \<Rightarrow> ('a \<times> 'a) set"   and i :: "'b" assumes "profile A Is P"   and "i \<in> Is" shows "rpr A (P i)"
pol_coeff_pre: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (Suc n, f)" shows "pol_coeff S (n, f)"
set_annul_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True s1)) \<and> t2 = snd (fst (set_annul True s2))" shows "low_equal t1 t2"
simps_4: fixes i :: "32 word" shows "typeof (Intg i) = \<lfloor>Integer\<rfloor>"
Imp_cong: fixes H :: "fm set"   and A :: "fm"   and A' :: "fm"   and B :: "fm"   and B' :: "fm" assumes "H \<turnstile> A IFF A'"   and "H \<turnstile> B IFF B'" shows "H \<turnstile> (A IMP B) IFF A' IMP B'"
real_Icc_closed_segment: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "{a..b} = closed_segment a b"
assms_2: assumes "\<And>x. x \<in>\<^sub>\<circ> r \<Longrightarrow> \<exists>a b. x = [a, b]\<^sub>\<circ>" shows "P"
bits_ident: fixes n :: "nat"   and a :: "'a" shows "push_bit n (drop_bit n a) + take_bit n a = a"
g0: fixes x :: "'b" assumes "x \<in> vs1.span B"   and "g x = (0::'c)" shows "x = (0::'b)"
RWRNext_enabled: fixes rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan" shows "\<turnstile> rs!p = #NotAResult \<and> (\<forall>l. MemInv mm l) \<and> Enabled (Read ch mm rs p \<or> (\<exists>l. Write ch mm rs p l)) \<longrightarrow> Enabled (<RNext ch mm rs p>_(rtrner ch!p, rs!p))"
fps_compose_divide_distrib: fixes g :: "'a fps"   and f :: "'a fps"   and h :: "'a fps" assumes "g dvd f"   and "fps_nth h 0 = (0::'a)"   and "g oo h \<noteq> 0" shows "f / g oo h = (f oo h) / (g oo h)"
Inf_many_def: fixes P :: "'b \<Rightarrow> bool" shows "Inf_many P = infinite {x. P x}"
ZFUnionNatInRangeExplode: fixes f :: "nat \<Rightarrow> ZF set" assumes "\<And>n. f n \<in> range explode" shows "\<Union> (range f) \<in> range explode"
unit_factor_mod_ring_def: fixes x :: "'a mod_ring" shows "unit_factor x = x"
assms_1: shows "y \<in> Fvars (rawpsubst \<phi> txs)"
hyps_1: shows "\<guillemotleft>\<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>r\<guillemotright>"
R_def: shows "R \<equiv> \<lambda>N n. EM_remainder (2 * N + 1) (\<lambda>x. - fact (2 * N + 2) / x ^ (2 * N + 3)) (int n)"
homeomorphic_box_UNIV: fixes a :: "'a"   and b :: "'a" assumes "box a b \<noteq> {}" shows "box a b homeomorphic UNIV"
runit_in_hom_2: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>B.runit f : f \<star>\<^sub>B src\<^sub>B f \<rightarrow>\<^sub>B f\<guillemotright>"
that: fixes l :: "'a \<Rightarrow> 'b"   and u :: "'a \<Rightarrow> 'b" assumes "finite I"   and "\<And>x. \<lbrakk>a < x; x < b; x \<notin> I\<rbrakk> \<Longrightarrow> isCont f x"   and "\<And>x. \<lbrakk>a < x; x \<le> b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> l x) (at_left x)"   and "\<And>x. \<lbrakk>a \<le> x; x < b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> u x) (at_right x)"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b; x \<notin> I\<rbrakk> \<Longrightarrow> f x = l x"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b; x \<notin> I\<rbrakk> \<Longrightarrow> f x = u x" shows "thesis"
ww'_1: shows "- 1 \<le> w \<and> w * exp w = x"
IO: fixes x :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "IO x c \<in> set_spmf (the_gpv gpv)" shows "x \<in> outs_gpv \<I> gpv"
eNorm_uminus: fixes N :: "'c quasinorm"   and x :: "'c" shows "eNorm N (- x) = eNorm N x"
closed_segment_commute: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b = closed_segment b a"
assms_2: shows "4 / (1 + 2 ^ j) \<le> d * ln 2"
RL: fixes c :: "com"   and s\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HTset_r (\<lambda>c' s'. ((c', s'), c, s\<^sub>0) \<in> R) \<pi> \<Theta>"   and "(P, c, Q) \<in> \<Theta>"   and "P s\<^sub>0" shows "wp \<pi> c (Q s\<^sub>0) s\<^sub>0"
elims_2: fixes x :: "'b TESL_atomic" assumes "positive_atom x" obtains uu_ :: "clock" and uv_ :: "'b tag_const" and uw_ :: "clock" where "x = uu_ sporadic uv_ on uw_"
sel_rews_1: shows "shead\<cdot>\<bottom> = \<bottom>"