hcju/isabelleps huggingface dataset:
reachable_snapshot_inv_def: fixes m :: "'e" shows "mut_m.reachable_snapshot_inv m = (\<lambda>s. \<forall>x. mut_m.reachable m x s \<longrightarrow> in_snapshot x s)"
max_ip_addr_def: shows "max_ip_addr \<equiv> word_of_nat (2 ^ LENGTH('a) - 1)"
iptables_bigstep_induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "\<And>t. P [] t t"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Accept\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalAllow)"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Drop \<or> a = Reject\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalDeny)"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Log \<or> a = Empty\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a. \<not> matches \<gamma> m p \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>rs X. P rs (Decision X) (Decision X)"   and "\<And>rs rs\<^sub>1 rs\<^sub>2 t t'. \<lbrakk>rs = rs\<^sub>1 @ rs\<^sub>2; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t; P rs\<^sub>1 Undecided t; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'; P rs\<^sub>2 t t'\<rbrakk> \<Longrightarrow> P rs Undecided t'"   and "\<And>m a chain rs\<^sub>1 m' rs\<^sub>2. \<lbrakk>matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2); matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided; P rs\<^sub>1 Undecided Undecided\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a chain rs t. \<lbrakk>matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t; P rs Undecided t\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided t" shows "P rs s t"
new_tv_subst: fixes n :: "nat"   and S :: "nat \<Rightarrow> typ" shows "new_tv n S = ((\<forall>m\<ge>n. S m = TVar m) \<and> (\<forall>l<n. new_tv n (S l)))"
SUP_L_P_def: fixes X :: "nat \<Rightarrow> I \<Rightarrow> 'a"   and u :: "I \<times> nat"   and i :: "I" shows "SUP_L_P X u i = (SUP v\<in>{v. (i, v) \<sqsubset> u}. X v i)"
irreducible_imp_prime_elem_gcd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "comm_semiring_1.irreducible times one zero x" shows "comm_semiring_1.prime_elem times one zero x"
all_bij: fixes f :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1" shows "bij f"
PreEnvironmentJView_axioms: shows "PreEnvironmentJView jkbp clock_jview"
it_paths_butlast: fixes "is" :: "nat list"   and t :: "'a rose_tree" assumes "is \<in> it_paths t" shows "butlast is \<in> it_paths t"
trans_FD: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q"   and "Q \<sqsubseteq>\<^sub>F\<^sub>D S" shows "P \<sqsubseteq>\<^sub>F\<^sub>D S"
induct: fixes P :: "nat \<Rightarrow> ('dyn, 'c, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr \<Rightarrow> Unboxed.type option list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "('dyn, 'c, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and a2 :: "Unboxed.type option list" assumes "\<And>uu_ d \<Sigma>. P uu_ (Ubx.instr.IPush d) \<Sigma>"   and "\<And>uv_ n \<Sigma>. P uv_ (IPushUbx1 n) \<Sigma>"   and "\<And>uw_ b \<Sigma>. P uw_ (IPushUbx2 b) \<Sigma>"   and "\<And>ux_ uy_ \<Sigma>. P ux_ Ubx.instr.IPop (uy_ # \<Sigma>)"   and "\<And>n x \<Sigma>. P n (Ubx.instr.ILoad x) (None # \<Sigma>)"   and "\<And>uz_ \<tau> x \<Sigma>. P uz_ (ILoadUbx \<tau> x) (None # \<Sigma>)"   and "\<And>va_ x \<Sigma>. P va_ (Ubx.instr.IStore x) (None # None # \<Sigma>)"   and "\<And>vb_ x \<tau> \<Sigma>. P vb_ (Ubx.instr.IStore x) (None # Some \<tau> # \<Sigma>)"   and "\<And>vc_ \<tau>\<^sub>1 x \<tau>\<^sub>2 \<Sigma>. P vc_ (IStoreUbx \<tau>\<^sub>1 x) (None # Some \<tau>\<^sub>2 # \<Sigma>)"   and "\<And>vd_ op \<Sigma>. P vd_ (Ubx.instr.IOp op) \<Sigma>"   and "\<And>ve_ opinl \<Sigma>. P ve_ (Ubx.instr.IOpInl opinl) \<Sigma>"   and "\<And>vf_ opubx \<Sigma>. P vf_ (IOpUbx opubx) \<Sigma>"   and "\<And>vg_ f \<Sigma>. P vg_ (Ubx.instr.ICall f) \<Sigma>"   and "\<And>vh_. P vh_ Ubx.instr.IPop []"   and "\<And>vh_ v. P vh_ (Ubx.instr.ILoad v) []"   and "\<And>vh_ v vc vb. P vh_ (Ubx.instr.ILoad v) (Some vc # vb)"   and "\<And>vh_ v va. P vh_ (ILoadUbx v va) []"   and "\<And>vh_ v va vd vc. P vh_ (ILoadUbx v va) (Some vd # vc)"   and "\<And>vh_ v. P vh_ (Ubx.instr.IStore v) []"   and "\<And>vh_ v vc vb. P vh_ (Ubx.instr.IStore v) (Some vc # vb)"   and "\<And>vh_ v va. P vh_ (Ubx.instr.IStore v) [va]"   and "\<And>vh_ v vb ve vd. P vh_ (Ubx.instr.IStore v) (Some vb # Some ve # vd)"   and "\<And>vh_ v va. P vh_ (IStoreUbx v va) []"   and "\<And>vh_ v va vd vc. P vh_ (IStoreUbx v va) (Some vd # vc)"   and "\<And>vh_ v va vb. P vh_ (IStoreUbx v va) [vb]"   and "\<And>vh_ v va vb ve. P vh_ (IStoreUbx v va) (vb # None # ve)"   and "\<And>vh_ v vj_. P vh_ (Ubx.instr.ICJump v) vj_" shows "P a0 a1 a2"
simps_20: fixes x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf" shows "Pr x51 x52 x53 \<noteq> S"
lexordp_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "class.linorder less_eq less"   and "ord.lexordp less xs ys"   and "ord.lexordp less ys zs" shows "ord.lexordp less xs zs"
minimal_QuadRes_neg1: obtains x :: "nat" where "0 < x" and "x \<le> (p - 1) div 2" and "[x\<^sup>2 + 1 = 0] (mod p)"
write_cpu_PSR_icc_privilege: fixes s' :: "'b sparc_state"   and n_val :: "word1"   and z_val :: "word1"   and v_val :: "word1"   and c_val :: "word1"   and s :: "'b sparc_state" assumes "s' = snd (fst (write_cpu (update_PSR_icc n_val z_val v_val c_val (cpu_reg_val PSR s)) PSR s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
exec1I: fixes c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and P :: "instr list"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list" assumes "c' = iexec (P !! i) (i, s, stk)"   and "0 \<le> i"   and "i < size P" shows "P \<turnstile> (i, s, stk) \<rightarrow> c'"
that: fixes c :: "'b" assumes "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)" shows "thesis"
hyps: fixes r :: "'a rexp" shows "Derivs s (lang r) = \<Union> (lang ` pderivs s r)"
induct: fixes P :: "('b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and a1 :: "'b list"   and a2 :: "'b list" assumes "\<And>f. P f [] []"   and "\<And>f uu_ uv_. P f (uu_ # uv_) []"   and "\<And>f uw_ ux_. P f [] (uw_ # ux_)"   and "\<And>f a as b bs. (\<And>x xa y. \<lbrakk>x = f a b; (xa, y) = x; \<not> xa; y\<rbrakk> \<Longrightarrow> P f as bs) \<Longrightarrow> P f (a # as) (b # bs)" shows "P a0 a1 a2"
IH_2: fixes va :: "nat"   and vd :: "'a list \<times> 'a list"   and x :: "'a list"   and y :: "'a list"   and ve :: "'a list" assumes "va = Time_Monad.val (length_tm (x__ # y__ # xs'))"   and "vd = Time_Monad.val (split_at_tm (va div 2) (x__ # y__ # xs'))"   and "(x, y) = vd"   and "ve = Time_Monad.val (mergesort_tm f x)" shows "real (time (mergesort_tm f y)) \<le> mergesort_recurrence (length y)"
not_less_iff_gr_or_eq: fixes ko :: "'a nat_term_order"   and x :: "'a"   and y :: "'a" shows "(\<not> ko_ntm.lt ko x y) = (ko_ntm.lt ko y x \<or> x = y)"
fun_relI: fixes A :: "('b \<times> 'c) set"   and f :: "'b \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'e"   and B :: "('d \<times> 'e) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
K'_def: shows "(K', S') = widest_spread (k\<^sub>1 # ks) ps"
bound2: fixes z :: "complex"   and u :: "real" assumes "Re z \<in> {x..y}"   and "u \<in> {exp (- 1)<..<1}" shows "cmod (f z u) \<le> g2 u"
twiddleVal_Null: fixes \<beta> :: "(Location \<times> Location) set" shows "(\<beta>, RVal Nullref, RVal Nullref) \<in> twiddleVal"
redundant_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" shows "uninst.redundant u p g \<equiv> \<exists>v\<in>uninst.allVars u p g. uninst.trivial u p g v"
init_ode_ops_def: fixes poincare :: "bool"   and c1 :: "bool"   and ode_ops :: "ode_ops" shows "init_ode_ops poincare c1 ode_ops = (ode_ops, ode_slps_of ode_ops, if poincare then Some (solve_poincare_slp ode_ops) else None, if c1 then Some (ode_slps_of (approximate_sets_ode'.var_ode_ops ode_ops)) else None)"
replace5: fixes f :: "'a"   and oldF :: "'a"   and newfs :: "'a list"   and fs :: "'a list" assumes "f \<in> set (replace oldF newfs fs)" shows "f \<in> set fs \<or> f \<in> set newfs"
scale_one: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(1::'a) *s x = x"
simps_1: fixes ts\<^sub>1 :: "'b tree list" shows "merge ts\<^sub>1 [] = ts\<^sub>1"
sel_empt: shows "\<forall>i<n. S_G (CAs ! i) = {#}"
poly_mapping_eqI: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and g :: "'a \<Rightarrow>\<^sub>0 'b" assumes "\<And>k. lookup f k = lookup g k" shows "f = g"
invalidate_kD_inv: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option" shows "kD (invalidate rt dests) = kD rt"
a_inv_op_closed1: fixes x :: "'a"   and h :: "'a" assumes "x \<in> carrier G"   and "h \<in> H" shows "\<ominus> x \<oplus> h \<oplus> x \<in> H"
ex_rotation_mapping_u_to_positive_y_axis: fixes u :: "complex_homo" assumes "u \<noteq> 0\<^sub>h"   and "u \<noteq> \<infinity>\<^sub>h" shows "\<exists>\<phi>. moebius_pt (moebius_rotation \<phi>) u \<in> positive_y_axis"
that: fixes a :: "real" assumes "A < a"   and "f \<in> \<Theta>(\<lambda>x. real x powr p * (1 + integral (\<lambda>u. g' u / u powr (p + 1)) a (real x)))" shows "thesis"
runit_simps_1: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.arr (S.runit f)"
F_4: fixes x :: "'a" shows "(SUP i. F i x) = ennreal (f x)"
oldest_token_rec: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q (Suc n) = (if semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n \<noteq> {} then Some (Min (semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n)) else None)"
start_proof_sound: fixes S :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "sound (start_proof S)"
igSubstBinpIPresIGWlsBinp_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstBinpIPresIGWlsBinp MOD \<equiv> \<forall>ys y Y delta binp. igWls MOD (asSort ys) Y \<and> igWlsBinp MOD delta binp \<longrightarrow> igWlsBinp MOD delta (igSubstBinp MOD ys Y y binp)"
second_countable_metric_dense_subset: shows "\<exists>A. countable A \<and> closure A = UNIV"
is_regular_spair_alt_spp: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "is_regular_spair p q = is_regular_spair_spp (spp_of p) (spp_of q)"
less_numeral_extra_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
inf_absorb: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> - y) \<sqinter> - x = - x"
FOREACH_rule_insert: fixes S :: "'g set"   and I :: "'g set \<Rightarrow> 'h \<Rightarrow> bool"   and s :: "'h"   and P :: "'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h nres" assumes "finite S"   and "I {} s"   and "\<And>s. I S s \<Longrightarrow> P s"   and "\<And>T x s. \<lbrakk>T \<subseteq> S; I T s; x \<in> S; x \<notin> T\<rbrakk> \<Longrightarrow> f x s \<le> SPEC (I (insert x T))" shows "FOREACH S f s \<le> SPEC P"
distinct_25: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'a"   and x51 :: "nat"   and x52 :: "nat" shows "x31\<not>x32:arrive(x33) \<noteq> disconnect(x51, x52)"
WTrtCons: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and es :: "char list exp list"   and Ts :: "ty list" assumes "P,E,h,sh \<turnstile> e : T"   and "P,E,h,sh \<turnstile> es [:] Ts" shows "P,E,h,sh \<turnstile> e # es [:] T # Ts"
stl: fixes s :: "'a stream" shows "f (stl s) = stl (f s)"
cases: fixes a1 :: "'loc"   and a2 :: "'loc"   and a3 :: "('loc \<times> 'sum \<times> 'loc) list" assumes "path a1 a2 a3" obtains l1 :: "'loc" and l2 :: "'loc" where "a1 = l1b__" and "a2 = l2b__" and "a3 = []" and "l1b__ = l2b__"   | l1 :: "'loc" and l2 :: "'loc" and xs :: "('loc \<times> 'sum \<times> 'loc) list" and lbl :: "'sum" and l3 :: "'loc" where "a1 = l1" and "a2 = l3" and "a3 = xs @ [(l2, lbl, l3)]" and "path l1 l2 xs" and "lbl \<in>\<^sub>A summary l2 l3"
cone_smult: fixes a :: "'a"   and Vs :: "'a vec set"   and x :: "'a vec" assumes "(0::'a) \<le> a"   and "Vs \<subseteq> carrier_vec n"   and "x \<in> cone Vs" shows "a \<cdot>\<^sub>v x \<in> cone Vs"
simps_4: fixes x :: "'b"   and y :: "'b"   and p :: "'c"   and xs :: "('b, 'c) Combinators list" shows "net_list_aux (AllowPortFromTo x y p # xs) = x # y # net_list_aux xs"
SDA0_to_SMDA: shows "S DA0 \<cdot> \<eta>0 = S_M S M DA"
rel_witness_fun1: fixes A :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and A' :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "rel_fun (A OO A') B f g"   and "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'" shows "rel_fun A (\<lambda>x (x', y). x = x' \<and> B x' y) f (rel_witness_fun A A' (f, g))"
mono_id: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.order less_eq less" shows "mono id"
\<phi>_\<psi>: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<phi> y (\<psi> x g) = g"
le_iff_add: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq a b = (\<exists>c. b = plus a c)"
setc_def_to_intro: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "A = {x. P x}"   and "P x" shows "x \<in> A"
HEndPhase1_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4b s q" shows "HInv4b s' q"
steps_reds: fixes prog :: "cmd"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and l :: "nat"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and l' :: "nat" assumes "prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto>* \<langle>c',s',l'\<rangle>" shows "\<langle>c,s\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>"
solution_uminus_right: fixes D :: "nat"   and x :: "'a"   and y :: "'a" assumes "pell D" shows "pell.solution D (x, - y) = pell.solution D (x, y)"
map_converter_id_move_left: fixes f :: "'i \<Rightarrow> 'm"   and g :: "'n \<Rightarrow> 'j"   and f' :: "'m \<Rightarrow> 'k"   and g' :: "'l \<Rightarrow> 'n" shows "map_converter f g f' g' 1\<^sub>C = map_converter (f' \<circ> f) (g \<circ> g') id id 1\<^sub>C"
lift_subst_dummy: fixes s :: "db"   and i :: "nat"   and dummy :: "db" assumes "\<not> freedb s i" shows "lift (s<dummy/i>) i = s"
perfect_hiding_ind_cpa_def: fixes \<A> :: "('pub_input \<times> 'witness \<Rightarrow> (('challenge \<times> 'challenge) \<times> 'b) spmf) \<times> ('msg \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "abstract_com.perfect_hiding_ind_cpa \<A> \<equiv> abstract_com.hiding_advantage_ind_cpa \<A> = 0"
components_are_iso: fixes a :: "'a \<times> 'a \<times> 'a" assumes "VVV.ide a" shows "local.iso \<a>[fst a, fst (snd a), snd (snd a)]"
rep_cod: fixes f :: "'c local.term set" assumes "arr f" shows "rep (cod f) = \<^bold>\<parallel>COD f\<^bold>\<parallel>"
preserves_dom: fixes f :: "'c" assumes "C.arr f" shows "D.dom (F f) = F (C.dom f)"
llist_all2_cons_def: fixes sg :: "'sg"   and sh :: "'sh" shows "local.llist_all2_cons sg sh = llist_all2 P (lunstream g sg) (lunstream h sh)"
in_atlas_order_le: fixes l :: "enat"   and c :: "('b, 'f) chart" assumes "l \<le> k"   and "c \<in> dest.atlas" shows "c \<in> c_manifold.atlas charts2 l"
intros_3: fixes F :: "\<Pi>\<^sub>2"   and y :: "\<kappa>" shows "SimpleExOrEnc (exe2 F y)"
inverse_functors_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" shows "inverse_functors A B F G \<equiv> (category A \<and> category B) \<and> functor B A F \<and> functor A B G \<and> inverse_functors_axioms A B F G"
llcp_lappend_same: fixes xs :: "'a llist"   and ys :: "'a llist"   and zs :: "'a llist" shows "llcp (xs @\<^sub>l ys) (xs @\<^sub>l zs) = llength xs + llcp ys zs"
map2_zip: fixes f :: "'d \<rightarrow> 'c"   and xs :: "['b]"   and ys :: "['d]" shows "map\<cdot>(\<Lambda> \<langle>x, y\<rangle>. \<langle>x, f\<cdot>y\<rangle>)\<cdot>(zip\<cdot>xs\<cdot>ys) = zip\<cdot>xs\<cdot>(map\<cdot>f\<cdot>ys)"
simps_10: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and x3 :: "'b" shows "(case Match x3 of No_match \<Rightarrow> f1 | Match_type_error \<Rightarrow> f2 | Match x \<Rightarrow> f3 x) = f3 x3"
rec_inseparable_symmetric: fixes A :: "nat set"   and B :: "nat set" assumes "rec_inseparable A B" shows "rec_inseparable B A"
A_splay_ub2: fixes t :: "'b tree"   and x :: "'b" assumes "bst t"   and "x \<in> set_tree t" shows "A_splay x t \<le> 3 * (\<phi> t - 1) + 1"
convert_aNeg: fixes var :: "nat"   and A :: "atom"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and A' :: "atomUni" assumes "convert_atom var A (xs' @ x # xs) = Some A'"   and "length xs' = var" shows "aEval (aNeg A) (xs' @ x # xs) = aEvalUni (aNegUni A') x"
poly_add_length_eq: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "polynomial K p1"   and "polynomial K p2"   and "length p1 \<noteq> length p2" shows "length (poly_add p1 p2) = max (length p1) (length p2)"
le_opt_antisym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b option"   and y :: "'b option" assumes "order r"   and "x \<sqsubseteq>\<^bsub>Opt.le r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>Opt.le r\<^esub> x" shows "x = y"
WT1_expr_locks: fixes P :: "'a J1_prog"   and E :: "ty list"   and e :: "'a expr1"   and T :: "ty" assumes "P,E \<turnstile>1 e :: T" shows "expr_locks e = (\<lambda>a. 0)"
inc': fixes n :: "nat" shows "c \<cdot>\<^sub>m (f n - f 0) \<le>\<^sub>L c \<cdot>\<^sub>m (f (Suc n) - f 0)"
closure_of_openin_Int_superset: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S \<and> S \<subseteq> X closure_of T" shows "X closure_of (S \<inter> T) = X closure_of S"
ex_orthogonal_current': assumes "\<And>fa. \<lbrakk>current (quotient_web \<Gamma> f) fa; wave (quotient_web \<Gamma> f) fa; loose (quotient_web (quotient_web \<Gamma> f) fa)\<rbrakk> \<Longrightarrow> linkable (quotient_web (quotient_web \<Gamma> f) fa)" shows "\<exists>fa S. web_flow (quotient_web \<Gamma> f) fa \<and> separating (quotient_web \<Gamma> f) S \<and> orthogonal_current (quotient_web \<Gamma> f) fa S"
nilTrans_2: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and M :: "'d"   and xvec :: "name list"   and N :: "'d"   and P :: "('d, 'e, 'f) psi"   and K :: "'d"   and yvec :: "name list"   and N' :: "'d"   and P' :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> (M\<lparr>\<lambda>*xvec N\<rparr>.P) (K\<lparr>\<nu>*yvec\<rparr>\<langle>N'\<rangle> \<prec> P')" shows "False"
mfinalE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x2"   and ln :: "'a \<Rightarrow>f nat" assumes "r2.mfinal s"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" obtains "final2 x" and "ln = no_wait_locks" and "wset s t = None"
hom_power: fixes hom :: "'b \<Rightarrow> 'c"   and x :: "'b"   and n :: "nat" assumes "monoid_mult_hom hom" shows "hom (x ^ n) = hom x ^ n"
flatf_fixp_transfer: fixes tr :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'c"   and B :: "('d \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'c"   and fp' :: "'e \<Rightarrow> 'b"   and B' :: "('e \<Rightarrow> 'b) \<Rightarrow> 'e \<Rightarrow> 'b"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and x' :: "'e" assumes "\<And>x'. tr b x'"   and "flatf_mono b B"   and "fp' = B' fp'"   and "P x x'"   and "\<And>f f' x x'. \<lbrakk>\<And>x x'. P x x' \<Longrightarrow> tr (f x) (f' x'); P x x'; fp' = f'\<rbrakk> \<Longrightarrow> tr (B f x) (B' f' x')" shows "tr (flatf_fp b B x) (fp' x')"
norm_less: fixes x :: "nat"   and y :: "nat" assumes "x < y" shows "x + 1 \<le> y"
field_differentiable_at_Ln: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "Ln field_differentiable at z"
eval\<^sub>1_final: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "final e'"