hcju/isabelleps huggingface dataset:
inf_wf_to_step_inf_wf: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'b"   and x :: "'c" assumes "well_founded order"   and "inf_wf r order n x" shows "\<exists>y m. r x y \<and> inf_wf r order m y"
intros_2: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and p :: "'b"   and S :: "'b set"   and p' :: "'b"   and Sl :: "'b set"   and p'' :: "'b" assumes "stellar slices W"   and "stellar.reachable_through slices W p S p'"   and "p' \<in> W"   and "Sl \<in> slices p'"   and "Sl \<subseteq> S"   and "p'' \<in> Sl" shows "stellar.reachable_through slices W p S p''"
pred_fun_def: fixes A :: "'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> bool" shows "pred_fun A B = (\<lambda>f. \<forall>x. A x \<longrightarrow> B (f x))"
IH: assumes "purgeIdle sl = []" shows "\<exists>n st. 0 < n \<and> sl = replicate n (Idle st)"
fun_upd_upd: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c"   and z :: "'c" shows "f(x := y, x := z) = f(x := z)"
distinct_29: fixes x51 :: "nat"   and x52 :: "recf"   and x53 :: "recf"   and x61 :: "nat"   and x62 :: "recf" shows "Pr x51 x52 x53 \<noteq> Mn x61 x62"
fusion_lfp_leq: fixes H :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and G :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" assumes "mono (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "dist_over_sup (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "\<And>x::'b::refinement_lattice. ((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) x \<sqsubseteq> ((H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> F) x" shows "(F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (lfp (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) \<sqsubseteq> lfp (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"
unicoherent_def: fixes U :: "'a set" shows "unicoherent U \<equiv> \<forall>S T. connected S \<and> connected T \<and> S \<union> T = U \<and> closedin (top_of_set U) S \<and> closedin (top_of_set U) T \<longrightarrow> connected (S \<inter> T)"
IH: shows "fst (interp_ops ops) = interp_ins (insertions ops)"
filtermap_linear_at_left: fixes c :: "'a"   and b :: "'a"   and x :: "'a" assumes "c \<noteq> (0::'a)" shows "filtermap (\<lambda>x. c * x + b) (at_left x) = (if (0::'a) < c then at_left (c * x + b) else at_right (c * x + b))"
RObject_ext_def: fixes nothing :: "unit"   and more :: "'m" shows "\<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons RObject_ext_Tuple_Iso nothing more"
univalent_char: fixes R :: "('b \<times> 'c) set" shows "univalent R = (R\<inverse> O R \<subseteq> Id)"
assms_1: shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y"
tight_aux: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "\<exists>a b. a < b \<and> (\<forall>n. 1 - \<epsilon> < Mn.prob n {a<..b})"
sgn_div_norm: fixes sgn :: "'d \<Rightarrow> 'd"   and norm :: "'d \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" assumes "class.sgn_div_norm sgn norm scaleR" shows "sgn x = scaleR (inverse (norm x)) x"
add_mono_thms_linordered_semiring_1: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i \<le> j \<and> k \<le> l" shows "i + k \<le> j + l"
ide_char': fixes a :: "'a \<times> 'a" shows "VoV.ide a = (VoV.arr a \<and> (VoV.dom a = a \<or> VoV.cod a = a))"
ca: fixes x :: "'a"   and s :: "'s" assumes "x \<in> supp (p s)" shows "bd_cts (wp (a x))"
assms: shows "take DIM('a) (xs @ replicate (DIM('a) - length xs) 0) = take DIM('a) (ys @ replicate (DIM('a) - length ys) 0)"
fminus_finter_distrib2: fixes A :: "'c fset"   and B :: "'c fset"   and C :: "'c fset" shows "A |-| B |\<inter>| C = A |\<inter>| C |-| (B |\<inter>| C)"
lp_times: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "lpp (p * q) = lpp p + lpp q"
pivot_positions_form: fixes A :: "'b mat" assumes "row_echelon_form A"   and "dim_row A \<le> dim_col A" shows "\<forall>i<length (pivot_positions A). fst (pivot_positions A ! i) = i"
assms_1: shows "v\<^sub>1 \<in> carrier_vec n"
comp2_def: fixes h :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e" shows "h o2 (f,g) \<equiv> \<lambda>x y. h (f x) (g y)"
module_hom_uminus: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "module scale" shows "module_hom scale scale uminus"
simps_11: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and l2 :: "value" shows "local.less_aexp_aux (Plus e1 e2) (L l2) = False"
slp_of_fa_rev_bin_def: fixes Binop :: "floatarith \<Rightarrow> floatarith \<Rightarrow> floatarith"   and a :: "floatarith"   and b :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and slpl :: "nat"   and M2 :: "(floatarith, nat) mapping"   and slp2 :: "floatarith list"   and slpl2 :: "nat" shows "slp_of_fa_rev_bin Binop a b M slp slpl M2 slp2 slpl2 = (case Mapping.lookup M (Binop a b) of None \<Rightarrow> (Mapping.update (Binop a b) slpl2 M2, Binop (Var (slp_indexl_lookup slpl2 M2 a)) (Var (slp_indexl_lookup slpl2 M2 b)) # slp2, Suc slpl2) | Some i \<Rightarrow> (Mapping.update (Binop a b) slpl M, Var (slp_indexl slpl i) # slp, Suc slpl))"
\<Delta>_simps_1: shows "seq (\<pi> \<cdot> \<mu> \<star> r\<^sub>0) \<rho>"
dot_cross: fixes w :: "(real, 3) vec"   and x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "w \<times> x \<bullet> y \<times> z = w \<bullet> y * (x \<bullet> z) - w \<bullet> z * (x \<bullet> y)"
SPN_rs_leg0_eq: shows "SPN_rs.leg0 = \<lbrakk>\<lbrakk>tab\<^sub>0 (r \<star> s)\<rbrakk>\<rbrakk>"
lm055: fixes bids :: "'d \<times> 'e set \<Rightarrow> 'f"   and N :: "'d set"   and G :: "'e set"   and a :: "('d \<times> 'e set) set" shows "runiq (summedBidVectorRel bids N G || a)"
hcomp_reassoc_1: fixes \<tau> :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.arr \<tau>"   and "B.arr \<mu>"   and "B.arr \<nu>"   and "src\<^sub>B \<tau> = trg\<^sub>B \<mu>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "(\<tau> \<star>\<^sub>B \<mu>) \<star>\<^sub>B \<nu> = B.\<a>' (B.cod \<tau>) (B.cod \<mu>) (B.cod \<nu>) \<cdot>\<^sub>B (\<tau> \<star>\<^sub>B \<mu> \<star>\<^sub>B \<nu>) \<cdot>\<^sub>B \<a>\<^sub>B[B.dom \<tau>, B.dom \<mu>, B.dom \<nu>]"
filter_M_reads: fixes xs :: "'l list"   and P :: "'l \<Rightarrow> ('m, 'n, bool) prog"   and h :: "'m"   and h' :: "'m"   and S :: "('m \<Rightarrow> 'm \<Rightarrow> bool) set" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h'"   and "\<And>x. x \<in> set xs \<Longrightarrow> reads S (P x) h h'"   and "\<forall>P\<in>S. reflp P \<and> transp P" shows "reads S (filter_M P xs) h h'"
is_monomial_monomial_ordered: fixes p :: "'t \<Rightarrow>\<^sub>0 'd" assumes "is_monomial p" obtains c :: "'d" and v :: "'t" where "c \<noteq> (0::'d)" and "lc p = c" and "lt p = v" and "p = monomial c v"
Sup_fold_sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A" shows "Sup A = Finite_Set.fold sup bot A"
Class_eq: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E" shows "Class a = Class b"
cong_mod_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b (modulo c a) a = unique_euclidean_semiring.cong modulo b c a"
le_sup_iff: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "less_eq (sup x y) z = (less_eq x z \<and> less_eq y z)"
pmf_of_set_def: fixes S :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "local.pmf_of_set S \<equiv> embed_pmf (\<lambda>x. indicat_real S x / real (card S))"
\<Psi>_simp: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "\<Psi> yx = \<Psi>o yx"
Proc_CFG_sourcelabel_less_num_nodes: fixes prog :: "cmd"   and l :: "nat"   and et :: "p_edge_kind"   and n' :: "label" assumes "prog \<turnstile> Label l -et\<rightarrow>\<^sub>p n'" shows "l < #:prog"
compliant_stateful_ACS_no_side_effects: shows "\<forall>E\<subseteq>backflows (flows_state \<T>). \<forall>F\<in>get_offending_flows (get_ACS M) \<lparr>nodes = hosts \<T>, edges = flows_fix \<T> \<union> E\<rparr>. F \<subseteq> E"
transpD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "transp r"   and "r x y"   and "r y z" shows "r x z"
hom_one: shows "f \<one>\<^bsub>A\<^esub> = \<one>\<^bsub>B\<^esub>"
length_norm_stack: fixes xs :: "('dyn, 'ubx1, 'ubx2) unboxed list" shows "length (norm_stack xs) = length xs"
distinct_m: shows "distinct_mset (mset_set (\<V> - {x}))"
pfinite_gpv_ContD: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and out :: "'h"   and c :: "('j, 'h, 'i) rpv"   and input :: "'i" assumes "pfinite_gpv \<I> gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "\<I> \<turnstile>g gpv \<surd>" shows "pfinite_gpv \<I> (c input)"
append_cons_eq: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "xs @ y # ys = (xs @ [y]) @ ys"
less_le_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.preorder less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"
right_angle_def: fixes p :: "proj2"   and a :: "proj2"   and q :: "proj2" shows "right_angle p a q \<equiv> p \<in> S \<and> q \<in> S \<and> a \<in> hyp2 \<and> M_perp (proj2_line_through p a) (proj2_line_through a q)"
update_conv: fixes k :: "'d"   and v :: "'c"   and al :: "('d \<times> 'c) list"   and k' :: "'d" shows "map_of (AList.update k v al) k' = (map_of al(k \<mapsto> v)) k'"
has_vderiv_on_def: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and S :: "real set" shows "(f has_vderiv_on f') S = (\<forall>x\<in>S. (f has_vector_derivative f' x) (at x within S))"
z_def: fixes n :: "nat" shows "z n = (f ^^ n) z0"
rel_converter_conversep: fixes A :: "'e \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'c \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_converter A\<inverse>\<inverse> B\<inverse>\<inverse> C\<inverse>\<inverse> R\<inverse>\<inverse> = (rel_converter A B C R)\<inverse>\<inverse>"
rel_3: shows "0 \<le> int_of_uint64 x'"
certificate_valid_pubEK: fixes C :: "agent"   and EKi :: "nat"   and evs :: "event list" assumes "cert C EKi onlyEnc (priSK RCA) \<in> parts (knows Spy evs)"   and "evs \<in> set_cr" shows "EKi = pubEK C"
face_cycle_succ_inI: fixes x :: "'b"   and y :: "'b" assumes "x \<in> face_cycle_set y" shows "face_cycle_succ x \<in> face_cycle_set y"
inf_assoc: fixes x :: "'e multiset"   and y :: "'e multiset"   and z :: "'e multiset" shows "x \<inter># y \<inter># z = x \<inter># (y \<inter># z)"
prems_1: fixes f :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "(f, g) \<in> set (p # ps) \<or> (g, f) \<in> set (p # ps)"   and "h = monom_mult ((1::'b) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f)" shows "thesis"
Diff_mono: fixes A :: "'a set"   and C :: "'a set"   and D :: "'a set"   and B :: "'a set" assumes "A \<subseteq> C"   and "D \<subseteq> B" shows "A - B \<subseteq> C - D"
assms_4: shows "\<guillemotleft>h : a \<rightarrow> exp b c\<guillemotright>"
space_embed_measure: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'a" shows "space (embed_measure M f) = f ` space M"
split_face_edges_f_vs2: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and vs1 :: "nat list" assumes "pre_split_face f ram1 ram2 vs"   and "vs1 = between (vertices f) ram1 ram2"   and "vs1 \<noteq> []"   and "between (vertices f) ram2 ram1 = []" shows "\<E> f = {(ram2, ram1), (ram1, hd vs1), (last vs1, ram2)} \<union> Edges vs1"
tagged_division_ofD_4: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set"   and x :: "'c"   and K :: "'c set" assumes "s tagged_division_of i"   and "(x, K) \<in> s" shows "\<exists>a b. K = cbox a b"
cases: fixes l :: "color list"   and a :: "nat" assumes "valid_point l a" obtains i :: "nat" where "a = i__" and "i__ + 2 < length l" and "l ! i__ = R" and "l ! (i__ + 1) = R" and "l ! (i__ + 2) = R"   | i :: "nat" where "a = ia__" and "1 \<le> ia__" and "ia__ + 1 < length l" and "l ! (ia__ - 1) = R" and "l ! ia__ = R" and "l ! (ia__ + 1) = R"   | i :: "nat" where "a = ib__" and "2 \<le> ib__" and "ib__ < length l" and "l ! (ib__ - 2) = R" and "l ! (ib__ - 1) = R" and "l ! ib__ = R"   | i :: "nat" where "a = i" and "i < length l" and "l ! i = B"
ball_conj_distrib: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>A. P x \<and> Q x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"
prime_decomposition: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize" shows "times (unit_factor x) (comm_monoid_mult.prod_mset times one (factorial_semiring.prime_factorization zero times one normalize x)) = x"
simps_2: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and r :: "'b memref"   and rs :: "'b memref list" shows "sharing_consistent \<S> \<O> (r # rs) = (case r of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then A \<subseteq> dom \<S> \<union> \<O> \<and> L \<subseteq> A \<and> A \<inter> R = {} \<and> R \<subseteq> \<O> \<and> sharing_consistent (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) (\<O> \<union> A - R) rs else sharing_consistent \<S> \<O> rs | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> A \<subseteq> dom \<S> \<union> \<O> \<and> L \<subseteq> A \<and> A \<inter> R = {} \<and> R \<subseteq> \<O> \<and> sharing_consistent (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) (\<O> \<union> A - R) rs | _ \<Rightarrow> sharing_consistent \<S> \<O> rs)"
strategyproofness_1: shows "(pmf (sds R12) b + pmf (sds R12) d < pmf (sds R10) c + pmf (sds R10) a \<or> pmf (sds R12) b + (pmf (sds R12) d + pmf (sds R12) a) < pmf (sds R10) c + (pmf (sds R10) a + pmf (sds R10) d)) \<or> pmf (sds R12) b + pmf (sds R12) d = pmf (sds R10) c + pmf (sds R10) a \<and> pmf (sds R12) b + (pmf (sds R12) d + pmf (sds R12) a) = pmf (sds R10) c + (pmf (sds R10) a + pmf (sds R10) d)"
subprob_space_dens: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and \<rho> :: "nat \<Rightarrow> val" assumes "density_context V V' \<Gamma> \<delta>"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "subprob_space (dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho>)"
SigmaD2: fixes a :: "'b"   and b :: "'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "(a, b) \<in> Sigma A B" shows "b \<in> B a"
wi_hom_syms_4: fixes a :: "int" shows "word_of_int (- a) = - word_of_int a"
ground_subst: fixes tm :: "tm"   and i :: "name"   and S :: "atom set"   and t :: "tm" assumes "ground_aux tm (insert (atom i) S)"   and "ground t" shows "ground_aux (subst i t tm) S"
tC_\<Phi>_satPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "TE.satPB \<xi> (tC ` \<Phi>) = Ik.satPB \<xi> \<Phi>"
tree_type: fixes t :: "'b bro"   and h :: "nat" assumes "t \<in> T (h + 1)" shows "tree t \<in> B (h + 1) \<union> B h"
fun_left_comm: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<in> A"   and "x \<in> A"   and "z \<in> A" shows "f y (f x z) = f x (f y z)"
distinct_4: fixes x3 :: "key"   and x1 :: "agent" shows "aKey x3 \<noteq> aAgt x1"
FORWEAK_mono_rule: fixes S :: "'e set"   and d :: "'f nres"   and P :: "'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f nres"   and I :: "'e set \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'f \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "S = {} \<Longrightarrow> d \<le> SPEC P"   and "\<And>s. s \<in> S \<Longrightarrow> f s \<le> SPEC (I {s})"   and "\<And>it it' \<sigma>. \<lbrakk>I it \<sigma>; it' \<subseteq> it; it \<subseteq> S\<rbrakk> \<Longrightarrow> I it' \<sigma>"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> S; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<le> SPEC (\<lambda>f'. c \<sigma> f' \<le> SPEC (I (insert x it)))"   and "\<And>\<sigma>. I S \<sigma> \<Longrightarrow> P \<sigma>" shows "FORWEAK S d f c \<le> SPEC P"
ballots_not_zero: fixes b :: "nat"   and p :: "Proc" assumes "b \<in> Ballot p" shows "0 < b"
rev_rev_ident: fixes xs :: "'f list" shows "rev (rev xs) = xs"
ext_inject: fixes block :: "DiskBlock"   and proc :: "Proc"   and more :: "'a"   and block' :: "DiskBlock"   and proc' :: "Proc"   and more' :: "'a" shows "(\<lparr>block = block, proc = proc, \<dots> = more\<rparr> = \<lparr>block = block', proc = proc', \<dots> = more'\<rparr>) = (block = block' \<and> proc = proc' \<and> more = more')"
meet_ord_def: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "(rel_ad x \<subseteq> rel_ad y) = (rel_ad x ; rel_ad y = rel_ad x)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and negation_type :: "'b negation_type" shows "P (case negation_type of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = ((\<forall>x1. negation_type = Pos x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. negation_type = Neg x2 \<longrightarrow> P (f2 x2)))"
add_mset_eq_single: fixes b :: "'k"   and M :: "'k multiset"   and a :: "'k" shows "(add_mset b M = {#a#}) = (b = a \<and> M = {#})"
that: fixes z' :: "('a \<times> 'c) H" assumes "Q z'"   and "set_H z' \<subseteq> set_H z"   and "map_H fst z' = map_H fst z"   and "map_H snd z' = map_H snd z" shows "thesis"
bet_cop__tsp: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and X :: "'p"   and P :: "'p"   and Y :: "'p" assumes "\<not> Coplanar A B C X"   and "P \<noteq> Y"   and "Coplanar A B C P"   and "Bet X P Y" shows "A B CTSP X Y"
residualFreshChainSimp2_2: fixes xvec :: "name list"   and M :: "'d"   and B :: "('d, 'e, 'f) boundOutput" shows "xvec \<sharp>* ROut M B = (xvec \<sharp>* M \<and> xvec \<sharp>* B)"
indicator_def: fixes S :: "'f set"   and x :: "'f" shows "indicator S x = (if x \<in> S then 1::'e else (0::'e))"
simps: fixes e :: "'b DBMEntry"   and t :: "'b" shows "norm_upper e t = (if Le t \<prec> e then \<infinity> else e)"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_reader return bind ask" shows "monad return bind"
simps: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" shows "rightPrincipal a1 a2 = (\<exists>C F Fs Ps. a1 = (Ps, C) \<and> a2 = Compound F Fs \<and> C = ( \<Empt> \<Rightarrow>* Compound F Fs))"
cases: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'a \<times> 'c"   and a2 :: "'b \<times> 'd" assumes "rel_prod R1 R2 a1 a2" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'d" where "a1 = (a, c)" and "a2 = (b, d)" and "R1 a b" and "R2 c d"
hcomp_in_hom\<^sub>W\<^sub>C: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "\<guillemotleft>\<nu> \<star> \<mu> : local.dom \<nu> \<star> local.dom \<mu> \<Rightarrow> cod \<nu> \<star> cod \<mu>\<guillemotright>"
poincare_line_cvec_cmat_nonzero: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" assumes "z \<noteq> vec_zero"   and "w \<noteq> vec_zero" shows "poincare_line_cvec_cmat z w \<noteq> mat_zero"
InterpEq: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and B :: "'a"   and g1 :: "'d"   and e2 :: "('a, 'b) Expression"   and g2 :: "'d" assumes "L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g1"   and "L\<lbrakk>Vx : A \<turnstile> e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g2" shows "L\<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IBool (g1 = g2)"
UNIF1_singleton_Var_left: fixes x :: "'c"   and t :: "('d, 'c) Term.term" assumes "x \<notin> vars_term t" shows "UNIF1 (subst x t) {#(Var x, t)#} {#}"
const_list_cons_last_1: fixes es :: "e list"   and e :: "e" assumes "const_list (es @ [e])" shows "const_list es"
part_equiv_refl_1: fixes y :: "'a" assumes "part_equiv R"   and "(x, y) \<in> R" shows "(x, x) \<in> R"
Standard_add: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Standard"   and "y \<in> Standard" shows "x + y \<in> Standard"
simps_2: fixes f :: "'c"   and Tl :: "'c trm list" shows "nvT (Fn f Tl) = {}"
exec_callAbrupt: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and t :: "'d"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Abrupt t" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> \<Rightarrow> Abrupt (return s t)"
Eats_Subset_I: fixes H :: "fm set"   and x :: "tm"   and z :: "tm"   and y :: "tm" assumes "H \<turnstile> x SUBS z"   and "H \<turnstile> y IN z" shows "H \<turnstile> Eats x y SUBS z"
residual_blocks_def: shows "\<B>\<^sup>R \<equiv> {#b - bl. b \<in># remove1_mset bl \<B>#}"
h1: shows "((TVarB X T1 # \<Gamma>) @ \<Delta>)\<turnstile>S2<:T2"
hpsafe_Test_simps: fixes p :: "('a, 'b, 'c) formula" shows "hpsafe (? p) = fsafe p"
Interp_imp_INTERP: fixes C :: "'b literal multiset"   and D :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "C \<le> D"   and "Interp N D \<Turnstile> C" shows "INTERP N \<Turnstile> C"
hom_mult: fixes x :: "'b poly"   and y :: "'b poly"   and a :: "'b" shows "poly (x * y) a = poly x a * poly y a"
fresh_3: fixes a :: "NominalSequents.var"   and x3 :: "char list"   and x1 :: "NominalSequents.var"   and x2 :: "form" shows "a \<sharp> x3 \<nabla> [x1].x2 = (a \<sharp> x3 \<and> a \<sharp> [x1].x2)"
lepoll_trans1: fixes A :: "'b set"   and B :: "'c set"   and C :: "'d set" assumes "A \<approx> B"   and "B \<lesssim> C" shows "A \<lesssim> C"