hcju/isabelleps huggingface dataset:
Tgs_authenticates_A: fixes authK :: "nat"   and A :: "agent"   and T2 :: "nat"   and evs :: "event list"   and Ta :: "nat" assumes "Crypt authK \<lbrace>Agent A, Number T2\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "\<exists>B. Says A Tgs \<lbrace>Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"
assms_3: fixes i :: "nat" assumes "i < dim_vec b" shows "(A *\<^sub>v x) $ i \<le> b $ i"
rep_eq: fixes x :: "'a \<Rightarrow> PreSimplyTyped.type option"   and xa :: "'a trm" shows "infer x xa = ptrm_infer_type x (rep_trm xa)"
cut_less_eq_imp_f_join_eq: fixes A :: "nat set"   and xs :: "'b list"   and B :: "nat set" assumes "A \<down>< length xs = B \<down>< length xs" shows "xs \<Join>\<^sub> A = xs \<Join>\<^sub> B"
diff_single_trivial: fixes x :: "'k"   and M :: "'k multiset" assumes "x \<notin># M" shows "M - {#x#} = M"
Done_bind_gpv: fixes a :: "'d"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "Generative_Probabilistic_Value.Done a \<bind> f = f a"
perpendicular_intersects: fixes H1 :: "circline"   and H2 :: "circline" assumes "is_poincare_line H1"   and "is_poincare_line H2"   and "perpendicular H1 H2" shows "\<exists>z. z \<in> unit_disc \<and> z \<in> circline_set H1 \<inter> circline_set H2"
rref_right_mul: fixes A :: "'b mat" assumes "row_echelon_form A"   and "dim_row A \<le> dim_col A" shows "take_cols A (map snd (pivot_positions A)) * take_rows A [0..<length (pivot_positions A)] = A"
simps_2: fixes n :: "nat"   and x :: "dB" shows "unlift_vars n (Pure x) = liftn n x 0"
tabulate2_def: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'd" shows "tabulate2 m n f = IArray.of_fun (\<lambda>i. IArray.of_fun (f i) n) m"
verts_H: shows "verts H = verts G \<union> {w}"
local_peak_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and p :: "('a \<times> ('b \<times> 'a) list) \<times> 'a \<times> ('b \<times> 'a) list" shows "local_peak ars p = (let (\<tau>, \<sigma>) = p in peak ars p \<and> length (snd \<tau>) = 1 \<and> length (snd \<sigma>) = 1)"
simps_1: fixes x :: "'a" shows "max_list_non_empty [x] = x"
sheaf_spec_is_presheaf: shows "presheaf_of_rings pi.spectrum pi.is_zariski_open pi.sheaf_spec pi.sheaf_spec_morphisms pi.\<O>b pi.add_sheaf_spec pi.mult_sheaf_spec pi.zero_sheaf_spec pi.one_sheaf_spec"
prems_4: shows "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"
assms: shows "\<forall>x\<in>set xs. qfree (f x)"
preserves_reflects_arr: fixes f :: "'c" shows "C.arr (\<epsilon> f) = C.arr f"
PRJ\<^sub>0_def: shows "Maps.PRJ\<^sub>0 \<equiv> \<lambda>K H. if Maps.cospan K H then \<lbrakk>\<lbrakk>prj\<^sub>0 (Maps.REP H) (Maps.REP K)\<rbrakk>\<rbrakk> else Maps.null"
some_geodesic_segment_between_def: shows "some_geodesic_segment_between = (SOME f. \<forall>x y S. f x S y = f y S x \<and> (if \<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S then geodesic_segment_between (f x S y) x y \<and> f x S y \<subseteq> S else f x S y = {x, y}))"
boundedI: fixes a :: "'b upred"   and b :: "'b upred"   and c :: "'b upred" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> a" shows "(b \<and> c) \<sqsubseteq> a"
coprime_crossproduct: fixes a :: "'a"   and d :: "'a"   and b :: "'a"   and c :: "'a" assumes "coprime a d"   and "coprime b c" shows "(normalize a * normalize c = normalize b * normalize d) = (normalize a = normalize b \<and> normalize c = normalize d)"
InternalSets_def: shows "InternalSets = {X. \<exists>As. X = *sn* As}"
arctan_lower_34_def: shows "arctan_lower_34 \<equiv> \<lambda>x. pi / 2 - (64 + 735 * x\<^sup>2 + 945 * x ^ 4) / (15 * x * (15 + 70 * x\<^sup>2 + 63 * x ^ 4))"
infdist_triangle_abs: fixes x :: "'b"   and A :: "'b set"   and y :: "'b" shows "\<bar>infdist x A - infdist y A\<bar> \<le> dist x y"
embP_def: fixes P :: "(char list \<Rightarrow> int option) \<Rightarrow> bool" shows "embP P = (\<lambda>(s, n). P s \<and> n = 0)"
UN_constant: fixes c :: "'a set"   and A :: "'b set" shows "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
RS_simps_14: shows "Maps.Dom S\<^sub>0 = src (tab\<^sub>0 s)"
is_2sati_correct: fixes cnfi :: "(lit \<times> lit) list" shows "is_2sati cnfi = is_2sat (cnf_\<alpha> cnfi)"
swapidseq_cases: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b" shows "swapidseq n p = (n = 0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id \<circ> q \<and> swapidseq m q \<and> a \<noteq> b))"
simps_45: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> While x61 x62"
Un_ac_3: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B = B \<union> A"
converse_rtrancl_into_rtrancl: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>*"
assms_2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t I"
bit_eqI: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "\<And>n. power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<Longrightarrow> bit a n = bit b n" shows "a = b"
lesseq_vecD: fixes w :: "'b vec"   and n :: "nat"   and v :: "'b vec"   and i :: "nat" assumes "w \<in> carrier_vec n"   and "v \<le> w"   and "i < n" shows "v $ i \<le> w $ i"
left_commute: fixes b :: "'a multiset"   and a :: "'a multiset"   and c :: "'a multiset" shows "b \<union># (a \<union># c) = a \<union># (b \<union># c)"
affine_hull_open: fixes S :: "'b set" assumes "open S"   and "S \<noteq> {}" shows "affine hull S = UNIV"
lincomb_closed: fixes S :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "S \<subseteq> carrier_vec n" shows "lincomb a S \<in> carrier_vec n"
of_nat_dvd_of_nat: fixes a :: "nat"   and b :: "nat" assumes "a dvd b" shows "of_nat a dvd of_nat b"
Iio_eq_empty_iff: fixes n :: "'a" shows "({..<n} = {}) = (n = bot)"
coeff_add: fixes p :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and q :: "'b mpoly" shows "coeff p m + coeff q m = coeff (p + q) m"
oghoare_seq_imp_ann_matches: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "'e set"   and a :: "('e, 'd, 'f) ann"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P a c Q, A" shows "ann_matches \<Gamma> \<Theta> a c"
exec_cons: fixes \<delta> :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and s :: "'a"   and r :: "'b \<times> 'c"   and rs :: "('b \<times> 'c) list" shows "pre_RDR.exec \<delta> s (Cons r rs) = \<delta> (pre_RDR.exec \<delta> s rs) r"
2: fixes x :: "brick list"   and xs :: "wall"   and y :: "brick list" shows "x * xs \<otimes> basic y = (if codomain_block y = 0 then (x \<otimes> y) * xs else (x \<otimes> y) * (xs \<otimes> basic (make_vert_block (nat (codomain_block y)))))"
octo_mult_scaleR_right: fixes x :: "octo"   and r :: "real"   and y :: "octo" shows "x * r *\<^sub>R y = r *\<^sub>R (x * y)"
osafe_OProd_simps: fixes ODE1 :: "('a, 'b) ODE"   and ODE2 :: "('a, 'b) ODE" shows "osafe (OProd ODE1 ODE2) = (osafe ODE1 \<and> osafe ODE2 \<and> ODE_dom ODE1 \<inter> ODE_dom ODE2 = {})"
MN_eq_nextl: shows "MyhillNerode language eq_nextl"
less_funE: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "less_fun f g" obtains k :: "'b" where "f k < g k" and "\<And>k'. k' < k \<Longrightarrow> f k' = g k'"
fv: shows "f ` V = T \<inter> {x. g x \<in> V}"
card_Suc_Diff1: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "Suc (card (A\<setminus>{x})) = card A"
perfect_imp_surjective_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "perfect_map X Y f" shows "f ` topspace X = topspace Y"
sup_idem: 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" assumes "class.semilattice_sup sup less_eq less" shows "sup x x = x"
Init_JN: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program" shows "Init (JOIN I F) = (\<Inter>i\<in>I. Init (F i))"
avl_update_1: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and x :: "'c"   and y :: "'d" assumes "avl t" shows "avl (update x y t)"
prod_orD_def: fixes p1 :: "'d \<Rightarrow> 'f decision option"   and p2 :: "'e \<Rightarrow> 'g decision option" shows "p1 \<Otimes>\<^sub>\<or>\<^sub>D p2 = (\<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>)"
\<phi>: shows "\<guillemotleft>\<phi> : w \<rightarrow>\<^sub>B a\<guillemotright> \<and> B.iso \<phi> \<and> \<guillemotleft>\<phi> : w \<Rightarrow> a\<guillemotright> \<and> local.iso \<phi>"
rep_eq: fixes x :: "'b"   and xa :: "'b poly" shows "coeff (smult x xa) = (\<lambda>n. x * coeff xa n)"
trace_eq_lazy: assumes "0 < \<eta>" shows "(local.valid_insecQ <+> nlists UNIV (id' \<eta>) <+> UNIV) \<turnstile>\<^sub>R RES (local.lazy_channel_insec \<oplus>\<^sub>O lazy_channel_send \<oplus>\<^sub>O local.lazy_channel_recv) (Void, None, Map.empty) \<approx> RES (\<dagger>\<dagger>insec_channel.insec_oracle \<oplus>\<^sub>O local.rorc_channel_send \<oplus>\<^sub>O local.rorc_channel_recv) ((False, ()), Map.empty, Void)"
solvable_imp_solvable_img: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd"   and K :: "'b set" assumes "group_hom G H h"   and "solvable_seq G K" shows "solvable_seq H (h ` K)"
order_uminus: fixes x :: "'b"   and p :: "'b poly" shows "order x (- p) = order x p"
simps_2: fixes \<xi> :: "nat \<Rightarrow> 'b set" shows "\<xi> \<Turnstile>\<^sub>n false\<^sub>n = False"
a_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.a p q = card (QR.A p q)"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "slast\<cdot>(x :# xs) = (case xs of [::] \<Rightarrow> x | y :# ys \<Rightarrow> slast\<cdot>xs)"
cases: fixes a1 :: "'b literal multiset"   and a2 :: "'b literal multiset"   and a3 :: "'b literal multiset" assumes "unord_resolve a1 a2 a3" obtains C :: "'b literal multiset" and n :: "nat" and A :: "'b" and D :: "'b literal multiset" where "a1 = C + replicate_mset (Suc n) (Pos A)" and "a2 = add_mset (Neg A) D" and "a3 = C + D"
num_params_tm_norm': fixes t :: "taylor_model"   and X :: "nat"   and prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list" assumes "num_params (tm_poly t) \<le> X" shows "num_params (tm_poly (tm_norm' prec ord I a t)) \<le> X"
vectorize_zero: shows "local.punit.vectorize_poly 0 = 0"
3: fixes c :: "instr"   and s :: "item list"   and cs :: "instr list" assumes "\<And>l. c = Label l \<Longrightarrow> isFresh l s" shows "unwind (c # cs) s = unwind cs s"
assms_6: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
simps: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "Optimal_BST2 w" shows "Optimal_BST2.opt_bst_wpl2 w i j = (if j < i then (\<langle>\<rangle>, 0) else if i = j then (\<langle>\<langle>\<rangle>, i, \<langle>\<rangle>\<rangle>, w i i) else let l = root (fst (Optimal_BST2.opt_bst_wpl2 w i (j - 1))); r = root (fst (Optimal_BST2.opt_bst_wpl2 w (i + 1) j)) in argmin snd (map (\<lambda>k. let (tl, wl) = Optimal_BST2.opt_bst_wpl2 w i (k - 1); (tr, wr) = Optimal_BST2.opt_bst_wpl2 w (k + 1) j in (\<langle>tl, k, tr\<rangle>, wl + wr + w i j)) [l..r]))"
antisym: fixes b :: "'b"   and a :: "'b" assumes "(b::'b) \<le> (a::'b)"   and "(a::'b) \<le> (b::'b)" shows "(a::'b) = (b::'b)"
assms_4: shows "\<phi> (inv (SOME h. h \<in> R1)) x = \<phi> (inv (SOME h. h \<in> R2)) x"
"1_2": shows "col \<in> {a..b} \<rightarrow> {..<1}"
prems: fixes c' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate" assumes "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> (c', s')" shows "\<Gamma>\<turnstile>c' \<down> s'"
Ints_cases: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and q :: "'d" assumes "class.ring_1 one times plus zero minus uminus"   and "q \<in> ring_1.Ints one plus zero minus" obtains z :: "int" where "q = ring_1.of_int one plus zero minus z"
root_order_tree\<^sub>i: fixes k :: "nat"   and t :: "'a up\<^sub>i" shows "root_order_up\<^sub>i (Suc k) t = root_order (Suc k) (tree\<^sub>i t)"
Linear_order_in_diff_Id: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "Linear_order r"   and "a \<in> Field r"   and "b \<in> Field r" shows "((a, b) \<in> r) = ((b, a) \<notin> r - Id)"
induct: fixes P :: "nat rexp \<Rightarrow> nat rexp \<Rightarrow> bool"   and a0 :: "nat rexp"   and a1 :: "nat rexp" assumes "\<And>uu_. P Zero uu_"   and "P One Zero"   and "\<And>v. P (Atom v) Zero"   and "\<And>v va. P (Plus v va) Zero"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v. P (Star v) Zero"   and "\<And>v. P (rexp.Not v) Zero"   and "\<And>v va. P (rexp.Inter v va) Zero"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Plus v va)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P One (rexp.Not v)"   and "\<And>v va. P One (rexp.Inter v va)"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Atom v) (Atom va)"   and "\<And>v va vb. P (Atom v) (Plus va vb)"   and "\<And>v va vb. P (Atom v) (Times va vb)"   and "\<And>v va. P (Atom v) (Star va)"   and "\<And>v va. P (Atom v) (rexp.Not va)"   and "\<And>v va vb. P (Atom v) (rexp.Inter va vb)"   and "\<And>v va. P (Plus v va) One"   and "\<And>v va vb. P (Plus v va) (Atom vb)"   and "\<And>v va vb vc. P (Plus v va) (Plus vb vc)"   and "\<And>v va vb vc. P (Plus v va) (Times vb vc)"   and "\<And>v va vb. P (Plus v va) (Star vb)"   and "\<And>v va vb. P (Plus v va) (rexp.Not vb)"   and "\<And>v va vb vc. P (Plus v va) (rexp.Inter vb vc)"   and "\<And>v va. P (Times v va) One"   and "\<And>v va vb. P (Times v va) (Atom vb)"   and "\<And>v va vb vc. P (Times v va) (Plus vb vc)"   and "\<And>v va vb vc. P (Times v va) (Times vb vc)"   and "\<And>v va vb. P (Times v va) (Star vb)"   and "\<And>v va vb. P (Times v va) (rexp.Not vb)"   and "\<And>v va vb vc. P (Times v va) (rexp.Inter vb vc)"   and "\<And>v. P (Star v) One"   and "\<And>v va. P (Star v) (Atom va)"   and "\<And>v va vb. P (Star v) (Plus va vb)"   and "\<And>v va vb. P (Star v) (Times va vb)"   and "\<And>v va. P (Star v) (Star va)"   and "\<And>v va. P (Star v) (rexp.Not va)"   and "\<And>v va vb. P (Star v) (rexp.Inter va vb)"   and "\<And>v. P (rexp.Not v) One"   and "\<And>v va. P (rexp.Not v) (Atom va)"   and "\<And>v va vb. P (rexp.Not v) (Plus va vb)"   and "\<And>v va vb. P (rexp.Not v) (Times va vb)"   and "\<And>v va. P (rexp.Not v) (Star va)"   and "\<And>v va. P (rexp.Not v) (rexp.Not va)"   and "\<And>v va vb. P (rexp.Not v) (rexp.Inter va vb)"   and "\<And>v va. P (rexp.Inter v va) One"   and "\<And>v va vb. P (rexp.Inter v va) (Atom vb)"   and "\<And>v va vb vc. P (rexp.Inter v va) (Plus vb vc)"   and "\<And>v va vb vc. P (rexp.Inter v va) (Times vb vc)"   and "\<And>v va vb. P (rexp.Inter v va) (Star vb)"   and "\<And>v va vb. P (rexp.Inter v va) (rexp.Not vb)"   and "\<And>v va vb vc. P (rexp.Inter v va) (rexp.Inter vb vc)" shows "P a0 a1"
b3: fixes x :: "'b"   and y :: "'a" shows "\<langle>cnj a *\<^sub>C (A\<^sup>\<dagger>) x, y\<rangle> = \<langle>x, a *\<^sub>C A y\<rangle>"
map_bit_interval_eq: fixes w :: "'b word"   and n :: "nat" shows "map (bit w) [0..<n] = takefill False n (rev (to_bl w))"
zmod_trivial_iff: fixes i :: "int"   and k :: "int" shows "(i mod k = i) = (k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i)"
proj_arcs_H_def: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and x :: "'d" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM" shows "subdiv1_contr.proj_arcs_H rev_G rev_H uv uw vw x \<equiv> if x = uw \<or> x = rev_H vw then uv else if x = vw \<or> x = rev_H uw then rev_G uv else x"
hoare_augment_context: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and p :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and \<Theta>' :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P p Q,A"   and "\<Theta> \<subseteq> \<Theta>'" shows "\<Gamma>,\<Theta>'\<turnstile>\<^bsub>/F \<^esub>P p Q,A"
simps_1: fixes N :: "nat" shows "ladder_shift_n N [] = []"
ltln_expand_Until: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> or\<^sub>n (\<phi> and\<^sub>n X\<^sub>n (\<phi> U\<^sub>n \<psi>))"
GodExistenceImpliesNecExistence: shows "\<lfloor>(\<lambda>w. (\<exists>x. \<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w) \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v))) w)\<rfloor>"
mat_of_row_mult_append_rows: fixes y1 :: "'a vec"   and nr1 :: "nat"   and y2 :: "'a vec"   and nr2 :: "nat"   and A1 :: "'a mat"   and nc :: "nat"   and A2 :: "'a mat" assumes "y1 \<in> carrier_vec nr1"   and "y2 \<in> carrier_vec nr2"   and "A1 \<in> carrier_mat nr1 nc"   and "A2 \<in> carrier_mat nr2 nc" shows "mat_of_row (y1 @\<^sub>v y2) * (A1 @\<^sub>r A2) = mat_of_row y1 * A1 + mat_of_row y2 * A2"
horizontal_chop_non_empty: fixes v :: "view" shows "0 < \<parallel>ext v\<parallel> \<longrightarrow> (\<exists>u w. v=u\<parallel>w \<and> 0 < \<parallel>ext u\<parallel> \<and> 0 < \<parallel>ext w\<parallel>)"
LstSeq_append_app2: fixes s1 :: "hf"   and k1 :: "hf"   and y1 :: "hf"   and s2 :: "hf"   and k2 :: "hf"   and y2 :: "hf"   and l :: "hf"   and j :: "hf" assumes "LstSeq s1 k1 y1"   and "LstSeq s2 k2 y2"   and "l = succ k1 @+ j" shows "app (seq_append (succ k1) s1 s2) l = app s2 j"
nths_vars1'_comp: shows "nths dims0 (- vars2') = dims1"
non_empty_range: fixes y :: "'index \<times> 'trap" shows "y \<in> set_spmf I \<longrightarrow> range (fst y) \<noteq> {}"
line_extension_mem_iff: fixes u :: "'a"   and K :: "'a set"   and a :: "'a"   and E :: "'a set" shows "(u \<in> line_extension K a E) = (\<exists>k\<in>K. \<exists>v\<in>E. u = k \<otimes> a \<oplus> v)"
path_in_verts: fixes l :: "nat list"   and v :: "nat" assumes "path_entry (g_E G) l v" shows "set l \<subseteq> set (g_V G)"
e2: shows "pdevs_val e (fst (trunc_bound_pdevs p (add_pdevs (snd (fst X)) (snd (fst Y))))) = pdevs_val e (add_pdevs (snd (fst X)) (snd (fst Y))) + e2"
gInduct_gConsIndif_recAll_isSurjAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) Recursion.model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.presWlsAll Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"   and "FixSyn.presCons Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"   and "FixSyn.gConsIndif Delta MOD"   and "FixSyn.gInduct Delta MOD" shows "FixSyn.isSurjAll Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"
Linear_order_under_aboveS_Field: fixes r :: "('c \<times> 'c) set"   and a :: "'c" assumes "Linear_order r"   and "a \<in> Field r" shows "Field r = under r a \<union> aboveS r a"
tickFree_implies_front_tickFree_single: fixes t :: "'c event list"   and a :: "'c event" assumes "tickFree t" shows "front_tickFree (t @ [a])"
acquire_locks_conv: fixes L :: "('c \<times> nat) option"   and t :: "'c"   and n :: "nat" shows "acquire_locks L t n = (case L of None \<Rightarrow> case n of 0 \<Rightarrow> None | Suc m \<Rightarrow> \<lfloor>(t, m)\<rfloor> | \<lfloor>(t', m)\<rfloor> \<Rightarrow> \<lfloor>(t', n + m)\<rfloor>)"
call_result: fixes \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'a match_expr"   and p :: "'b"   and \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and chain :: "char list"   and rs :: "'a rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"
simps_2: fixes uu :: "'b"   and uv :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "set_kdt (Node uu uv l r) = set_kdt l \<union> set_kdt r"
aint_S: shows "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
content_cball_pos: fixes r :: "real"   and c :: "'c" assumes "0 < r" shows "0 < content (cball c r)"
LeftDerivation_take_derive: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and n :: "nat" assumes "LeftDerivation a D b" shows "LeftDerivation a (take n D) (Derive a (take n D))"
add_rel_class_of: fixes U :: "'a set"   and x :: "'b"   and V :: "'a set"   and y :: "'b"   and W :: "'a set" assumes "(U, x) \<in> Sigma I \<FF>1"   and "(V, y) \<in> Sigma I \<FF>1"   and "W \<in> I"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "stk1.add_rel (stk1.class_of U x) (stk1.class_of V y) = stk1.class_of W (add_str1 W (\<rho>1 U W x) (\<rho>1 V W y))"
convex_closure: fixes S :: "'b set" assumes "convex S" shows "convex (closure S)"
path_image_rectpath_subset_cbox: fixes a :: "complex"   and b :: "complex" assumes "Re a \<le> Re b"   and "Im a \<le> Im b" shows "path_image (rectpath a b) \<subseteq> cbox a b"
sort_key_conv_fold: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "inj_on f (set xs)" shows "sort_key f xs = fold (insort_key f) xs []"
matrix_inv_idempotent: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "invertible A" shows "matrix_inv (matrix_inv A) = A"
mat_of_cols_cols: fixes A :: "'a mat" shows "mat_of_cols (dim_row A) (cols A) = A"
ExitBlockAbrupt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and abrs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> ([], (nrms, abrs) # css, Abrupt s) \<rightarrow> (abrs, css, Normal s)"
absorb_iff1: fixes b :: "'b multiset"   and a :: "'b multiset" shows "(b \<subseteq># a) = (a \<union># b = a)"
unique_factorization_m_alt_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "q.unique_factorization_m f cfs = (q.factorization_m f cfs \<and> (\<forall>dgs. q.factorization_m f dgs \<longrightarrow> q.Mf dgs = q.Mf cfs))"
hyps: assumes "abc_list_crsp (lm1 @ 0 \<up> n) lm2" shows "abc_list_crsp lm1 lm2"