hcju/isabelleps huggingface dataset:
prime_imp_not_one: fixes x :: "'a" assumes "ASSUMPTION (prime x)" shows "x \<noteq> (1::'a)"
results_gpv_catch_gpv: fixes \<I> :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, 'f) gpv" shows "results_gpv \<I> (catch_gpv gpv) = Some ` results_gpv \<I> gpv \<union> (if colossless_gpv \<I> gpv then {} else {None})"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "S.ide (S.UP a)"
subset_preserves_o: fixes v :: "'d \<Rightarrow> 'e"   and w :: "'e \<Rightarrow> 'f" shows "preserves v \<subseteq> preserves (w \<circ> v)"
sp: shows "splay x rb = \<langle>lu, u, ru\<rangle>"
callee_invariant': fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and I :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>"   and y :: "'f"   and s' :: "'d"   and s :: "'d"   and x :: "'e" assumes "callee_invariant_on callee I \<I>"   and "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "x \<in> outs_\<I> \<I>" shows "I s' \<and> y \<in> responses_\<I> \<I> x"
reflp_tllist_all2: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "reflp R"   and "reflp Q" shows "reflp (tllist_all2 R Q)"
\<tau>move1CastRed: fixes P :: "'a prog"   and h :: "'heap"   and U :: "ty"   and v :: "'addr val" shows "\<tau>move1 P h (Cast U (Val v))"
inv_comp: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "CC.iso f"   and "CC.iso g"   and "CC.seq g f" shows "CC.inv (CC.comp g f) = CC.comp (CC.inv f) (CC.inv g)"
sep_disj_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.pre_sep_algebra plus zero sep_disj" shows "sep_disj x y = sep_disj y x"
in_P2R_Im_singl: fixes y :: "'c"   and \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'d" shows "(y \<in> P2R \<phi> `` {x}) = \<phi> x y"
fin: shows "finite {n. c n \<noteq> (0::'a)}"
assms: shows "run_one_step d i (s, vs, ves, $Call x12) = (s', vs', res)"
infinite_iff_card_of_nat: fixes A :: "'f set" shows "infinite A = (|UNIV| \<le>o |A|)"
and_bin_mask_conv_mod: fixes x :: "int"   and n :: "nat" shows "x AND mask n = x mod 2 ^ n"
assms_2: shows "4 \<le> rk {A, B, P, a}"
solution_nat_abs_nat_abs: fixes x :: "int"   and y :: "int" shows "solution (nat \<bar>x\<bar>, nat \<bar>y\<bar>) = solution (x, y)"
closed: fixes A :: "'b sat set" assumes "finite A"   and "A \<noteq> {}"   and "\<And>x y. max x y \<in> {x, y}" shows "Sup A \<in> A"
elims_1: fixes x :: "nat"   and y :: "bool" assumes "Prime x = y" obtains xa :: "nat" where "x = xa" and "y = (1 < xa \<and> (\<forall>u<xa. \<forall>v<xa. u * v \<noteq> xa))"
isomorphism_expand: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" shows "(f \<circ> g = id \<and> g \<circ> f = id) = ((\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x))"
upper_le_binomial: fixes k :: "nat"   and n :: "nat" assumes "0 < k"   and "k < n" shows "n \<le> n choose k"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b list" shows "[] \<noteq> Cons x21 x22"
keys_update: fixes k :: "'a"   and v :: "'b"   and f :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (Poly_Mapping.update k v f) = (if v = (0::'b) then keys f - {k} else insert k (keys f))"
AB: fixes i :: "'a set"   and j :: "'a" assumes "i \<in> \<I>"   and "j \<in> J i" shows "A i < B j"
sumlist_rw: shows "M.sumlist (map (\<lambda>i. f (ws ! i) \<cdot>\<^sub>v ws ! i) [0..<length ws]) = M.sumlist (map (\<lambda>v. f v \<cdot>\<^sub>v v) ws)"
imirror_bounds_Max: fixes I :: "nat set"   and l :: "nat"   and r :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "Max I \<le> l + r" shows "Max (imirror_bounds I l r) = l + r - iMin I"
isPath_distD: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "isPath u p v" shows "dist u (length p) v"
real_transfer_operator_abs: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M" shows "AE x in M. ennreal \<bar>real_transfer_operator f x\<bar> \<le> nn_transfer_operator (\<lambda>x. ennreal \<bar>f x\<bar>) x"
assms_2: fixes gs :: "(('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" shows "ab gs bs [] data = bs"
h2: shows "lose t = [True] \<longrightarrow> ack t = [x] \<and> st_out t = x"
Delta1_N1_Delta2_N2_subset_Delta: shows "\<Delta>\<^bsub>\<Gamma>1\<^esub> \<inter> N\<^bsub>\<V>1\<^esub> \<union> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> \<subseteq> \<Delta>\<^bsub>\<Gamma>\<^esub>"
zero_closed: shows "0\<^sub>v d \<in> carrier_vec d"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>\<And>x xa. P p; \<And>x xa. P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>\<And>x xa. P p; \<And>x xa. P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CN 0 c e))"   and "\<And>c e. P (NEq (CN 0 c e))"   and "\<And>c e. P (Lt (CN 0 c e))"   and "\<And>c e. P (Le (CN 0 c e))"   and "\<And>c e. P (Gt (CN 0 c e))"   and "\<And>c e. P (Ge (CN 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Lt (C va))"   and "\<And>va. P (Lt (Bound va))"   and "\<And>vd vb vc. P (Lt (CN (Suc vd) vb vc))"   and "\<And>va. P (Lt (Neg va))"   and "\<And>va vb. P (Lt (Add va vb))"   and "\<And>va vb. P (Lt (Sub va vb))"   and "\<And>va vb. P (Lt (Mul va vb))"   and "\<And>va. P (Le (C va))"   and "\<And>va. P (Le (Bound va))"   and "\<And>vd vb vc. P (Le (CN (Suc vd) vb vc))"   and "\<And>va. P (Le (Neg va))"   and "\<And>va vb. P (Le (Add va vb))"   and "\<And>va vb. P (Le (Sub va vb))"   and "\<And>va vb. P (Le (Mul va vb))"   and "\<And>va. P (Gt (C va))"   and "\<And>va. P (Gt (Bound va))"   and "\<And>vd vb vc. P (Gt (CN (Suc vd) vb vc))"   and "\<And>va. P (Gt (Neg va))"   and "\<And>va vb. P (Gt (Add va vb))"   and "\<And>va vb. P (Gt (Sub va vb))"   and "\<And>va vb. P (Gt (Mul va vb))"   and "\<And>va. P (Ge (C va))"   and "\<And>va. P (Ge (Bound va))"   and "\<And>vd vb vc. P (Ge (CN (Suc vd) vb vc))"   and "\<And>va. P (Ge (Neg va))"   and "\<And>va vb. P (Ge (Add va vb))"   and "\<And>va vb. P (Ge (Sub va vb))"   and "\<And>va vb. P (Ge (Mul va vb))"   and "\<And>va. P (Eq (C va))"   and "\<And>va. P (Eq (Bound va))"   and "\<And>vd vb vc. P (Eq (CN (Suc vd) vb vc))"   and "\<And>va. P (Eq (Neg va))"   and "\<And>va vb. P (Eq (Add va vb))"   and "\<And>va vb. P (Eq (Sub va vb))"   and "\<And>va vb. P (Eq (Mul va vb))"   and "\<And>va. P (NEq (C va))"   and "\<And>va. P (NEq (Bound va))"   and "\<And>vd vb vc. P (NEq (CN (Suc vd) vb vc))"   and "\<And>va. P (NEq (Neg va))"   and "\<And>va vb. P (NEq (Add va vb))"   and "\<And>va vb. P (NEq (Sub va vb))"   and "\<And>va vb. P (NEq (Mul va vb))"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
cong_prod: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "\<And>x. x \<in> A \<Longrightarrow> unique_euclidean_semiring.cong modulo (f x) (g x) a" shows "unique_euclidean_semiring.cong modulo (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A) a"
add_fract: fixes b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<noteq> (0::'b)"   and "d \<noteq> (0::'b)" shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
upto': shows "\<pi>' =\<^bsub>m'\<^esub> \<pi>\<^sub>1'"
inv2: shows "\<exists>hs. rev v @ rev u @ h = [y, y] @ hs"
simps_2: fixes C :: "'b set set"   and S :: "'b set"   and n :: "nat" shows "pcp_seq (C::'b set set) (S::'b set) (Suc (n::nat)) = (let Sn::'b set = pcp_seq C S n; Sn1::'b set = from_nat n \<triangleright> Sn in if Sn1 \<in> C then Sn1 else Sn)"
le_iff_le_annos: fixes c1 :: "'b acom"   and c2 :: "'b acom" shows "(c1 \<sqsubseteq> c2) = ((annos c1, annos c2) \<in> listrel {(x, y). x \<sqsubseteq> y} \<and> strip c1 = strip c2)"
integrable_split_2: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and k :: "'a"   and c :: "real" assumes "f integrable_on cbox a b"   and "k \<in> Basis" shows "f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
pne34: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "pid3 \<noteq> pid4"
InvariantsNoDecisionsWhenConflictNorUnitAfterApplyBackjump_1: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantUniqC (getC state)"   and "getC state = [opposite (getCl state)]"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "getConflictFlag state"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)" shows "let state' = applyBackjump state in InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state')) \<and> InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state'))"
qSwap_preserves_qFresh: fixes ys :: "'a"   and zs :: "'a"   and z1 :: "'b"   and z2 :: "'b"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" shows "qFresh ys (y @ys[z1 \<and> z2]_zs) (X #[[z1 \<and> z2]]_zs) = qFresh ys y X"
assms: fixes s :: "'n list" assumes "s \<in># csp" shows "\<exists>p. s = [entry fg p] \<and> initialproc fg p"
set_tree_ins': fixes k :: "'c"   and t :: "('c \<times> 'd) tree"   and p :: "'d" assumes "k \<notin> keys t" shows "{(k, p)} \<union> set_tree t \<subseteq> set_tree (ins k p t)"
simps: fixes l1 :: "32 Word.word" shows "wabs l1 = wmax l1 (wneg l1)"
map_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "diagonal_functor J C" shows "diagonal_functor.map J C f = (if partial_magma.arr C f then \<chi>.\<Phi>.Ya.Cop_S.MkArr (constant_functor.map J C (partial_magma.dom C f)) (constant_functor.map J C (partial_magma.cod C f)) (constant_transformation.map J C f) else partial_magma.null (concrete_category.COMP (Collect (functor J C)) (\<lambda>F G. Collect (natural_transformation J C F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map J C \<sigma> \<tau>)))"
order_listI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "order r" shows "order (Listn.le r)"
enc_satisfies_pred_iff_source_target_satisfies_pred_2: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>P Q R. Pred (P, Q) \<and> Pred (Q, R) \<longrightarrow> Pred (P, R)"   and "\<forall>P. Pred (P, P)" shows "(\<forall>S. Pred (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q)\<in>Rel. Pred (P, Q)) \<and> preorder Rel)"
rpi: shows "rp \<in> {flow0 x t1<--<flow0 x t2}"
ag_inv_inv: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> (-\<^sub>a\<^bsub>A\<^esub> x) = x"
simps_2: fixes x :: "'a"   and l :: "'a tree"   and r :: "'a tree"   and h :: "nat" shows "ht (MKT x l r h) = h"
bal_list_merge: fixes as :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and x :: "'b"   and bs :: "('b btree \<times> 'b) list"   and b :: "'b btree" assumes "bal_up\<^sub>i (Up\<^sub>i (Node as a) x (Node bs b))" shows "bal (Node (as @ (a, x) # bs) b)"
SUP_upper2: fixes i :: "'d"   and A :: "'d set"   and u :: "'e \<times> 'f"   and f :: "'d \<Rightarrow> 'e \<times> 'f" assumes "i \<in> A"   and "less_eq_prod u (f i)" shows "less_eq_prod u (Sup_prod (f ` A))"
assms_2: shows "(r, s) \<in> R \<times> S"
s_max: fixes x :: "real" shows "other.s x \<le> other.s_stop"
UN_Int_distrib: fixes f :: "'b \<Rightarrow> 'a set"   and A :: "'a set" shows "\<Union> (range f) \<inter> A = (\<Union>i. f i \<inter> A)"
coeff_of_int_poly: fixes p :: "int poly"   and n :: "nat" shows "poly.coeff (of_int_poly p) n = of_int (poly.coeff p n)"
pr_rec_scheme: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "h \<in> PrimRec3"   and "\<forall>x. f 0 x = g x"   and "\<forall>x y. f (Suc y) x = h y (f y x) x" shows "f \<in> PrimRec2"
standard_uniqueness_dbl: fixes W :: "'c set set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a"   and C :: "'c set"   and D :: "'c set" assumes "ChamberComplexMorphism W X f"   and "ChamberComplexMorphism W X g"   and "SimplicialComplex.maxsimp W C"   and "SimplicialComplex.maxsimp W D"   and "C \<sim> D"   and "f ` D \<noteq> f ` C"   and "g ` D \<noteq> g ` C"   and "chamber (g ` D)"   and "fun_eq_on f g C" shows "fun_eq_on f g D"
float_distinct_2: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_nan a \<and> is_normal a)"
inverse_distrib_swap: fixes a :: "'b perm"   and b :: "'b perm" shows "inverse (a * b) = inverse b * inverse a"
lF_simps_5: fixes f :: "'c" assumes "C.ide f" shows "D.cod (lF f) = F f"
simps_2: fixes x :: "'c"   and f :: "'c \<Rightarrow> ('a, 'b) result" shows "Ok x \<bind> f = f x"
elims_2: fixes x :: "nat \<times> cell list \<times> cell list" assumes "is_final x" obtains s :: "nat" and l :: "cell list" and r :: "cell list" where "x = (s, l, r)" and "s = 0"
assms_1: shows "smaller_lits_are_false (t \<lhd> \<sigma>) (subst_cl C \<sigma>) S"
dependent_nat_choice: fixes P :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<exists>x. P 0 x"   and "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y" shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
add_poly_l'_add_poly_p: fixes pq :: "(char list list \<times> int) list \<times> (char list list \<times> int) list"   and pq' :: "(char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset" assumes "(pq, pq') \<in> sorted_poly_rel \<times>\<^sub>r sorted_poly_rel" shows "\<exists>r. (add_poly_l' pq, r) \<in> sorted_poly_rel \<and> add_poly_p\<^sup>*\<^sup>* (fst pq', snd pq', {#}) ({#}, {#}, r)"
legs_form_cospan_1: shows "C.cospan \<nu>\<pi>.leg0 \<rho>.leg1"
divide_inverse: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a b = times a (inverse b)"
path_appendI: fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and v :: "'a"   and v' :: "'a"   and pth1 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list"   and v'' :: "'a"   and pth2 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list" assumes "Pre_Port_Graph.path edges v v' pth1"   and "Pre_Port_Graph.path edges v' v'' pth2" shows "Pre_Port_Graph.path edges v v'' (pth1 @ pth2)"
m: shows "map \<eta> (take (length lA) qC') = lA"
assms_2: shows "subgroup U (G\<lparr>carrier := H\<rparr>)"
orderI: fixes a :: "'e"   and b :: "'e" assumes "a = sup a b" shows "b \<le> a"
assms_4: fixes j :: "'b" assumes "j \<in> I" shows "f j \<in> R"
converseE: fixes yx :: "'c \<times> 'd"   and r :: "('d \<times> 'c) set" assumes "yx \<in> r\<inverse>" obtains x :: "'d" and y :: "'c" where "yx = (y, x)" and "(x, y) \<in> r"
surj_TM: fixes v :: "'b"   and tm :: "'b tm"   and t :: "nat" assumes "v = Time_Monad.val tm"   and "t = time tm" shows "tm = ??.Time_Monad.tm.TM v t"
bridge_nominals: fixes v :: "nat"   and v' :: "nat"   and xs :: "(nat \<times> nat) set"   and k :: "'b"   and j :: "'b"   and p :: "('c, 'b) fm" shows "nominals (bridge k j xs v v' p) \<union> {k, j} = nominals p \<union> {k, j}"
x3: shows "f\<^sub>1[x::=f\<^sub>2] \<Down> t\<^sub>2"
simps_7: fixes x1 :: "'b"   and x31 :: "'a"   and x32 :: "('a, 'b) process_term" shows "VAR x1 \<noteq> ACT x31 x32"
writes_singleton2: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a" shows "writes {f} f h h'"
less_imp_2: fixes x :: "real"   and y :: "real" assumes "x \<in> C"   and "y \<in> C"   and "x < y" shows "f' y * (x - y) \<le> f x - f y"
Pxyz_nn: fixes x :: "'b"   and y :: "'c"   and z :: "'d" assumes "x \<in> space S"   and "y \<in> space T"   and "z \<in> space P" shows "0 \<le> Pxyz (x, y, z)"
assms: shows "(c, ictxt, fctxt) \<in> decomposeLS s"
bin_cat_num: fixes n :: "nat"   and b :: "int"   and a :: "int" shows "concat_bit n b a = a * 2 ^ n + take_bit n b"
t_field: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and C0 :: "nat"   and Cf :: "varDef list"   and fi :: "nat"   and fDef :: "varDef"   and Ci :: "nat" assumes "CT;\<Gamma> \<turnstile> e0 : C0"   and "fields(CT,C0) = Cf"   and "lookup Cf (\<lambda>fd. vdName fd = fi) = Some fDef"   and "vdType fDef = Ci" shows "CT;\<Gamma> \<turnstile> FieldProj e0 fi : Ci"
surjective: fixes r :: "'a point_scheme" shows "r = \<lparr>xpos = xpos r, ypos = ypos r, \<dots> = point.more r\<rparr>"
summable_norm_comparison_test: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> real" assumes "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n"   and "summable g" shows "summable (\<lambda>n. norm (f n))"
distinct_3: fixes x11 :: "'g"   and x12 :: "'d \<Rightarrow> 'd set"   and x13 :: "('d, 'e, 'f, 'g) seqp"   and x31 :: "('d, 'e, 'f, 'g) seqp"   and x32 :: "('d, 'e, 'f, 'g) seqp" shows "{x11}\<langle>x12\<rangle> x13 \<noteq> x31 \<oplus> x32"
case1b: shows "sint 0xFFFFFFFF80000001 < 0"
simps_8: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c X\<^sub>c \<phi> = suffix 1 \<xi> \<Turnstile>\<^sub>c \<phi>"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and sum :: "'b + 'c" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((sum = Inl (projl sum) \<longrightarrow> P (f1 (projl sum))) \<and> (sum = Inr (projr sum) \<longrightarrow> P (f2 (projr sum))))"
keys_is_none_rep: fixes k :: "'a"   and m :: "('a, 'b) mapping" shows "(k \<in> Mapping.keys m) = (\<not> Option.is_none (Mapping.lookup m k))"
integrable_norm: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" shows "integrable M (\<lambda>x. norm (f x))"
Rep: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'b" assumes "type_definition Rep Abs A" shows "Rep x \<in> A"
left_v1: shows "left (len v1 ts c) = left (ext v1)"
that: fixes s :: "'a net_state"   and t :: "'a net_state" assumes "st = SubnetS s t" shows "thesis"
total_on_converse: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "total_on A (r\<inverse>) = total_on A r"
gSwapAllPresGWlsAll_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSwapAllPresGWlsAll MOD \<equiv> gSwapPresGWls MOD \<and> gSwapAbsPresGWlsAbs MOD"
pick_member: fixes i :: "natural"   and xs :: "(natural \<times> 'b) list" assumes "i < sum_list (map fst xs)" shows "pick xs i \<in> set (map snd xs)"
insert_noteq_member: fixes b :: "'b"   and B :: "'b multiset"   and c :: "'b"   and C :: "'b multiset" assumes "b, B = c, C"   and "b \<noteq> c" shows "c \<in># B"
assoc_props_4: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "src \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' (\<cdot>) prj0 prj1 f g h, Dom = Cod (f \<star> g \<star> h), Cod = Dom ((f \<star> g) \<star> h)\<rparr> = src h"
simps_17: fixes cache' :: "cpu_cache \<Rightarrow> cpu_cache"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "cache_update cache' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache' cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
elims: fixes x :: "Unboxed.type"   and y :: "'dyn \<Rightarrow> ('dyn, 'ubx1, 'ubx2) unboxed option" assumes "unbox x = y" obtains "x = Ubx1" and "y = map_option OpUbx1 \<circ> unbox_ubx1"   | "x = Ubx2" and "y = map_option OpUbx2 \<circ> unbox_ubx2"
covector_conv_vector: fixes v :: "'a" shows "(top \<sqinter> v = v) = (id v \<sqinter> top = id v)"
llast_worker_def: shows "llast_worker \<equiv> \<Lambda> r x yys. case yys of lnil \<Rightarrow> x | y :@ ys \<Rightarrow> r\<cdot>y\<cdot>ys"
assms_2: fixes i :: "nat" assumes "i < n" shows "algebraic (\<alpha> i)"
assms_2: fixes n :: "nat" shows "g n \<in> \<real>"
pref_pow_take: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u \<le>p r \<^sup>@ k" shows "u \<le>p take \<^bold>|r\<^bold>| u \<cdot> u"
c_pair_strict_mono1: fixes x1 :: "nat"   and x2 :: "nat"   and y :: "nat" assumes "x1 < x2" shows "c_pair x1 y < c_pair x2 y"
lr_r2l: fixes R :: "('a \<Rightarrow> ValD) set"   and fs :: "'a \<Rightarrow> ValD \<rightarrow> ValD" assumes "logical_relation R"   and "\<forall>xs\<in>R. (\<lambda>j. fs j\<cdot>(xs j)) \<in> R" shows "(\<lambda>i. ValF\<cdot>(fs i)) \<in> R"
app3_inv: fixes n :: "nat"   and t1 :: "('c, 'd) FingerTreeStruc"   and t2 :: "('c, 'd) FingerTreeStruc"   and xs :: "('c, 'd) Node list" assumes "is_leveln_ftree n t1"   and "is_leveln_ftree n t2"   and "is_measured_ftree t1"   and "is_measured_ftree t2"   and "\<forall>x\<in>set xs. is_leveln_node n x \<and> is_measured_node x" shows "is_leveln_ftree n (app3 t1 xs t2) \<and> is_measured_ftree (app3 t1 xs t2)"
set_enqueue_new_threads: fixes queue :: "'a list"   and ntas :: "('a, 'b, 'c) new_thread_action list" shows "set (Round_Robin.enqueue_new_threads queue ntas) = set queue \<union> {t. \<exists>x m. NewThread t x m \<in> set ntas}"
is1_valid: shows "is1 \<lhd> dims (tensors_from_net (witness_l (tl rs)) $ 0)"
simps_1: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_top_var (IF v t e) = Some v"