hcju/isabelleps huggingface dataset:
differentiable_imp_continuous_within: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" assumes "f differentiable at x within s" shows "continuous (at x within s) f"
sig_trd_spp_body_alt_Some_2: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and v :: "'t"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and i :: "nat"   and r :: "'a \<Rightarrow>\<^sub>0 'b" assumes "find_sig_reducer (map spp_of bs) v (punit.lt p) 0 = Some i" shows "sig_trd_spp_body (map spp_of bs) v (p, r) = (p - local.punit.monom_mult (punit.lc p / punit.lc (rep_list (bs ! i))) (punit.lt p - punit.lt (rep_list (bs ! i))) (rep_list (bs ! i)), r)"
next_plane0_via_subdivFace': fixes g :: "graph"   and g' :: "graph"   and p :: "nat"   and P :: "graph \<Rightarrow> graph \<Rightarrow> bool" assumes "minGraphProps g"   and "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "\<And>f v' v n g ovs. \<lbrakk>minGraphProps g; pre_subdivFace' g f v' v n ovs; f \<in> \<F> g\<rbrakk> \<Longrightarrow> P g (subdivFace' g f v n ovs)" shows "P g g'"
theta_bound: fixes l :: "nat"   and p :: "nat"   and i :: "nat"   and m :: "nat"   and \<delta>S :: "real" assumes "correct l (te p (i + 1))"   and "correct m (te p (i + 1))"   and "correct p (te p (i + 1))"   and "\<bar>IC l i (max (te l i) (te m i)) - IC m i (max (te l i) (te m i))\<bar> \<le> \<delta>S" shows "\<bar>\<theta> p (i + 1) l - \<theta> p (i + 1) m\<bar> \<le> 2 * \<Lambda> + \<delta>S + 2 * \<rho> * (rmax + \<beta>)"
vimage_vconst_on_ne: fixes B :: "V"   and A :: "V"   and c :: "V" assumes "B \<inter>\<^sub>\<circ> A \<noteq> 0" shows "vconst_on A c `\<^sub>\<circ> B = set {c}"
Edka_Impl_def: fixes c :: "nat \<times> nat \<Rightarrow> int"   and s :: "nat"   and t :: "nat"   and N :: "nat" shows "Edka_Impl c s t N \<equiv> Network_Impl c s t \<and> Edka_Impl_axioms c N"
INF_lower2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq (f i) u" shows "less_eq (Inf (f ` A)) u"
reindex_bij_witness_not_neutral: fixes S' :: "'a set"   and T' :: "'b set"   and S :: "'a set"   and i :: "'b \<Rightarrow> 'a"   and j :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and g :: "'a \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "finite S'"   and "finite T'"   and "\<And>a. a \<in> S - S' \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S - S' \<Longrightarrow> j a \<in> T - T'"   and "\<And>b. b \<in> T - T' \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T - T' \<Longrightarrow> i b \<in> S - S'"   and "\<And>a. a \<in> S' \<Longrightarrow> g a = (0::'c)"   and "\<And>b. b \<in> T' \<Longrightarrow> h b = (0::'c)"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "sum g S = sum h T"
map_add_map_of_foldr: fixes m :: "'c \<Rightarrow> 'd option"   and ps :: "('c \<times> 'd) list" shows "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
fun_eq_iff: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" shows "(f = g) = (\<forall>x. f x = g x)"
exec_meth_instr: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and stk' :: "'a val list"   and loc' :: "'a val list"   and pc' :: "nat"   and xcp :: "'a option" shows "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h (stk, loc, pc, None) ta h' (stk', loc', pc', xcp) = ((ta, xcp, h', [(stk', loc', undefined, undefined, pc')]) \<in> JVM_heap_base.exec_instr addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write (ins ! pc) P t h stk loc undefined undefined pc [] \<and> pc < length ins \<and> ci_app ci (ins ! pc) P h stk loc undefined undefined pc [])"
det_bezout_matrix: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and j :: "'d" assumes "is_bezout_ext bezout"   and "a < b"   and "A $h a $h j \<noteq> (0::'b)" shows "Determinants.det (bezout_matrix A a b j bezout) = (1::'b)"
\<sigma>_2: fixes v :: "'a" assumes "v \<in> winning_region p" shows "winning_strategy p \<sigma> v"
simps_1: fixes r :: "rat" shows "real_alg_1 (Rational r) = of_rat_1 r"
ass: shows "\<forall>x\<in>set (sorted_list_of_set (succs 0)). [0] \<sqsubseteq>\<^bsub>r\<^esub> start ! x"
components_are_equivalences: fixes a :: "'c" assumes "C.obj a" shows "D.equivalence_map (\<sigma>'.map\<^sub>0 a)"
simps_1: fixes x21 :: "('b btree \<times> 'b) list"   and x22 :: "'b btree"   and y21 :: "('b btree \<times> 'b) list"   and y22 :: "'b btree" shows "(Node x21 x22 = Node y21 y22) = (x21 = y21 \<and> x22 = y22)"
elims: fixes x :: "'a \<Rightarrow> 'b"   and xa :: "'a set"   and y :: "'a set" assumes "argmax x xa = y" obtains f :: "'a \<Rightarrow> 'b" and A :: "'a set" where "x = f" and "xa = A" and "y = {x \<in> A. f x = Max (f ` A)}"
between_def: shows "between = (\<lambda>(a, b) x. x \<in> closed_segment a b)"
surj_fun_eq: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set"   and g1 :: "'c \<Rightarrow> 'e"   and g2 :: "'c \<Rightarrow> 'e" assumes "\<P> f X = UNIV"   and "\<forall>x\<in>X. (g1 \<circ> f) x = (g2 \<circ> f) x" shows "g1 = g2"
entailed_formula_entails_implicates: fixes atom_ordering :: "('a \<times> 'a) set"   and S1 :: "'a Literal set set"   and S2 :: "'a Literal set set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.entails_formula S1 S2"   and "propositional_atoms.entails S2 C" shows "propositional_atoms.entails S1 C"
DDH_advantage_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "local.ddh.DDH_advantage \<A> = \<bar>spmf (local.ddh.DDH0 \<A>) True - spmf (local.ddh.DDH1 \<A>) True\<bar>"
mult_le_cancel_left1: fixes c :: "'b"   and b :: "'b" shows "(c \<le> c * b) = (((0::'b) < c \<longrightarrow> (1::'b) \<le> b) \<and> (c < (0::'b) \<longrightarrow> b \<le> (1::'b)))"
\<sigma>: shows "\<sigma> = \<theta>1 \<lozenge> \<theta>2"
eP_def: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and n :: "nat"   and \<omega> :: "nat \<Rightarrow> 'b" assumes "Ionescu_Tulcea P M" shows "Ionescu_Tulcea.eP P M n \<omega> = distr (P n \<omega>) (Pi\<^sub>M {0..<Suc n} M) (fun_upd \<omega> n)"
cset_count_infinite_bij: fixes A :: "'b cset" assumes "infinite (rcset A)" shows "bij_betw (cset_count A) (rcset A) UNIV"
remove_child_parent_child_rel_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "parent_child_rel h' \<subseteq> parent_child_rel h"
decode_plan_def: fixes \<Pi> :: "'a strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" shows "\<Phi>\<inverse> \<Pi> \<A> t \<equiv> map (decode_plan' \<Pi> \<A>) [0..<t]"
resumption_ord_is_DoneD: fixes r :: "('f, 'g, 'h) Resumption.resumption"   and r' :: "('f, 'g, 'h) Resumption.resumption" assumes "resumption_ord r r'"   and "Resumption.resumption.is_Done r'" shows "Resumption.resumption.is_Done r"
assms_5: fixes x :: "'c" assumes "x \<in> A \<inter> B" shows "g x = h x"
set_subset_Cons: fixes xs :: "'d list"   and x :: "'d" shows "set xs \<subseteq> set (x # xs)"
equal_image_mset: fixes E :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>E. f x = g x" shows "image_mset f (mset_set E) = image_mset g (mset_set E)"
trapped_forward_def: fixes x :: "'a"   and K :: "'a set" shows "rev.trapped_forward x K = (rev.flow0 x ` (rev.existence_ivl0 x \<inter> {0..}) \<subseteq> K)"
c: shows "3 * 2 ^ (7 - 1) - 1 = 191"
that: fixes B :: "'a set" assumes "B \<subseteq> S"   and "pairwise orthogonal B"   and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"   and "independent B"   and "card B = dim S"   and "span B = S" shows "thesis"
abs_eq: fixes x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "\<lparr>make\<o> x\<rparr> = make\<o> x"
foldpairs_def: shows "foldpairs \<equiv> {(f, g). \<exists>C. OpposedThinChamberComplexFoldings X f g C}"
spectrum_imp_pr: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<pp> :: "'c set" assumes "comm_ring R addition multiplication zero unit"   and "\<pp> \<in> comm_ring.spectrum R addition multiplication zero unit" shows "pr_ideal R \<pp> addition multiplication zero unit"
pm_eq: shows "poincare_map p2 x0 = poincare_map p2 (poincare_map p1 x0)"
prems: shows "\<Gamma>\<turnstile>sequence Seq (flatten (Seq c1 c2)) \<down> s"
range_subset_upper: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b"   and x :: "'b" assumes "{a..b} \<subseteq> {c..d}"   and "x \<in> {a..b}" shows "b \<le> d"
depends\<^sub>R_def: fixes a :: "atom" shows "depends\<^sub>R a = (hd_coeff a \<noteq> 0)"
preserves_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "src\<^sub>D (G \<mu>) = G.map\<^sub>0 (src\<^sub>C \<mu>)"
suf_ruler_le: fixes ps :: "'b list"   and xs :: "'b list"   and qs :: "'b list" assumes "ps \<le>s xs"   and "qs \<le>s xs"   and "\<^bold>|ps\<^bold>| \<le> \<^bold>|qs\<^bold>|" shows "ps \<le>s qs"
pderivs_lang_snoc: fixes s :: "'a list"   and c :: "'a"   and r :: "'a rexp" shows "pderivs_lang (PSuf s @@ {[c]}) r = pderiv_set c (pderivs_lang (PSuf s) r)"
compactin_path_image: fixes X :: "'a topology"   and g :: "real \<Rightarrow> 'a" assumes "pathin X g" shows "compactin X (g ` {0..1})"
vars_\<C>_insert: fixes x :: "'Var"   and Y :: "'Var set" shows "vars_\<C> (insert x Y) = vars_\<C> {x} \<union> vars_\<C> Y"
simps_7: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b BinTree \<Rightarrow> 'b BinTree \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x1 :: "'b" shows "rec_BinTree f1 f2 (BinTree.Leaf x1) = f1 x1"
assms_7: shows "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"
pmf_pair: fixes M :: "'b pmf"   and N :: "'c pmf"   and a :: "'b"   and b :: "'c" shows "pmf (pair_pmf M N) (a, b) = pmf M a * pmf N b"
fmpred_id: fixes f :: "'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap" shows "fmpred (\<lambda>_. id) (fmmap f m) = fmpred (\<lambda>_. f) m"
auth_pubkey_owner_ii: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Owner m, \<lbrace>Num 1, Auth_PubKey n\<rbrace>) \<in> s" shows "False"
assms_6: shows "4 \<le> rk {A, A', C, a}"
alpha_qVar_iff: fixes X :: "('f, 'g, 'h, 'i, 'j) qTerm"   and xs' :: "'h"   and x' :: "'i" assumes "FixVars TYPE('d) TYPE('e)" shows "(X #= qVar xs' x') = (X = qVar xs' x')"
same_lappend_eq: fixes r :: "'b llist"   and A :: "'b set"   and s :: "'b llist"   and t :: "'b llist" assumes "r \<in> A\<^sup>\<star>" shows "(r @@ s = r @@ t) = (s = t)"
update_i: fixes i :: "nat"   and ba :: "'a list \<times> 'a list"   and xs :: "'a list"   and x :: "'a" assumes "list_repr i ba xs"   and "i < length xs" shows "list_repr i (update_i ba x) (xs[i := x])"
strict_trans1: fixes b :: "'d list"   and a :: "'d list"   and c :: "'d list" assumes "prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"
closed_real_atLeastAtMost: fixes a :: "real"   and b :: "real" shows "closed {a..b}"
err: shows "x = aform_val (e(degree_aform_err X := err)) (aform_err_to_aform X (degree_aform_err X))"
expectation_indicator: fixes A :: "'a set" assumes "A \<in> events" shows "expectation (rind A) = prob A"
f: shows "(\<lambda>(x, y). f x y) \<in> count_space UNIV \<Otimes>\<^sub>M count_space UNIV \<rightarrow>\<^sub>M M"
u_in_\<W>: shows "u \<in> \<W> P k"
F: fixes n :: "nat" shows "F n \<noteq> [] \<and> sorted_wrt (\<subseteq>) (F n) \<and> set (F n) \<subseteq> {S. infinite S} \<and> set (F n) \<subseteq> Pow M"
const_of_const_list: fixes cs :: "e list" assumes "length cs = 1"   and "const_list cs" shows "\<exists>v. cs = [$C v]"
ms_xs_ne_ys: shows "mset xs \<noteq> mset ys"
open_Times: fixes S :: "'d set"   and T :: "'e set" assumes "open S"   and "open T" shows "open (S \<times> T)"
sum_upto_plus1: fixes x :: "real"   and f :: "nat \<Rightarrow> 'a" assumes "0 \<le> x" shows "sum_upto f (x + 1) = sum_upto f x + f (Suc (nat \<lfloor>x\<rfloor>))"
prob_space_T': fixes I :: "'s pmf" shows "prob_space (T' I)"
PR_CONST_def: fixes x :: "'e" shows "PR_CONST x \<equiv> x"
\<Delta>'_sel_4: shows "sink \<Delta>' = Inner (sink \<Delta>)"
bij_betw_comp_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and A' :: "'d set"   and f' :: "'d \<Rightarrow> 'e"   and A'' :: "'e set" assumes "bij_betw f A A'" shows "bij_betw f' A' A'' = bij_betw (f' \<circ> f) A A''"
x: shows "Derives1 v (fst d) (snd d) x \<and> Derivation x D w"
le_add_same_cancel1: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># a + b) = ({#} \<subseteq># b)"
bool_of_specT: fixes e :: "'c exp" assumes "bool_of e = \<lfloor>True\<rfloor>" shows "e = true"
app_beta_exhaust: fixes c :: "'a"   and s :: "Lambda_Z.term" obtains x :: "name" and s' :: "Lambda_Z.term" where "atom x \<sharp> c" and "s = Abs x s'"   | "\<And>t. app_beta s t = App s t"
select_rec_threeway_partition: fixes d :: "'b"   and k :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "(0::'b) < d"   and "k < length xs" shows "select k xs = (let (ls, es, gs) = threeway_partition x xs; nl = length ls; ne = length es in if k < nl then select k ls else if k < nl + ne then x else select (k - nl - ne) gs)"
poly_fun_eqI_ge: fixes f :: "int \<Rightarrow> int"   and g :: "int \<Rightarrow> int"   and b :: "int" assumes "poly_fun f"   and "poly_fun g"   and "\<And>a. b \<le> a \<Longrightarrow> f a = g a" shows "f = g"
InvariantsNoDecisionsWhenConflictNorUnitAfterExhaustivePropagate: fixes state :: "State" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))" shows "let state' = exhaustiveUnitPropagate state in InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state')) \<and> InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state'))"
simps: fixes l1 :: "lit"   and l2 :: "lit" shows "edges_of_clause (Upair l1 l2) = {(negate l1, l2), (negate l2, l1)}"
normal_restrict_supergroup: fixes S :: "'a set"   and N :: "'a set" assumes "subgroup S (G\<lparr>carrier := H\<rparr>)"   and "N \<lhd> G\<lparr>carrier := H\<rparr>"   and "N \<subseteq> S" shows "N \<lhd> G\<lparr>carrier := H, carrier := S\<rparr>"
mat_addrow_gen_def: fixes ad :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "mat_addrow_gen ad mul a k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then ad (mul a (A $$ (l, j))) (A $$ (i, j)) else A $$ (i, j))"
aux2: assumes "\<theta> < 0" shows "LBINT x. indicat_real {0<..<T} x * sin (x * \<theta>) / x = - (LBINT x. indicat_real {0<..<- (T * \<theta>)} x * sin x / x)"
rep_eq: fixes x :: "uint" shows "Rep_uint32 (uint32_of_uint x) = ucast (Rep_uint x)"
map_of_clearjunk: fixes al :: "('c \<times> 'd) list" shows "map_of (AList.clearjunk al) = map_of al"
parts_mpair: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "parts (insert \<lbrace>X, Y\<rbrace> H) = insert \<lbrace>X, Y\<rbrace> (parts ({X, Y} \<union> H))"
not_call_def: fixes p :: "('a, 'b, 'c, 'd) seqp" shows "not_call p \<equiv> \<forall>pn. p \<noteq> call(pn)"
sg_condTr1: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<forall>a b. a \<in> H \<and> b \<in> H \<longrightarrow> a \<cdot> \<rho> b \<in> H" shows "\<one> \<in> H"
exec_move_Cast: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and T :: "ty"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (Cast T e) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
simps_1: fixes \<sigma> :: "'c"   and y :: "'d"   and \<tau> :: "'c" shows "cSnoc \<lbrace> \<sigma> \<rbrace> y \<tau> = \<lbrace> \<sigma> \<rbrace> \<cdot> y \<cdot> \<lbrace> \<tau> \<rbrace>"
\<gamma>: shows "\<guillemotleft>\<gamma> : g' \<Rightarrow> g\<guillemotright> \<and> local.iso \<gamma>"
fullAbs: shows "\<forall>S1 S2. ((S1, S2) \<in> (symcl (SRel\<^sup>=))\<^sup>+) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> (symcl (TRel\<^sup>=))\<^sup>+)"
iso_cancel_right: fixes g :: "'a"   and f :: "'a"   and f' :: "'a" assumes "local.iso g"   and "f \<cdot> g = f' \<cdot> g"   and "seq f g"   and "local.iso g" shows "f = f'"
intros_4: fixes m :: "bool list"   and a :: "bool list" assumes "length m = id' \<eta>" shows "local.S' (return_spmf (Inl (Some (a, m)), (), Store m)) (return_spmf (Store m, None, [m \<mapsto> a]))"
PDG_path_Append: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and control_dependence :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n'' :: "'g"   and n' :: "'g"   and n :: "'g" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n'' n'"   and "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n''" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n'"
zero: fixes x :: "'a" shows "0 *\<^sub>R x = (0::'a)"
simps_8: shows "nnf (\<^bold>\<not> \<bottom>) = \<^bold>\<not> \<bottom>"
finestpart_def: fixes X :: "'b set" shows "finestpart X = (\<lambda>x. {x}) ` X"
L2_set_mult_ineq_lemma: fixes a :: "real"   and c :: "real"   and b :: "real"   and d :: "real" shows "2 * (a * c) * (b * d) \<le> a\<^sup>2 * d\<^sup>2 + b\<^sup>2 * c\<^sup>2"
min_simps_1: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "min a b = a"
commutes_with_unit: shows "\<phi>\<^sub>f V (one_str\<^sub>Y V) = one_im_sheaf V"
image_add_atLeast: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "'b"   and i :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "plus k ` ord.atLeast less_eq i = ord.atLeast less_eq (plus k i)"
legacy_Complex_simps_25: fixes r :: "real" shows "cor r = Complex r 0"
prems_1: shows "P \<turnstile> Ts Casts x # xs to vs' "
case_prodI: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and a :: "'k"   and b :: "'l" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
dims2_alter: fixes dims :: "nat list"   and A :: "nat set"   and vars :: "nat set" assumes "{0..<length dims} \<subseteq> A" shows "partial_state.dims2 dims vars = nths dims (A - vars)"
param_plus_def: fixes dict_plus :: "'a dict_plus" shows "param_plus dict_plus = (case dict_plus of mk_plus x \<Rightarrow> x)"
simps: fixes l :: "'b"   and u :: "'b" shows "is_interval (l, u) = (l < u)"
f_inv_f: fixes x :: "'b" shows "f (inv f x) = x"
def_scheme: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "profile_mono_scheme_defined f g" shows "f x \<equiv> \<lambda>\<tau>. if (\<delta> x) \<tau> = true \<tau> then g (x \<tau>) else invalid \<tau>"
cross_triple: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "x \<times> y \<bullet> z = y \<times> z \<bullet> x"
limit_is_suffixE: fixes x :: "nat \<Rightarrow> 'b" assumes "finite (range x)" obtains xa :: "nat" where "limit x = range (suffix xa x)"
div_comp_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and rs1 :: "word5"   and rd :: "word5"   and operand2 :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (div_comp instr rs1 rd operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
orderE: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># b" obtains "a = a \<inter># b"
ntrs_xchange_context: fixes s :: "'n list"   and c :: "'n list multiset"   and ww :: "('p, 'ba) label list list"   and s' :: "'n list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "((s, c), ww, s', c') \<in> trcl (ntrs fg)"   and "mon_c fg cn \<subseteq> mon_c fg c" shows "\<exists>csp. c' = csp + c \<and> ((s, cn), ww, s', csp + cn) \<in> trcl (ntrs fg)"
preserves_iso: fixes f :: "'d" assumes "D.iso f" shows "D.iso (D.R f)"
resumption_corec_def: fixes f :: "'j \<Rightarrow> ('f, 'g, 'h, 'i, 'j) resumption_corec"   and e :: "'j" shows "resumption_corec f e = Abs_resumption (RESUMPTION_corec f e)"
weight_le_def: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" shows "weight_le s t \<equiv> vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t) \<and> weight s \<le> weight t"
leadsETo_subset_leadsTo: fixes A :: "'d set"   and CC :: "'d set set"   and B :: "'d set" shows "A leadsTo[CC] B \<subseteq> A \<longmapsto> B"
product_prob_space_def: fixes M :: "'b \<Rightarrow> 'c measure" shows "product_prob_space M \<equiv> product_sigma_finite M \<and> product_prob_space_axioms M"