hcju/isabelleps huggingface dataset:
nodes_le_def: fixes G :: "nat graph_rec"   and xs :: "nat list"   and ys :: "nat list" assumes "cfg_doms G" shows "nodes_le xs ys \<equiv> Sorted_Less.sorted (rev ys) \<and> Sorted_Less.sorted (rev xs) \<and> set ys \<subseteq> set xs \<or> xs = ys"
assms_2: shows "\<Gamma>(chain \<mapsto> rs1 @ rs'),\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided"
cast_Boolean2: fixes G :: "prog"   and S :: "ty" assumes "G\<turnstile>S\<preceq>? PrimT Boolean" shows "S = PrimT Boolean"
subgroup_is_submonoid: shows "submonoid H (add_monoid M)"
senior_token_run: fixes x :: "nat"   and n :: "nat"   and y :: "nat" shows "(senior x n = senior y n) = (token_run x n = token_run y n)"
vec_def: fixes A :: "'b tensor" shows "vec A = snd (Rep_tensor A)"
simps_7: fixes t :: "'a local.term" shows "Arr \<^bold>\<r>\<^bold>[t\<^bold>] = Arr t"
disj_imp_eq_proj_exec: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and vs :: "'d set"   and s :: "('d, 'e) fmap" assumes "fmdom' (snd a) \<inter> vs = {}" shows "fmrestrict_set vs s = fmrestrict_set vs (state_succ s a)"
prog_unclocked_2: fixes env :: "v sem_env"   and cnt1 :: "nat"   and s :: "'b state"   and prog :: "top0 list"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and cnt2 :: "nat" assumes "evaluate_prog False env (update_clock (\<lambda>_. cnt1) s) prog (update_clock (\<lambda>_. cnt1) s', r)" shows "evaluate_prog False env (update_clock (\<lambda>_. cnt2) s) prog (update_clock (\<lambda>_. cnt2) s', r)"
le1: shows "tensor_mat m1 m2 \<le>\<^sub>L tensor_mat (1\<^sub>m d1) m2"
anycast_simps_3: fixes P :: "'a \<Rightarrow> bool"   and ip :: "nat"   and m :: "'a" shows "anycast P (unicast ip m) = P m"
is_empty_delete: fixes k :: "'b"   and m :: "('b, 'c) mapping" shows "is_empty (delete k m) = (is_empty m \<or> keys m = {k})"
atLeastatMost_empty_iff: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(subset_mset.atLeastAtMost a b = {}) = (\<not> a \<subseteq># b)"
power2_less_eq_zero_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times a 2) zero = (a = zero)"
r'_bigo: shows "r' \<in> O(\<lambda>x. x)"
RangepE: fixes r :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and b :: "'l" assumes "Rangep r b" obtains a :: "'k" where "r a b"
cases: fixes a1 :: "'c \<Rightarrow> 'c list"   and a2 :: "('c, 'd) Term.term"   and a3 :: "('c, 'd) Term.term" assumes "term_variants_pred a1 a2 a3" obtains P :: "'c \<Rightarrow> 'c list" and x :: "'d" where "a1 = Pb__" and "a2 = Var x" and "a3 = Var x"   | T :: "('c, 'd) Term.term list" and S :: "('c, 'd) Term.term list" and P :: "'c \<Rightarrow> 'c list" and g :: "'c" and f :: "'c" where "a1 = Pc__" and "a2 = Fun fa__ Ta__" and "a3 = Fun g Sa__" and "length Ta__ = length Sa__" and "\<And>i. i < length Ta__ \<Longrightarrow> term_variants_pred Pc__ (Ta__ ! i) (Sa__ ! i)" and "g \<in> set (Pc__ fa__)"   | T :: "('c, 'd) Term.term list" and S :: "('c, 'd) Term.term list" and P :: "'c \<Rightarrow> 'c list" and f :: "'c" where "a1 = P" and "a2 = Fun f T" and "a3 = Fun f S" and "length T = length S" and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)"
minus_plus: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and s :: "'a"   and t :: "'a"   and u :: "'a" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus (minus t s) u = minus (plus t u) s"
filter_nb_def: fixes b :: "'b \<Rightarrow> bool"   and I :: "'b nres" shows "filter_nb b I \<equiv> I \<bind> (\<lambda>s. ASSUME (\<not> b s) \<bind> (\<lambda>_. RETURN s))"
test_bit_uint16_code: fixes x :: "uint16"   and n :: "nat" shows "bit x n = (n < 16 \<and> uint16_test_bit x (integer_of_nat n))"
a: shows "pi \<bullet> a \<sharp> pi \<bullet> x"
borel_measurable_diff_null: fixes N :: "'a set"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "N \<in> null_sets (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "(f \<in> borel_measurable (lebesgue_on (S - N))) = (f \<in> borel_measurable (lebesgue_on S))"
BK4_imp_Gets: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Says B A (Crypt K (Number Ta)) \<in> set evs"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "\<exists>Tk. Gets B \<lbrace>Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs"
bernoulli_odd_numeral_eq_0: fixes n :: "num" shows "bernoulli (numeral (num.Bit1 n)) = 0"
stack_rel_def: fixes lrel :: "('d \<times> 'e) set \<Rightarrow> ('a \<times> 'c list) set"   and srel :: "('d \<times> 'e) set \<Rightarrow> ('b \<times> 'c set) set"   and vrel :: "('d \<times> 'e) set" shows "\<langle>vrel\<rangle>stack_rel lrel srel \<equiv> {((li, si), l). (li, l) \<in> \<langle>vrel\<rangle>lrel \<and> (si, set l) \<in> \<langle>vrel\<rangle>srel \<and> distinct l}"
proper_interval_set_Compl_aux: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "proper_intrvl.proper_interval_set_Compl_aux less proper_interval None 0 xs ys = (\<exists>A. ord.set_less_aux less_eq (set xs) A \<and> ord.set_less_aux less_eq A (- set ys))"
some_child_sub_2: fixes sub :: "'b"   and sep :: "'c"   and t :: "('b \<times> 'c) list" assumes "(sub, sep) \<in> set t" shows "sep \<in> set (separators t)"
kth_obs_mono: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat"   and l :: "nat"   and j :: "nat" assumes "is_kth_obs \<pi> k i"   and "is_kth_obs \<pi> l j"   and "k < l" shows "i < j"
spoly_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" shows "punit.spoly p q = (let v1 = punit.lt p; v2 = punit.lt q in if local.punit.component_of_term v1 = local.punit.component_of_term v2 then let t1 = local.punit.pp_of_term v1; t2 = local.punit.pp_of_term v2; l = lcs t1 t2 in local.punit.monom_mult ((1::'d) / lookup p v1) (l - t1) p - local.punit.monom_mult ((1::'d) / lookup q v2) (l - t2) q else 0)"
extend_subj_subj_def: fixes f :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and s :: "state_t" shows "extend_subj_subj f s \<equiv> s\<lparr>sp_impl_subj_subj := extend_f f (sp_impl_subj_subj s)\<rparr>"
arr_map: fixes gf :: "'c \<times> 'c" assumes "CopxC.arr gf" shows "S.arr (Hom.map gf)"
path_input_containment: fixes M :: "('d, 'e, 'f) FSM"   and p :: "(('d \<times> 'e) \<times> 'f) list"   and q :: "'f" assumes "well_formed M"   and "path M p q" shows "set (map fst (map fst p)) \<subseteq> inputs M"
pdifference_def: fixes a :: "nat"   and b :: "nat" shows "pdifference a b = (if a < b then b - a else a - b)"
filter_base: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R" obtains Vs :: "'a list" where "set Vs \<subseteq> carrier R" and "ring.independent R K Vs" and "ring.Span R K Vs = ring.Span R K Us"
d1: assumes "d" shows "(s, \<alpha>, t) \<in> ars"
Rep_run_inverse: fixes x :: "'b run" shows "Abs_run (Rep_run x) = x"
Collect_mem_eq: fixes A :: "'f set" shows "{x. x \<in> A} = A"
simplex_like_has_bottom: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "ordering less_eq less"   and "ordering.simplex_like less_eq less P" shows "ordering.has_bottom less_eq P"
ennreal_suminf_lessD: fixes f :: "nat \<Rightarrow> ennreal"   and x :: "ennreal"   and i :: "nat" assumes "(\<Sum>i. f i) < x" shows "f i < x"
stream_smap_fromN: fixes s :: "'b stream"   and n :: "nat" shows "s = smap (\<lambda>j. Let (j - n) ((!!) s)) (fromN n)"
poly_add_comm: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "poly_add p1 p2 = poly_add p2 p1"
iT_Minus_cut_greater2: fixes t :: "nat"   and k :: "nat"   and I :: "nat set" assumes "t \<le> k" shows "(k \<ominus> I) \<down>> (k - t) = k \<ominus> I \<down>< t"
add: fixes x :: "'b"   and y :: "'b" shows "a *s (x + y) = a *s x + a *s y"
sg19a: shows "ack (t + 3 + k + d + d) = [sending_data]"
assms: shows "n \<in>\<^sub>\<circ> \<omega>"
axioms_2: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" assumes "effective_redundancy_criterion \<Gamma> Rf Ri" shows "effective_redundancy_criterion_axioms \<Gamma> Rf Ri"
that_1: assumes "x = 0"   and "y = 0" shows "thesis"
vconst_on_vsingleton: fixes a :: "V"   and c :: "V" shows "vconst_on (set {a}) c = set {\<langle>a, c\<rangle>}"
is_nextElem_rotate1_aux: fixes m :: "nat"   and ls :: "'a list"   and x :: "'a"   and y :: "'a" assumes "is_nextElem (rotate m ls) x y" shows "is_nextElem ls x y"
is_empty_rep_iff: fixes p :: "int extended \<times> int extended" shows "is_empty_rep p = (\<gamma>_rep p = {})"
mem_upd_commute: fixes x :: "'a"   and y :: "'a"   and mem :: "'a \<Rightarrow> 'b"   and v\<^sub>1 :: "'b"   and v\<^sub>2 :: "'b" assumes "x \<noteq> y" shows "mem(x := v\<^sub>1, y := v\<^sub>2) = mem(y := v\<^sub>2, x := v\<^sub>1)"
invt: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "\<not> unprot \<sigma>"   and "\<not> protFw \<sigma>"   and "a \<in> (eintF (Tag \<sigma>) \<circ> singl) ` {b. eintT \<sigma> b}" shows "eintT \<sigma> (invt \<sigma> a)"
homom3Tr2: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "gsurj\<^bsub>(\<natural>\<^bsub>G\<^esub>H),(\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N / N)\<^esub> (Pj (\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N) N \<circ>\<^bsub>(\<natural>\<^bsub>G\<^esub>H)\<^esub> \<iota>\<^bsub>(\<natural>\<^bsub>G\<^esub>H)\<^esub>)"
funion_fempty_right: fixes A :: "'a fset" shows "A |\<union>| {||} = A"
tdconc': fixes xs :: "int list" assumes "xs \<noteq> []" shows "\<exists>ys. xs = take_decr xs @ ys \<and> (ys \<noteq> [] \<longrightarrow> \<not> hd ys \<le> last (take_decr xs))"
drop_eq_map_nth: fixes n :: "nat"   and xs :: "'c list" shows "drop n xs = map ((!) xs) [n..<length xs]"
filter_id_conv: fixes P :: "'k \<Rightarrow> bool"   and xs :: "'k list" shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
sumpos: fixes X :: "'b set" shows "0 \<le> (\<Sum>x\<in>X. norm (f x))"
irreducible_represents_imp_degree: fixes f :: "int poly"   and x :: "'a"   and g :: "int poly" assumes "irreducible f"   and "f represents x"   and "g represents x" shows "degree f \<le> degree g"
model_cs: shows "\<forall>n. \<not> falsifies\<^sub>c\<^sub>s (f n) Cs"
inheritable_instance_methd: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and sig :: "sig"   and old :: "qtname \<times> methd" assumes "G\<turnstile>C\<preceq>\<^sub>C D"   and "is_class G D"   and "wf_prog G"   and "methd G D sig = Some old"   and "Protected \<le> accmodi old"   and "\<not> is_static old" shows "\<exists>new. methd G C sig = Some new \<and> (new = old \<or> G \<turnstile> qmdecl sig new overrides\<^sub>S qmdecl sig old)"
lt_plus_eq_maxI: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.lt p = punit.lt q \<Longrightarrow> punit.lc p + punit.lc q \<noteq> (0::'c)" shows "punit.lt (p + q) = ordered_powerprod_lin.max (punit.lt p) (punit.lt q)"
obtain_nonzero: fixes G :: "'a set" assumes "Group G"   and "G \<noteq> 0" obtains g :: "'a" where "g \<in> G" and "g \<noteq> (0::'a)"
image_image: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "\<P> f (\<P> g A) = \<P> (\<lambda>x. f (g x)) A"
matrix_vector_mult_in_columnspace_gen: fixes A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "A *v x \<in> vec.span (columns A)"
subscene_trans: fixes Y :: "'a scene"   and X :: "'a scene"   and Z :: "'a scene" assumes "idem_scene Y"   and "X \<subseteq>\<^sub>S Y"   and "Y \<subseteq>\<^sub>S Z" shows "X \<subseteq>\<^sub>S Z"
finiteI_independent: fixes B :: "'c set" assumes "vs2.independent B" shows "finite B"
B: shows "Pri_AgrK x \<in> analz (insert (Crypt (sesK (c * f)) (Number 0)) (A \<union> spies evsFR5))"
span_Times_sing1: fixes B :: "'c set" shows "p.span ({0::'b} \<times> B) = {0::'b} \<times> vs2.span B"
components_are_iso: fixes a :: "'b" assumes "ide a" shows "local.iso (\<ll>'.map a)"
has_integral_restrict_Int: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and T :: "'a set" shows "((\<lambda>x. if x \<in> S then f x else (0::'b)) has_integral i) T = (f has_integral i) (S \<inter> T)"
closure_subset_eq: fixes S :: "'c set" shows "(closure S \<subseteq> S) = closed S"
SMP_wf_trm: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> SMP M"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
may_lockE: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "may_lock l t" obtains "l = None"   | n :: "nat" where "l = \<lfloor>(t, n)\<rfloor>"
final_REV: fixes r :: "'c rexp" shows "final (REV r) = final r"
nearly_healthyI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<And>P Q. \<lbrakk>unitary P; unitary Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q" shows "nearly_healthy t"
simps_2: fixes va :: "int option" shows "is_empty (I None va) = False"
cInf_le_finite: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set"   and x :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "finite X"   and "x \<in> X" shows "less_eq (Inf X) x"
language_elim: fixes w :: "'c stream"   and A :: "('c, 'd) ngba" assumes "w \<in> language A" obtains r :: "'d stream" and p :: "'d" where "p \<in> initial A" and "run A (w ||| r) p" and "gen infs (accepting A) (p ## r)"
v_def: shows "v = (\<lambda>cfg\<in>valid_cfg. emeasure (T cfg) {x \<in> space St. (HLD S1 suntil HLD S2) (state cfg ## x)})"
le_halfspace_def: fixes sctn :: "'e sctn"   and x :: "'e" shows "le_halfspace sctn x = (x \<bullet> normal sctn \<le> pstn sctn)"
inner_divide_left: fixes a :: "'a"   and m :: "real"   and b :: "'a" shows "inner (a / of_real m) b = inner a b / m"
trms\<^sub>l\<^sub>s\<^sub>t_union: fixes A :: "('c strand_label \<times> ('a, 'b) strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>t A = (\<Union>l. trms\<^sub>l\<^sub>s\<^sub>t (proj l A))"
map_def: fixes f :: "'a" shows "\<tau>'.map f = (if A.arr f then B.inv (\<tau> (A.cod f)) \<cdot>\<^sub>B G f else B.null)"
IH: fixes vs :: "real list"   and VS :: "((real \<times> real pdevs) \<times> real) list"   and X :: "((real \<times> real pdevs) \<times> real) list" assumes "vs \<in> aforms_err e VS"   and "approx_slp p ras VS = Some X" shows "interpret_slp ras vs \<in> aforms_err e X"
bet_neq23__neq: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "B \<noteq> C" shows "A \<noteq> C"
span_span: fixes A :: "('a, 'b) vec set" shows "vec.span (vec.span A) = vec.span A"
simps_3: fixes test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex"   and s :: "'d regex" shows "rpd test i (Plus r s) = rpd test i r \<union> rpd test i s"
posD: shows "0 \<le> te p (i + 1) - te q (i + 1)"
inject_4: fixes x51 :: "'d set"   and x52 :: "('d, 'e, 'f) com"   and x53 :: "('d, 'e, 'f) com"   and y51 :: "'d set"   and y52 :: "('d, 'e, 'f) com"   and y53 :: "('d, 'e, 'f) com" shows "(Cond x51 x52 x53 = Cond y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
simps_2: fixes v :: "char list" shows "fromA (B v) = None"
nat_of_integer_integer_of_nat: fixes n :: "nat" shows "nat_of_integer (integer_of_nat n) = n"
char_split: fixes B :: "'c \<Rightarrow> 'b set"   and R :: "'c set"   and A :: "'b set"   and t :: "'b" assumes "\<Union> (B ` R) = UNIV"   and "finite R"   and "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> B j1 \<inter> B j2 = {}" shows "\<chi> A t = (\<Sum>j\<in>R. \<chi> (A \<inter> B j) t)"
bv_to_nat_dist_append: fixes l1 :: "bit list"   and l2 :: "bit list" shows "bv_to_nat (l1 @ l2) = bv_to_nat l1 * 2 ^ length l2 + bv_to_nat l2"
simps_2: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and base :: "'c"   and elements :: "'d list"   and n :: "nat" shows "my_subseqs.create_subseqs f base elements (Suc n) = my_subseqs.next_subseqs f (snd (my_subseqs.create_subseqs f base elements n))"
sumlist_carrier: fixes G :: "('e, 'f) ring_scheme"   and xs :: "'e list" assumes "abelian_monoid G"   and "set xs \<subseteq> carrier G" shows "abelian_monoid.sumlist G xs \<in> carrier G"
fst_map_prod: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and x :: "'d \<times> 'e" shows "fst (map_prod f g x) = f (fst x)"
hoare\<^sub>3_def: fixes P :: "'d \<Rightarrow> bool"   and M :: "'d \<Rightarrow> ('e \<times> 'd) option"   and Q :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> M \<lbrace>Q\<rbrace> \<equiv> \<forall>\<sigma>. P \<sigma> \<longrightarrow> (case M \<sigma> of None \<Rightarrow> False | Some (x, \<sigma>') \<Rightarrow> Q x \<sigma>')"
m_comm: fixes x :: "int"   and y :: "int" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "x + y = y + x"
star_n_eq_iff: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" shows "(star_n X = star_n Y) = (\<forall>\<^sub>F n in \<U>. X n = Y n)"
power_diff_power_eq: 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 a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "a \<noteq> zero" shows "divide (power.power one times a m) (power.power one times a n) = (if n \<le> m then power.power one times a (m - n) else divide one (power.power one times a (n - m)))"
exhaust_sel: fixes nba :: "('c, 'd) nba" obtains "nba = nba (alphabet nba) (initial nba) (transition nba) (accepting nba)"
MarkGuardsD_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'e set"   and f :: "'f"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P mark_guards f c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A"
LAnd2: fixes b :: "coname"   and a1 :: "coname"   and M1 :: "trm"   and a2 :: "coname"   and M2 :: "trm"   and N :: "trm"   and y :: "name"   and x :: "name" assumes "b \<sharp> ([a1].M1, [a2].M2, N, a1, a2)"   and "y \<sharp> ([x].N, M1, M2, x)"   and "x \<sharp> (M1, M2)"   and "a1 \<sharp> (M2, N)"   and "a2 \<sharp> (M1, N)"   and "a1 \<noteq> a2" shows "Cut <b>.AndR <a1>.M1 <a2>.M2 b y.AndL2 x.N y \<longrightarrow>\<^sub>l Cut <a2>.M2 x.N"
left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> D"   and "y \<in> D"   and "z \<in> D" shows "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
KeyWithNonceI: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and NB :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace> \<in> set evs" shows "KeyWithNonce K NB evs"
reduce_basis_impl: shows "LLL_Impl.reduce_basis \<alpha> fs_init = local.reduce_basis"
perm_remove_perm: fixes xs :: "'a list"   and ys :: "'a list"   and z :: "'a" assumes "xs <~~> ys" shows "remove1 z xs <~~> remove1 z ys"
one_mat_def: fixes n :: "nat" shows "1\<^sub>m n \<equiv> Matrix.mat n n (\<lambda>(i, j). if i = j then 1::'b else (0::'b))"
rrep_dip_in_vD: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:4..PRrep-:6} \<longrightarrow> dip \<xi> \<in> vD (rt \<xi>))"
set_of_k_words_length_m_def: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and k :: "nat"   and m :: "nat" assumes "source_code M b fi X enc dec" shows "source_code.set_of_k_words_length_m M X enc k m = {xk. xk \<in> {w. length w = k \<and> set w \<subseteq> source_code.L M X}} \<inter> (\<lambda>w. foldr (\<lambda>x. (+) (code_word_length enc x)) w 0) -` {m}"
distinct_47: fixes x4 :: "'a OCL_Types.type"   and x7 :: "'a OCL_Types.type" shows "Collection x4 \<noteq> Bag x7"
arthan: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "last xs \<noteq> (0::'a)"   and "sum_list xs = (0::'a)" shows "sign_changes (psums xs) < sign_changes xs \<and> odd (sign_changes xs - sign_changes (psums xs))"
assms_5: shows "S1 x \<noteq> S1 x \<triangle> S2 x"
sub_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "sub_list xs ys = (set xs \<subseteq> set ys)"
assms_2: shows "theta \<subseteq> Sretr (theta \<union> Sbis)"
stieltjes_gamma_aux1: fixes k :: "nat" shows "(\<lambda>x. complex_of_real ((- 1) ^ (x + 1) * ln (real (x + 1)) ^ k / real (x + 1))) sums ((- 1) ^ k * (deriv ^^ k) g 1)"
OBTAIN_def: fixes P :: "'b \<Rightarrow> bool" shows "OBTAIN P \<equiv> ASSERT (\<exists>x. P x) \<bind> (\<lambda>_. SPEC P)"
complex_mult_cnj_cmod: fixes z :: "complex" shows "z * cnj z = cor ((cmod z)\<^sup>2)"
assms: shows "fbrelation (vinsert \<langle>a, b\<rangle> r)"
push_pres_height_bound: fixes e :: "nat \<times> nat" assumes "push_precond f l e" shows "Height_Bounded_Labeling c s t (push_effect f e) l"
D_mono: fixes A :: "'b set option"   and A' :: "'b set option"   and e :: "'b exp" assumes "A \<sqsubseteq> A'"   and "\<D> e A" shows "\<D> e A'"
that: fixes C :: "'a set" assumes "C \<in> components (- (S \<union> T))"   and "C \<noteq> {}"   and "frontier C \<inter> S \<noteq> {}"   and "frontier C \<inter> T \<noteq> {}" shows "thesis"
left_commute: fixes b :: "'g multiset"   and a :: "'g multiset"   and c :: "'g multiset" shows "b \<union># (a \<union># c) = a \<union># (b \<union># c)"
simplify_preserves_redundancy: fixes S :: "'at Literal set set"   and C :: "'at Literal set" assumes "all_fulfill finite S"   and "redundant C S" shows "redundant C (simplify S)"
exhaust: fixes y :: "'d list" obtains "y = []"   | x21 :: "'d" and x22 :: "'d list" where "y = x21 # x22"
Krein_Milman: fixes S :: "'b set" assumes "compact S"   and "convex S" shows "S = closure (convex hull {x. x extreme_point_of S})"
surjective: fixes r :: "'a gState_scheme" shows "r = \<lparr>gState.vars = gState.vars r, channels = gState.channels r, timeout = timeout r, procs = procs r, \<dots> = gState.more r\<rparr>"
elims_2: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b stream" assumes "holds x xa" obtains P :: "'b \<Rightarrow> bool" and xs :: "'b stream" where "x = P" and "xa = xs" and "P (shd xs)"