hcju/isabelleps huggingface dataset:
InvariantUniqAfterBacktrack: fixes M :: "(Literal \<times> bool) list"   and M' :: "(Literal \<times> bool) list" assumes "InvariantUniq M"   and "InvariantConsistent M"   and "decisions M \<noteq> []"   and "M' = prefixBeforeLastDecision M @ [(opposite (lastDecision M), False)]" shows "InvariantUniq M'"
surjective: fixes r :: "('a, 'b, 'c) calculusWithBarbs_scheme" shows "r = \<lparr>Calculus = Calculus r, HasBarb = HasBarb r, \<dots> = calculusWithBarbs.more r\<rparr>"
split_word_eq_on_mask: fixes x :: "'b word"   and y :: "'b word"   and m :: "'b word" shows "(x = y) = (x AND m = y AND m \<and> x AND NOT m = y AND NOT m)"
Ord_linear_lt: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" obtains "k \<subset>\<^sub>\<circ> l"   | "k = l"   | "l \<subset>\<^sub>\<circ> k"
deg_smult_ring: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "deg R (a \<odot>\<^bsub>P\<^esub> p) \<le> (if a = \<zero> then 0 else deg R p)"
topological_tendstoI: fixes l :: "'a"   and f :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "\<And>S. \<lbrakk>open S; l \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<in> S" shows "(f \<longlongrightarrow> l) F"
case_prodE: fixes c :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and p :: "'g \<times> 'h" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'g" and y :: "'h" where "p = (x, y)" and "c x y"
PSQS: fixes s :: "'a"   and s' :: "'a"   and a :: "'c" assumes "PS s s' a" shows "QS s s' a"
K_def: shows "K = {B \<in> sets M. B \<subseteq> A}"
det_def': fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "Determinant.det A = (\<Sum>p | p permutes {0..<n}. signof p * (\<Prod>i = 0..<n. A $$ (i, p i)))"
ESem_simps_as_defined_5: fixes scrut :: "exp"   and e\<^sub>1 :: "exp"   and e\<^sub>2 :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrakk> (scrut ? e\<^sub>1 : e\<^sub>2) \<rbrakk>\<^bsub>\<rho>\<^esub> = B_project\<cdot>(\<lbrakk> scrut \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>(\<lbrakk> e\<^sub>1 \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>(\<lbrakk> e\<^sub>2 \<rbrakk>\<^bsub>\<rho>\<^esub>)"
hom_functor_axioms: shows "hom_functor (\<cdot>) (\<cdot>\<^sub>S) \<phi>"
wf_ciD2: fixes ci :: "'a instr \<Rightarrow> (nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'b \<Rightarrow> 'a val list \<Rightarrow> 'a val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'a frame list \<Rightarrow> bool"   and ins :: "'a 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 h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list" assumes "wf_ci ci"   and "ci ins P h stk loc C M pc frs" shows "check_instr' ins P h stk loc C M pc frs"
outpt': shows "\<forall>p. outpt s' p = (if phase s' p = 3 then inp (dblock s' p) else NotAnInput)"
adj_mat_def: fixes A :: "'a mat" shows "adj_mat A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). cofactor A j i)"
exec_gpv_invariant: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and x :: "'d"   and s' :: "'a"   and gpv :: "('d, 'b, 'c) gpv"   and s :: "'a" assumes "callee_invariant_on callee I \<I>"   and "(x, s') \<in> set_spmf (exec_gpv callee gpv s)"   and "I s"   and "\<I> \<turnstile>g gpv \<surd>" shows "I s'"
dterms_ctermsl: fixes q :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> dterms \<Gamma> p"   and "wellformed \<Gamma>" shows "q \<in> ctermsl p \<or> (\<exists>pn. q \<in> ctermsl (\<Gamma> pn))"
transitive_closurep_trans'_1: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "r\<^sup>+\<^sup>+ x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
Inv0: shows "case ([(R0, S0)], []) of (ws, bs) \<Rightarrow> distinct ws \<and> (\<forall>(R, S)\<in>set ws. R \<subseteq> PDERIVS R0 \<and> S \<subseteq> PDERIVS S0 \<and> (R, S) \<notin> set bs)"
spick_pseudo_proj_gen: fixes m :: "nat"   and n :: "nat"   and w :: "bool stream"   and v :: "bool" assumes "m < n" shows "spick (pseudo_proj_True n w) m v = spick w m v"
mult_aform'_def: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and y :: "(real \<times> real pdevs) \<times> real" shows "mult_aform' p x y = (let (fx, sx) = x; (fy, sy) = y; ex = \<bar>sx\<bar>; ey = \<bar>sy\<bar>; z0 = trunc_bound_eucl p (fst fx * fst fy); u = trunc_bound_pdevs p (scaleR_pdevs (fst fy) (snd fx)); v = trunc_bound_pdevs p (scaleR_pdevs (fst fx) (snd fy)); w = trunc_bound_pdevs p (add_pdevs (fst u) (fst v)); tx = Radius' p fx; ty = Radius' p fy; l = truncate_up p (tx * ty); ee = truncate_up p (ex * ey); e1 = truncate_up p (ex * truncate_up p (\<bar>fst fy\<bar> + ty)); e2 = truncate_up p (ey * truncate_up p (\<bar>fst fx\<bar> + tx)) in ((fst z0, fst w), sum_list' p [ee, e1, e2, l, snd z0, snd u, snd v, snd w]))"
\<psi>_in_terms_of_\<epsilon>: fixes x :: "'a"   and g :: "'b"   and y :: "'b" assumes "A.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>B G x\<guillemotright>" shows "\<psi> x g = FG.\<epsilon> x \<cdot>\<^sub>A F g"
splits_wrtD_3: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" assumes "splits_wrt (ps, qs) T F"   and "hU \<in> set qs" shows "cone hU \<subseteq> P[X]"
execute_parallel_operator_negative_effect_if: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and op :: "'a strips_operator"   and v :: "'a" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "op \<in> set ops"   and "v \<in> set (delete_effects_of op)" shows "execute_parallel_operator s ops v = Some False"
sum_bounded_below: fixes A :: "'b set"   and K :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> A \<Longrightarrow> K \<le> f i" shows "of_nat (card A) * K \<le> sum f A"
simps_1: fixes S :: "nat \<Rightarrow> typ"   and n :: "nat" shows "bound_typ_inst S (FVar n) = TVar n"
fixp_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "monotone (\<le>) (\<le>) f" shows "ccpo_class.fixp f = f (ccpo_class.fixp f)"
has_fields_b_fields: fixes P :: "'a prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "P \<turnstile> C has_fields FDTs" shows "TypeRel.fields P C = FDTs"
s1: shows "(prod_list (map ((!) (map_sgas sign)) (a # xa)) = 1) = ((sign ! a)\<^sup>2 = 0 \<and> prod_list (map ((!) (map_sgas sign)) xa) = 1)"
poly_offset_poly: fixes p :: "'a poly"   and h :: "'a"   and x :: "'a" shows "poly (offset_poly p h) x = poly p (h + x)"
post_rule: fixes P :: "assn"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and R :: "'b \<Rightarrow> assn" assumes "<P> f <Q>"   and "\<forall>x. Q x \<Longrightarrow>\<^sub>A R x" shows "<P> f <R>"
TimesLR_PlusL: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "TimesLR (Plus r s) t = Plus (TimesLR r t) (TimesLR s t)"
simps_1: fixes A :: "'a" shows "complement (Pos A) = Neg A"
simps_6: shows "OUTfromCh data6 = {data4}"
cSup_upper2: fixes x :: "'d multiset"   and X :: "'d multiset set"   and y :: "'d multiset" assumes "x \<in> X"   and "y \<subseteq># x"   and "subset_mset.bdd_above X" shows "y \<subseteq># Sup X"
Quotient3_endofun: shows "Quotient3 (=) abs_endofun rep_endofun"
eq: fixes a :: "'a"   and b :: "'b" shows "a \<sim> b = (b = f a)"
countably_additive_additive: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "countably_additive M f" shows "Measure_Space.additive M f"
P_lnull_ltl_LCons: assumes "lnull (ltl P')" shows "P' = LCons (P $ n) LNil"
sublist_appendI: fixes xs :: "'c list"   and ps :: "'c list"   and ss :: "'c list" shows "sublist xs (ps @ xs @ ss)"
emeasure_UN_eq_0: fixes M :: "'c measure"   and N :: "nat \<Rightarrow> 'c set" assumes "\<And>i. emeasure M (N i) = 0"   and "range N \<subseteq> sets M" shows "emeasure M (\<Union> (range N)) = 0"
inf_assoc: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" shows "((x \<and> y) \<and> z) = (x \<and> y \<and> z)"
apply_power_right_mult_order: fixes f :: "'a perm"   and a :: "'a"   and n :: "nat" shows "(f ^ (order f a * n)) \<langle>$\<rangle> a = a"
invertible_interchange_columns: fixes a :: "'c"   and b :: "'c" shows "invertible (interchange_columns (Finite_Cartesian_Product.mat (1::'b)) a b)"
ws_overrides_commonD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old"   and "ws_prog G" shows "declclass new \<noteq> Object \<and> \<not> is_static new \<and> \<not> is_static old \<and> accmodi new \<noteq> Private \<and> G\<turnstile>resTy new\<preceq>resTy old \<and> msig new = msig old \<and> G\<turnstile>declclass new\<prec>\<^sub>C declclass old \<and> G\<turnstile>Method new declared_in declclass new \<and> G\<turnstile>Method old declared_in declclass old"
elims: fixes x :: "char list \<times> hterm list"   and y :: "char list \<times> fterm list" assumes "fatom_of_hatom x = y" obtains p :: "char list" and ts :: "hterm list" where "x = (p, ts)" and "y = (p, fterms_of_hterms ts)"
opp_mult_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" shows "times.opp_mult times x y \<equiv> times y x"
p_ge_0: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "0 < int p"
vimage_set_eq_at: fixes A :: "V" shows "{b. \<exists>a\<in>elts (A \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b} = {b. \<exists>a\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
invariant_reaches: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow>* b"   and "Q a" shows "Q b"
monom_finsum_decomp: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "p = (\<Oplus>\<^bsub>K [X]\<^esub>t\<in>set (dense_repr p). local.monom (fst t) (snd t))"
is_nextElem_def: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" shows "is_nextElem xs x y \<equiv> is_sublist [x, y] xs \<or> xs \<noteq> [] \<and> x = last xs \<and> y = hd xs"
cases: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and a1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and a2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval"   and "(a1, a2) \<in> sifum_security_init.mm_equiv dma \<C>_vars eval" obtains \<R> :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set" and lc\<^sub>1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" and lc\<^sub>2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" where "a1 = lc\<^sub>1" and "a2 = lc\<^sub>2" and "sifum_security_init.strong_low_bisim_mm dma \<C>_vars eval \<R>" and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>"
border_take_lengths: fixes i :: "nat"   and s :: "'a list"   and j :: "nat" assumes "i \<le> length s"   and "border (take i s) (take j s)" shows "i \<le> j"
lower_circle_map: shows "{x. 1 / x \<in> (\<lambda>x. x + 1) ` {x. - sqrt 3 * Re x < Im x}} = lower_circle_01"
ChiPlus_ChiRel_Ex: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(S, T) \<in> ChiPlus A"   and "(S, T) \<notin> ChiRel A" shows "\<exists>U. (S, U) \<in> ChiPlus A \<and> (U, T) \<in> ChiRel A"
lseg_prec2: fixes p :: "'a node ref option"   and F1 :: "assn"   and F2 :: "assn"   and h :: "heap \<times> nat set" shows "\<forall>l l'. h \<Turnstile> os_list l p * F1 \<and>\<^sub>A os_list l' p * F2 \<longrightarrow> l = l'"
integral_return: fixes x :: "'e"   and M :: "'e measure"   and g :: "'e \<Rightarrow> 'f" assumes "x \<in> space M"   and "g \<in> borel_measurable M" shows "integral\<^sup>L (return M x) g = g x"
f'_inv: shows "f' (x, y) o\<^sub>L embed2_blinfun \<in> invertibles_blinfun"
monoD: fixes f :: "'a \<Rightarrow> 'd"   and x :: "'a"   and y :: "'a" assumes "mono f"   and "x \<le> y" shows "f x \<le> f y"
index_of_bound: fixes y :: "'c list"   and x :: "'c" assumes "y \<noteq> []"   and "x \<in> set y" shows "index_of x y < length y"
path_weightp_distinct_def: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a"   and s :: "'b" assumes "graph weights" shows "graph.path_weightp_distinct weights l1 l2 s \<equiv> \<exists>xs. distinct xs \<and> graph.path weights l1 l2 xs \<and> s = foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b)"
assms_1: shows "\<psi> \<in> subfrmlsn \<phi>"
mbisim_final_thread_inv: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "s1 \<approx>m s2" shows "r1.final_thread s1 t = r2.final_thread s2 t"
counit_simps_5: shows "trg \<psi>' = trg f"
continuous_map_snd: fixes X :: "'a topology"   and Y :: "'b topology" shows "continuous_map (prod_topology X Y) Y snd"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom \<d>[f] = local.map (local.dom f)"
below_rel_internal: fixes A :: "('c \<times> 'd) set" shows "below_rel A = \<langle>A\<rangle>sctn_rel O br below_halfspace top"
gcdof_exists: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "\<exists>c. c \<in> carrier G \<and> c gcdof a b"
fcomp_assoc: fixes f :: "'b \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd"   and h :: "'d \<Rightarrow> 'c" shows "fcomp (fcomp f g) h = fcomp f (fcomp g h)"
to_list: shows "SIDE_GEN_ALGO (is_set_to_list (A \<times>\<^sub>r B) Rs1 tol1)"
open_MInfty: fixes A :: "ereal set" assumes "open A"   and "- \<infinity> \<in> A" shows "\<exists>x. {..<ereal x} \<subseteq> A"
sigma_sets_eq: fixes M :: "'b measure" shows "sigma_sets (space M) (sets M) = sets M"
GuardFault: fixes s :: "'a"   and g :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and f :: "'c"   and c :: "('a, 'b, 'c) com" assumes "s \<notin> g" shows "\<Gamma>\<turnstile> (Guard f g c, Normal s) \<rightarrow> (Skip, Fault f)"
poly_power_inv: fixes p :: "('c monom \<times> 'd) list"   and n :: "nat" assumes "poly_inv p" shows "poly_inv (poly_power p n)"
ws_def: shows "ws = getFrN (xs @ us @ vs) ts [\<phi>] (length xs)"
neval_sched_Nil: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val" shows "(cms, mem) \<rightarrow>\<^bsub>[]\<^esub> (cms, mem)"
intros_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "CFG.path sourcenode targetnode valid_edge n [] n"
tt_seq: fixes p :: "'a"   and x :: "'a"   and y :: "'a"   and q :: "'a" shows "(\<exists>r. \<lparr>p\<rparr>x\<lparr>r\<rparr> \<and> \<lparr>r\<rparr>y\<lparr>q\<rparr>) = \<lparr>p\<rparr>x \<cdot> y\<lparr>q\<rparr>"
omega_coinduct_eq: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y = z + x \<cdot> y" shows "y \<le> x\<^sup>\<omega> + x\<^sup>\<star> \<cdot> z"
refines_def: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec" shows "refines R pi Sa Sc \<equiv> obs_consistent R pi Sa Sc \<and> PO_refines R Sa Sc"
add_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> S"   and "b \<in> S"   and "c \<in> S" shows "a \<oplus>\<^sub>o\<^sub>w b \<oplus>\<^sub>o\<^sub>w c = a \<oplus>\<^sub>o\<^sub>w (b \<oplus>\<^sub>o\<^sub>w c)"
fold_mset_empty: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a" shows "fold_mset f s {#} = s"
simps_5: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and h :: "'b"   and va :: "32 word" shows "heap_base.typeof_h typeof_addr h (Intg va) = typeof (Intg va)"
stieltjes_gamma_aux6: fixes k :: "nat" shows "(\<Sum>t<k. of_nat (k choose t) * Ln 2 ^ (k - t) * stieltjes_gamma t) - Ln 2 ^ (k + 1) / of_nat (k + 1) = (- 1) ^ k * fact k * (\<Sum>i = 1..k + 1. (- Ln 2) ^ i * A (k - (i - 1)) / fact i)"
comp_assoc: fixes h :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "CC.comp (CC.comp h g) f = CC.comp h (CC.comp g f)"
Always_def: fixes A :: "'c set" shows "Always A \<equiv> {F. Init F \<subseteq> A} \<inter> Stable A"
irreducibleI': fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "\<not> is_unit a"   and "\<And>b. b dvd a \<Longrightarrow> a dvd b \<or> is_unit b" shows "irreducible a"
path_is_forbidden: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent"   and es :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) stream" assumes "valid_in_port (v, p)"   and "ipath (edge_tree v p) es" shows "forbidden_path v es"
prems_1: shows "(a, b) \<in> set (arcs i j (x # xs))"
fmlookup_filter: fixes P :: "'e \<Rightarrow> bool"   and m :: "('e, 'd) fmap"   and x :: "'e" shows "fmlookup (fmfilter P m) x = (if P x then fmlookup m x else None)"
prv_prv_eql_sym: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "prv (eql t1 t2)" shows "prv (eql t2 t1)"
right_idem: fixes a :: "'a word"   and b :: "'a word" shows "signed.min (signed.min a b) b = signed.min a b"
assms_2: shows "\<Delta> : e'[y::=x] \<Down>\<^bsub>L\<^esub> \<Theta> : z"
Pi'_iff: fixes f :: "'d \<Rightarrow>\<^sub>F 'e"   and I :: "'d set"   and X :: "'d \<Rightarrow> 'e set" shows "(f \<in> Pi' I X) = (domain f = I \<and> (\<forall>i\<in>I. (f)\<^sub>F i \<in> X i))"
intros: fixes uu :: "nat" shows "isConst (Const uu)"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 (trg f) f"
one: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "R (arith_ops_record.one ops) (1::'c)"
absorb_iff1: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a \<sqinter> b = a)"
push_bit_xor: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "push_bit n (xor a b) = xor (push_bit n a) (push_bit n b)"
2: shows "\<forall>a\<in>{(i, t, s)} \<union> set D \<union> set Di. case a of (j, p) \<Rightarrow> \<forall>a\<in>{(i, t, s)} \<union> set D \<union> set Di. case a of (k, q) \<Rightarrow> (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"
safe_delayed: shows "\<O>s,\<R>s,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
assms: shows "\<rho> f|` (S \<union> fv \<Gamma> - domA \<Gamma>) = \<rho>' f|` (S \<union> fv \<Gamma> - domA \<Gamma>)"
nb: shows "op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)"
ev_wait_all_respects_policy: fixes tid :: "thread_id_t"   and u :: "partition_id_t"   and s :: "state_t"   and ipt :: "int_point_t"   and ev_wait_stage :: "ev_wait_stage_t" assumes "\<not> Policy.ifp (Step.partition tid) u"   and "atomic_step_invariant s"   and "atomic_step_precondition s tid ipt"   and "ipt = SK_EV_WAIT ev_wait_stage EV_CONSUME_ALL" shows "vpeq u s (atomic_step_ev_wait_all tid s)"
assms_1: shows "compare_expansions F G = (EQ, c, c')"
eq: shows "range (flow0 (flow0 y t)) = flow0 (flow0 y t) ` {0..t' - t}"
vertex_3_list_def: shows "vertex_3 \<equiv> [vertex_1, vertex_1]"
C_def: shows "C = Max {\<bar>u i\<bar> |i. i \<le> 2 * n}"
eq_add_iff1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and e :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.ring plus zero minus uminus times" shows "(plus (times a e) c = plus (times b e) d) = (plus (times (minus a b) e) c = d)"
is_cnf_encode_problem: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (\<Phi> \<Pi> t)"
secret_object1_def: shows "\<iota> = (SOME x. True)"
AE_count_space: fixes P :: "'k \<Rightarrow> bool"   and A :: "'k set" shows "almost_everywhere (count_space A) P = (\<forall>x\<in>A. P x)"
desargues_config_not_col_8: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col A C C'"
dghm_id_ArrMap_app: fixes x :: "V"   and \<CC> :: "V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "dghm_id \<CC>\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = x"
exhaust: fixes y :: "safety" obtains "y = Strict"   | "y = Lax"
ab_group_add_axioms: shows "class.ab_group_add (+) (0::'d) (-) uminus"
simps: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" shows "a1 \<rhd> a2 \<approx>\<^sub>w a3 = (\<exists>\<Psi> P Q. a1 = \<Psi> \<and> a2 = P \<and> a3 = Q \<and> \<Psi> \<rhd> P \<lessapprox>\<^sub>w<weakenBisim> Q \<and> \<Psi> \<rhd> P \<leadsto>\<^sub>w<weakenBisim> Q \<and> (\<forall>\<Psi>'. \<Psi> \<otimes> \<Psi>' \<rhd> P \<approx>\<^sub>w Q) \<and> \<Psi> \<rhd> Q \<approx>\<^sub>w P)"
zdvd_antisym_abs: fixes a :: "int"   and b :: "int" assumes "a dvd b"   and "b dvd a" shows "\<bar>a\<bar> = \<bar>b\<bar>"
Class_widen_Class: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" shows "P \<turnstile> Class C \<le> Class D = P \<turnstile> C \<preceq>\<^sup>* D"