hcju/isabelleps huggingface dataset:
map_filter: fixes f :: "'c \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool"   and xs :: "'c list" shows "map f (filter (\<lambda>x. P (f x)) xs) = filter P (map f xs)"
model: assumes "infinite (deriv s)"   and "init s" shows "\<forall>A m n. contains f n (m, A) \<longrightarrow> \<not> FEval mo ntou A"
eventually_def: fixes p :: "'a" shows "\<diamond> p = Sup (range ((!!) p))"
1: shows "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go (C.dom f)) (C.cod f) (f \<cdot>\<^sub>C \<epsilon>o (C.dom f))"
subtypings_trans: fixes CT :: "nat \<Rightarrow> classDef option"   and Cs :: "nat list"   and Ds :: "nat list"   and Es :: "nat list" assumes "CT \<turnstile>+ Cs <: Ds"   and "CT \<turnstile>+ Ds <: Es" shows "CT \<turnstile>+ Cs <: Es"
is_top_sorted_append: fixes R :: "('a \<times> 'a) set"   and l1 :: "'a list"   and l2 :: "'a list" shows "is_top_sorted R (l1 @ l2) = (set l1 \<times> set l2 \<inter> (R\<^sup>*)\<inverse> = {} \<and> is_top_sorted R l1 \<and> is_top_sorted R l2)"
simulation_Wakeup1: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x1' :: "'x1"   and m1' :: "'m1" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "x1 \<approx>w x2"   and "r1.r_syntax t x1 m1 ta1 x1' m1'"   and "Notified \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub> \<or> WokenUp \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub>" shows "\<exists>ta2 x2' m2'. r2.r_syntax t x2 m2 ta2 x2' m2' \<and> t \<turnstile> (x1', m1') \<approx> (x2', m2') \<and> ta1 \<sim>m ta2"
entropy_def: fixes b :: "real"   and S :: "'c measure"   and X :: "'a \<Rightarrow> 'c" shows "entropy b S X = - KL_divergence b S (distr M S X)"
open_Un: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open"   and "open S"   and "open T" shows "open (S \<union> T)"
smc_0_def: shows "smc_0 = [[]\<^sub>\<circ>, []\<^sub>\<circ>, []\<^sub>\<circ>, []\<^sub>\<circ>, []\<^sub>\<circ>]\<^sub>\<circ>"
rm_vars_fv\<^sub>s\<^sub>e\<^sub>t_subst: fixes x :: "'c"   and X :: "'c set"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and Y :: "'c set" assumes "x \<in> fv\<^sub>s\<^sub>e\<^sub>t (rm_vars X \<theta> ` Y)" shows "x \<in> fv\<^sub>s\<^sub>e\<^sub>t (\<theta> ` Y) \<or> x \<in> X"
eq: shows "xs ! n = e_nth (e2xs (encode_config c)) n"
make_mat_def: fixes n :: "nat"   and k :: "nat"   and B :: "'a mat" shows "make_mat n k B = Matrix.mat n n (\<lambda>(i, j). if i = 0 \<and> j = 0 then B $$ (0, 0) else if i = 0 \<and> j = k then B $$ (0, 1) else if i = k \<and> j = 0 then B $$ (1, 0) else if i = k \<and> j = k then B $$ (1, 1) else if i = j then 1::'a else (0::'a))"
Infinitesimal_approx_minus: fixes x :: "'a star"   and y :: "'a star" shows "(x - y \<in> Infinitesimal) = (x \<approx> y)"
protCl_resOf_arOf: fixes f :: "'fsym"   and i :: "nat" assumes "protCl (resOf f)"   and "i < length (arOf f)" shows "protCl (arOf f ! i)"
parAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<sim>\<^sub>e P \<parallel> Q \<parallel> R"
subst_self_idemp: fixes x :: "'a"   and t :: "('b, 'a) Term.term" assumes "x \<notin> vars_term t" shows "subst x t \<circ>\<^sub>s subst x t = subst x t"
C7: shows "rel_weakly_respects_barbs Rel (STCalWB SWB TWB)"
prems_7: shows "\<forall>l s1 s2. s1 = s2 on LVE \<longrightarrow> postList upds l s1 = postList upds l s2"
decompose_Max: fixes A :: "'a set" assumes "A \<subseteq> carrier" shows "restrict_relation A le = (\<lambda>x y. x \<in> A \<and> y \<in> Max_wrt_among le A \<or> y \<notin> Max_wrt_among le A \<and> restrict_relation (A - Max_wrt_among le A) le x y)"
iex_triv: fixes I :: "nat set"   and P :: "bool" shows "(\<diamond> t I. P) = ((\<exists>t. t \<in> I) \<and> P)"
strongSimWeakSim: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>[Rel] Q" shows "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
stOf_castOp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "stOf (fst3 (castOp s X)) = s"
e0_def: shows "e0 \<equiv> finfun_const 0"
diffs_def: fixes c :: "nat \<Rightarrow> 'a" shows "diffs c = (\<lambda>n. of_nat (Suc n) * c (Suc n))"
add_increasing2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero c"   and "less_eq b a" shows "less_eq b (plus a c)"
abs_mult: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (times a b) = times (abs a) (abs b)"
lalternate_llist_of: fixes xs :: "'b list" shows "lalternate (llist_of xs) = llist_of (alternate xs)"
encD_ex: fixes x :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list" assumes "x \<in> enc (w, I)"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "\<exists>n. x = map2 (enc_atom I) [0..<n] (stake n (w @- sconst any))"
subseteq_guards_Skip: fixes c :: "('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g Skip" shows "c = Skip"
add_diff_assoc2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b - a + c = b + c - a"
f: shows "f \<in> X \<rightarrow> carrier_vec n"
subformulas\<^sub>\<nu>_subfrmlsn: fixes \<phi> :: "'b ltln" shows "subformulas\<^sub>\<nu> \<phi> \<subseteq> subfrmlsn \<phi>"
components_eq_empty: fixes s :: "'c set" shows "(components s = {}) = (s = {})"
simps_2: fixes e :: "aexp"   and S :: "'a st" shows "aval'' e (Some S) = aval' e S"
app_hunion2: fixes x :: "hf"   and f :: "hf"   and g :: "hf" assumes "x \<^bold>\<notin> hdomain f" shows "app (f \<squnion> g) x = app g x"
admits_tupling_def: fixes I :: "'a set" shows "admits_tupling I \<equiv> \<exists>\<pi>. \<pi> \<in> (I \<rightarrow> Univ) \<inter> extensional' I \<rightarrow> Univ \<and> inj_on \<pi> ((I \<rightarrow> Univ) \<inter> extensional' I)"
kmp1_refine: fixes s :: "'a list"   and t :: "'a list" shows "kmp1 s t \<le> kmp s t"
lQuot_rQuot: fixes a :: "'b"   and b :: "'b"   and A :: "'b list set" shows "lQuot a (rQuot b A) = rQuot b (lQuot a A)"
subcategory_VVV: shows "subcategory VxVxV.comp (\<lambda>\<tau>\<mu>\<nu>. arr (fst \<tau>\<mu>\<nu>) \<and> VV.arr (snd \<tau>\<mu>\<nu>) \<and> src (fst \<tau>\<mu>\<nu>) = trg (fst (snd \<tau>\<mu>\<nu>)))"
factorization_lattice_def: fixes u :: "'a poly"   and k :: "nat"   and m :: "'a" shows "factorization_lattice u k m \<equiv> map (\<lambda>i. vec_of_poly_n (u * monom (1::'a) i) (degree u + k)) [k>..0] @ map (\<lambda>i. vec_of_poly_n (monom m i) (degree u + k)) [degree u>..0]"
r_minus: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "x \<otimes>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub> y = \<ominus>\<^bsub>S\<^esub> (x \<otimes>\<^bsub>S\<^esub> y)"
induct: fixes P :: "'b hp list \<Rightarrow> bool"   and a0 :: "'b hp list" assumes "P []"   and "\<And>h. P [h]"   and "\<And>h1 h2 hs. (\<And>x. x = link h1 h2 \<Longrightarrow> P hs) \<Longrightarrow> P (h1 # h2 # hs)" shows "P a0"
ffact_nat_triv: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "ffact n m = 0"
min_def_raw: shows "min \<equiv> \<lambda>a b. if a \<le> b then a else b"
Seq: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c\<^sub>1' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and c\<^sub>2 :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')" shows "\<Gamma>\<turnstile> (Seq c\<^sub>1 c\<^sub>2, s) \<rightarrow> (Seq c\<^sub>1' c\<^sub>2, s')"
subprob_not_empty: fixes s :: "'s" shows "space (K.T s) \<noteq> {}"
simps_1: fixes i :: "nat" shows "[i..<0] = []"
jumpF_polyR_coprime: fixes p :: "real poly"   and q :: "real poly"   and x :: "real" assumes "coprime p q" shows "jumpF_polyR q p x = (if p \<noteq> 0 \<and> q \<noteq> 0 \<and> poly p x = 0 then if sign_r_pos p x = (0 < poly q x) then 1 / 2 else - 1 / 2 else 0)"
reflection_and_closures_1: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> bool" assumes "rel_reflects_pred Rel Pred" shows "rel_reflects_pred (Rel\<^sup>=) Pred"
inject_3: fixes x31 :: "'b aexp"   and x32 :: "'b aexp"   and y31 :: "'b aexp"   and y32 :: "'b aexp" shows "(Gt x31 x32 = Gt y31 y32) = (x31 = y31 \<and> x32 = y32)"
same_cardinality_post_swap_1: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and j :: "nat"   and i :: "nat"   and n :: "nat" assumes "computation channel trans send recv init final"   and "computation.prerecording_event channel trans send recv init t j"   and "computation.postrecording_event channel trans send recv init t i"   and "i < j"   and "j < length t"   and "computation.count_violations channel trans send recv init t = Suc n"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "distributed_system.trace channel trans send recv init t final" shows "{k \<in> {0..<i}. computation.prerecording_event channel trans send recv init t k} = {k \<in> {0..<i}. computation.prerecording_event channel trans send recv init (swap_events i j t) k}"
lconjby_surj: fixes x :: "'a" shows "surj (lconjby x)"
induct: fixes P :: "('e, 'f) FingerTreeStruc \<Rightarrow> bool"   and a0 :: "('e, 'f) FingerTreeStruc" assumes "P Empty"   and "\<And>a. P (Single a)"   and "\<And>uu_ a b m sf. P (Deep uu_ (Two a b) m sf)"   and "\<And>uv_ a b c m sf. P (Deep uv_ (Three a b c) m sf)"   and "\<And>uw_ a b c d m sf. P (Deep uw_ (Four a b c d) m sf)"   and "\<And>ux_ a m sf. P m \<Longrightarrow> P (Deep ux_ (One a) m sf)" shows "P a0"
simps: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and t :: "taylor_model" shows "tm_abs prec I a t = (let bound = compute_bound_tm prec I a t; abs_bound = Ivl 0 (max \<bar>lower bound\<bar> \<bar>upper bound\<bar>) in TaylorModel (mid abs_bound)\<^sub>p (centered abs_bound))"
has_contour_integral_shiftpath: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex"   and a :: "real" assumes "(f has_contour_integral i) g"   and "valid_path g"   and "a \<in> {0..1}" shows "(f has_contour_integral i) (shiftpath a g)"
sp_in_extends_not_using_sub: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v :: "'d \<times> nat"   and v' :: "'d \<times> nat"   and subs' :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat" assumes "wf_sub_rel_of g subs"   and "SubExt.extends g subs (v, v') subs'"   and "SubRel.subpath g v1 es v2 subs'"   and "\<not> uses_sub v1 es v2 (v, v')" shows "SubRel.subpath g v1 es v2 subs"
div_less_conv: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "0 < m" shows "(n div m < k) = (n < k * m)"
act_inv_act: fixes g :: "cltn2"   and b :: "proj2" assumes "g \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "apply_cltn2 (apply_cltn2 b (inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> g)) g = b"
If_then_TT: fixes e :: "tr"   and b :: "tr" shows "If b then TT else e = (b orelse e)"
ord_Zp_ultrametric_eq'': fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "ord_Zp y < ord_Zp x" shows "ord_Zp (y \<ominus> x) = ord_Zp y"
euclidean_interior_of: fixes S :: "'b set" shows "euclidean interior_of S = interior S"
D_T_DT: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq>\<^sub>D Q"   and "P \<sqsubseteq>\<^sub>T Q" shows "P \<sqsubseteq>\<^sub>D\<^sub>T Q"
not_Iic_eq_empty: fixes h :: "'b multiset" shows "subset_mset.atMost h \<noteq> {}"
y: shows "norm (y - x) * (1 - e) < e * d"
rule1: fixes vs :: "'a list"   and a :: "'a"   and ys :: "'a list" assumes "distinct vs"   and "filter (\<lambda>v. v = a \<or> v \<in> set ys) vs = a # ys" shows "filter (\<lambda>v. v \<in> set ys) vs = ys"
fmrelD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('c, 'a) fmap"   and n :: "('c, 'b) fmap"   and x :: "'c" assumes "fmrel P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
inv_one: shows "inv\<^bsub>H'\<^esub> \<one>\<^bsub>H'\<^esub> = \<one>\<^bsub>H'\<^esub>"
abs_ast_variable_section_def: shows "abs_ast_variable_section = [0..<length astDom]"
assms_1: shows "IO out c \<in> set_spmf (the_gpv gpv)"
Domainp_set: fixes T :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
hashable_bhc_is_bhc: fixes eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "('g \<times> 'g) set" assumes "STRUCT_EQ_tag eq (=)"   and "REL_FORCE_ID R" shows "is_bounded_hashcode R eq bounded_hashcode_nat"
q: fixes i :: "'a" assumes "i \<in> d" shows "q i = (Q)\<^sub>F i"
sup_finite_4_def: fixes x :: "finite_4"   and y :: "finite_4" shows "x \<squnion> y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>1, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>1, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>4 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (a\<^sub>4, xa) \<Rightarrow> a\<^sub>4)"
Union_iff: fixes A :: "'e"   and C :: "'e set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
r_zero: fixes x :: "int" assumes "x \<in> carrier \<Z>" shows "x \<oplus>\<^bsub>\<Z>\<^esub> \<zero>\<^bsub>\<Z>\<^esub> = x"
lprefix_LNil: fixes xs :: "'c llist" shows "xs \<sqsubseteq> LNil = lnull xs"
meet_col_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and P :: "Points" assumes "meet_in (line A B) (line C D) P" shows "col C D P"
\<UU>\<bb>\<xx>_opubx_type: fixes opinl :: "'opinl"   and ts :: "Unboxed.type option list"   and opubx :: "'opubx" assumes "\<UU>\<bb>\<xx> opinl ts = Some opubx" shows "fst (\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> opubx) = ts"
PiE'_mono: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 's set"   and C :: "'a \<Rightarrow> 's set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi A B \<inter> extensional' A \<subseteq> Pi A C \<inter> extensional' A"
Buchberger_criterion_finite: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "finite F"   and "\<And>p q. \<lbrakk>p \<in> F; q \<in> F; p \<noteq> 0; q \<noteq> 0; p \<noteq> q; component_of_term (lt p) = component_of_term (lt q)\<rbrakk> \<Longrightarrow> (red F)\<^sup>*\<^sup>* (spoly p q) 0" shows "is_Groebner_basis F"
compact_def: fixes S :: "'c set" shows "compact S = (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
simps: fixes a1 :: "'state"   and a2 :: "(('act, 'pred) S_action, 'state) residual" shows "a1 \<rightarrow>\<^sub>S a2 = ((\<exists>P \<alpha> P'. a1 = P \<and> a2 = \<langle>S_action.Act \<alpha>,P'\<rangle> \<and> P \<rightarrow> \<langle>\<alpha>,P'\<rangle>) \<or> (\<exists>P \<phi>. a1 = P \<and> a2 = \<langle>S_action.Pred \<phi>,P\<rangle> \<and> P \<turnstile> \<phi>))"
span_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'd"   and S :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> phull S" shows "sum f A \<in> phull S"
saturated_model: fixes N :: "'a literal multiset set" assumes "saturated N"   and "{#} \<notin> N" shows "I_of N \<Turnstile>s N"
simps_8: fixes f :: "'e \<Rightarrow> 'd" shows "map_tree f \<langle>\<rangle> = \<langle>\<rangle>"
simps: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "deep_model Y r rs = Conv (Y, r) (deep_model' r rs)"
ipd': shows "ipd (\<pi> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> \<pi> m)) \<noteq> \<pi> m"
prod_simps_2: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "local.dom (local.prod f0 f1) = local.prod (local.dom f0) (local.dom f1)"
tr_def_3: shows "path M (xs || trX) (target (vs || trV) q1)"
geometric_sums: fixes c :: "'b" assumes "norm c < 1" shows "(^) c sums ((1::'b) / ((1::'b) - c))"
Bang: fixes P :: "pi"   and Rs :: "residual" assumes "P \<parallel> !P \<longmapsto> Rs" shows "!P \<longmapsto> Rs"
finite_Ord_omega: fixes \<alpha> :: "V" assumes "\<alpha> \<in>\<^sub>\<circ> \<omega>" shows "vfinite \<alpha>"
Bisim_Lang_eq: fixes as :: "'b list"   and ps :: "('b rexp set \<times> 'b rexp set) list"   and R :: "'b rexp set"   and S :: "'b rexp set" assumes "is_Bisimulation as ps"   and "(R, S) \<in> set ps" shows "Lang R = Lang S"
nxt: fixes q :: "hf"   and x :: "'a" assumes "q \<in> nfa.states (Reverse_nfa M)" shows "nfa.nxt (Reverse_nfa M) q x \<subseteq> nfa.states (Reverse_nfa M)"
ind: assumes "y \<le> x" shows "{\<lambda>nl. nl = xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ y # anything} [Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)] {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
simps_10: shows "get_axiom AdConst = diff_const_axiom"
T4_v2: shows "serial (\<lambda>w v. w r v \<and> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa w \<longrightarrow> xa x))) \<longrightarrow> universal (\<lambda>w v. w r v \<longrightarrow> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa w \<longrightarrow> xa x)))"
points_psubset: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "proper_sub_set_system \<U> \<A> \<V> \<B>" shows "\<U> \<subset> \<V>"
lang_rhs_set: fixes X :: "'b list set"   and P :: "'b rexp \<Rightarrow> bool" shows "lang_rhs {Trn X r |r. P r} = \<Union> {lang_trm (Trn X r) |r. P r}"
LIMSEQ_linear: fixes X :: "nat \<Rightarrow> 'a"   and x :: "'a"   and l :: "nat" assumes "X \<longlonglongrightarrow> x"   and "0 < l" shows "(\<lambda>n. X (n * l)) \<longlonglongrightarrow> x"
wf_Union_wf_init_segs: fixes R :: "('b \<times> 'b) set set" assumes "R \<in> Chains init_seg_of"   and "\<forall>r\<in>R. wf r" shows "wf (\<Union> R)"
exhaust: fixes y :: "('c, 'd) insert_message" obtains x1 :: "'c extended" and x2 :: "'c" and x3 :: "'c extended" and x4 :: "'d" where "y = InsertMessage x1 x2 x3 x4"
powers_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b + c = a + (b + c)"
assms_4: shows "c_k i j k' \<le> c_k i j k"
eoutKnowsECorrect_L1: fixes C :: "specID"   and e :: "Expression" assumes "eoutKnowsECorrect C e"   and "eout C e" shows "(\<exists>k. e = kE k \<and> k \<in> specKeys C) \<or> (\<exists>s. e = sE s \<and> s \<in> specSecrets C) \<or> knows C [e]"
FW_zone_equiv: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)" shows "[M]\<^bsub>v,n\<^esub> = [FW M n]\<^bsub>v,n\<^esub>"
inj_Inl: fixes A :: "'a set" shows "inj_on Inl A"
eqvtChainE: fixes perm :: "(name \<times> name) list"   and P :: "pi"   and P' :: "pi" assumes "perm \<bullet> P \<Longrightarrow>\<^sub>\<tau> perm \<bullet> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P'"
subalgebra_inter: fixes K :: "'a set"   and V :: "'a set"   and V' :: "'a set" assumes "subalgebra K V R"   and "subalgebra K V' R" shows "subalgebra K (V \<inter> V') R"
Collect_cong: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
diff: fixes v :: "'a" shows "(\<lambda>x. frechet_derivative f (at x) v) differentiable at a"
abs_eq: fixes xa :: "'a ltl"   and x :: "'a set" shows "af_G_abs.f_abs (Abs xa) x = Abs (af_G_letter xa x)"
closez_def: fixes s :: "char list"   and p :: "char list"   and t :: "sterm" shows "\<sigma>[s,p] t = {0 \<leftarrow> [s,p]} t"
children_eq_h: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "ptr' \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr" shows "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
mdijkstra_refines: shows "mdijkstra \<le> \<Down> (br \<alpha>r res_invarm) dijkstra'"
prod_map_map: fixes f1 :: "'g \<rightarrow> 'e"   and g1 :: "'h \<rightarrow> 'f"   and f2 :: "'i \<rightarrow> 'g"   and g2 :: "'j \<rightarrow> 'h"   and p :: "'i \<times> 'j" shows "prod_map\<cdot>f1\<cdot>g1\<cdot>(prod_map\<cdot>f2\<cdot>g2\<cdot>p) = prod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
d_IN_plus_web: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and \<Gamma> :: "('k, 'l) web_scheme"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and y :: "'k" shows "d_IN (f \<frown>\<^bsub>\<Gamma>\<^esub> g) y = d_IN f y + d_IN (g \<upharpoonleft> \<Gamma> / f) y"
entails_trans_strong: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N1 :: "'a set"   and N2 :: "'a set"   and N3 :: "'a set" assumes "consequence_relation Bot entails"   and "entails N1 N2"   and "entails (N1 \<union> N2) N3" shows "entails N1 N3"
refines_determ: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "determ (wp a)"   and "well_def a"   and "well_def b"   and "a \<sqsubseteq> b" shows "a \<simeq> b"
bchoice_Basis_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and A :: "real set"   and P :: "'a \<Rightarrow> real \<Rightarrow> bool" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(\<forall>i\<in>Basis. \<exists>x\<in>A. P i x) = (\<exists>x. \<forall>i\<in>Basis. inner x i \<in> A \<and> P i (inner x i))"
matrix_to_iarray_Gram_Schmidt_matrix: fixes A :: "((real, 'c) vec, 'd) vec" shows "matrix_to_iarray (Gram_Schmidt_matrix A) = Gram_Schmidt_matrix_iarrays (matrix_to_iarray A)"
Fix_1: fixes i :: "nat" assumes "i < lo" shows "xs' ! i = xs ! i"