hcju/isabelleps huggingface dataset:
dom_object: fixes f :: "'a" assumes "f \<in> Ar" shows "Dom f \<in> Ob"
list_update_code_1: fixes i :: "nat"   and y :: "'b" shows "[][i := y] = []"
all_Left: shows "list_all (\<lambda>x. x = dir.Left) p"
ctr: fixes p :: "'d spmf" shows "lift_spmf p = GPV (map_spmf Pure p)"
trivial_group_alt: shows "trivial_group G = (\<exists>a. carrier G \<subseteq> {a})"
unit_factor_mult: fixes a :: "'a"   and b :: "'a" shows "unit_factor (a * b) = unit_factor a * unit_factor b"
real_root_strict_increasing: fixes n :: "nat"   and N :: "nat"   and x :: "real" assumes "0 < n"   and "n < N"   and "0 < x"   and "x < 1" shows "root n x < root N x"
reflcl_supteq: shows "{\<unrhd>}\<^sup>= = {\<unrhd>}"
1: fixes x :: "nat"   and xs :: "nat list"   and s :: "nat" assumes "Max (set a) < x" shows "cond2 a b (x # xs) s = False"
contg': fixes x :: "real" assumes "a < ereal x"   and "ereal x < b" shows "isCont g' x"
usubstappt_func: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and f :: "char"   and r :: "trm"   and U :: "variable set"   and \<theta> :: "trm"   and \<sigma>\<theta> :: "trm" assumes "SFuncs \<sigma> f = Aterm r"   and "FVT r \<inter> U = {}"   and "usubstappt \<sigma> U \<theta> = Aterm \<sigma>\<theta>" shows "usubstappt \<sigma> U (trm.Func f \<theta>) = usubstappt (dotsubstt \<sigma>\<theta>) {} r"
good_qAbs_pickX: shows "qGoodAbs (qAbs xs x (pick X))"
Theorem_9_stable_Xd1': shows "Theorem_9.stable_on UNIV {X3.Xd1'}"
axioms_2: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and order :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "backward_simulation step1 step2 final1 final2 order match" shows "semantics step2 final2"
mapping_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.mapping f = (\<forall>i. \<exists>j. f (i, j) = top \<and> (\<forall>k. j \<noteq> k \<longrightarrow> f (i, k) = bot))"
tails_neq_Nil: fixes xs :: "['b]" shows "tails\<cdot>xs \<noteq> []"
fresh_psubst_E2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho"   and "rho ys y = Some Y"   and "fresh zs z X #[rho]" shows "fresh ys y X \<or> fresh zs z Y"
safe_regex_mono: fixes r :: "'a Regex.regex"   and safe :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and safe' :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and fv :: "'a \<Rightarrow> 'b set"   and m :: "modality"   and g :: "safety" assumes "\<And>g x. \<lbrakk>x \<in> regex.atms r; safe g x\<rbrakk> \<Longrightarrow> safe' g x"   and "Regex.safe_regex fv safe m g r" shows "Regex.safe_regex fv safe' m g r"
poly_smult: fixes a :: "'c"   and p :: "'c poly"   and x :: "'c" shows "poly (Polynomial.smult a p) x = a * poly p x"
inject_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(Cardholder x2 = Cardholder y2) = (x2 = y2)"
pseudo_divmod_def: fixes p :: "'a poly"   and q :: "'a poly" shows "pseudo_divmod p q \<equiv> if q = 0 then (0, p) else pseudo_divmod_main (Polynomial.lead_coeff q) 0 p q (Polynomial.degree p) (1 + length (Polynomial.coeffs p) - length (Polynomial.coeffs q))"
one_closed: shows "\<one>\<^bsub>tetrahedral_group\<^esub> \<in> carrier tetrahedral_group"
subcocycle_u_Tinv: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "I.subcocycle u"   and "I.invertible_qmpt" shows "mpt.subcocycle M I.Tinv (\<lambda>n x. u n ((I.Tinv ^^ n) x))"
simps_2: fixes \<sigma> :: "'tp"   and al :: "univ list" shows "GE_intF (Wit \<sigma>) al = pickT \<sigma>"
vars_sum: fixes f :: "'b \<Rightarrow> 'a mpoly"   and A :: "'b set" shows "vars (sum f A) \<subseteq> (\<Union>x\<in>A. vars (f x))"
inv_comp: fixes f :: "'b"   and g :: "'b" assumes "D.iso f"   and "D.iso g"   and "D.seq g f" shows "D.inv (g \<cdot>\<^sub>D f) = D.inv f \<cdot>\<^sub>D D.inv g"
ptrm_alpha_equiv_symmetric: fixes X :: "'b ptrm"   and Y :: "'b ptrm" assumes "X \<approx> Y" shows "Y \<approx> X"
subt_4: fixes C :: "('a, 'b) ctxt" assumes "C \<noteq> \<box>"   and "Fun f ss = C\<langle>t\<rangle>" shows "thesis"
agree_func: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and var :: "'e"   and args :: "'d \<Rightarrow> ('e, 'd) trm"   and i :: "'d" assumes "Vagree \<nu> \<nu>' (FVDiff ($f var args))" shows "Vagree \<nu> \<nu>' (FVDiff (args i))"
assms_3: shows "\<exists>es. char_poly D = (\<Prod>a\<leftarrow>es. [:- a, 1::'a:])"
vrange_VLambda: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "\<R>\<^sub>\<circ> (VLambda A f) = set (f ` elts A)"
flatten_plus: fixes p :: "'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b" shows "flatten (p + q) = flatten p + flatten q"
deg_an: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "deg R S X p = an (deg_n R S X p)"
ring_simprules_14: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R" shows "x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y"
Dependent_def: fixes x :: "e" shows "Dependent x \<equiv> mexistsActB ((dependsOn) x)"
Always_at_FP: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> A \<longmapsto>w B"   and "all_total F" shows "F \<in> Always (- (FP F \<inter> A \<inter> - B))"
neg: fixes S :: "'n set"   and s' :: "'n set" assumes "S \<in> \<D>"   and "s' \<in> \<D>"   and "S \<noteq> s'" shows "negligible (S \<inter> s')"
ord_iff_adds_univariate: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t \<in> .[X]" shows "(s \<preceq> t) = (s adds t)"
lit_ord_rhs: fixes t1 :: "'a trm"   and t2 :: "'a trm"   and x1 :: "'a literal"   and s :: "'a trm"   and p :: "sign"   and x2 :: "'a literal" assumes "(t1, t2) \<in> trm_ord"   and "orient_lit x1 s t1 p"   and "orient_lit x2 s t2 p"   and "vars_of_lit x1 = {}"   and "vars_of_lit x2 = {}" shows "(x1, x2) \<in> lit_ord"
weakBisimulationCoinductAux: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<(X \<union> weakBisimulation)> Q \<and> (Q, P) \<in> X" shows "P \<approx> Q"
strict_monoidal_extension_def: fixes C\<^sub>0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>C :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>C :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>C :: "'d"   and D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and T\<^sub>D :: "'e \<times> 'e \<Rightarrow> 'e"   and \<alpha>\<^sub>D :: "'e \<times> 'e \<times> 'e \<Rightarrow> 'e"   and \<iota>\<^sub>D :: "'e"   and I :: "'b \<Rightarrow> 'd"   and V :: "'b \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e" shows "strict_monoidal_extension C\<^sub>0 C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D I V F \<equiv> (category C\<^sub>0 \<and> monoidal_category C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C \<and> monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D) \<and> (functor C\<^sub>0 C I \<and> functor C\<^sub>0 D V) \<and> strict_monoidal_functor C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F \<and> strict_monoidal_extension_axioms C\<^sub>0 I V F"
vproduct_vsingleton_iff: fixes f :: "V"   and c :: "V"   and A :: "V \<Rightarrow> V" shows "(f \<in>\<^sub>\<circ> vproduct (ZFC_in_HOL.set {c}) A) = (f\<lparr>c\<rparr> \<in>\<^sub>\<circ> A c \<and> f = ZFC_in_HOL.set {\<langle>c, f\<lparr>c\<rparr>\<rangle>})"
rtrancl_beta_Upd: fixes u :: "sterm"   and u' :: "sterm"   and L :: "char list set"   and t :: "sterm"   and t' :: "sterm"   and l :: "Label" assumes "u ->> u'"   and "finite L"   and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> ->> t'' \<and> t' = \<sigma>[s,p] t'')"   and "lc u"   and "body t" shows "Upd u l t ->> Upd u' l t'"
lunit_coherence: fixes f :: "'a" assumes "C.ide f" shows "\<l>\<^sub>D[F f] = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f)"
rangeI: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "f x \<in> range f"
distinct_71: fixes x8 :: "'a"   and x9 :: "('a, String.literal) phantom" shows "\<langle>x8\<rangle>\<^sub>\<T> \<noteq> Enum x9"
multiset_concat: fixes xs :: "'c list list" shows "mset (concat xs) = fold (\<lambda>x M. M + mset x) xs {#}"
open_uniformity: fixes U :: "'a set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
Pi_pmf_insert: fixes A :: "'a set"   and x :: "'a"   and dflt :: "'b"   and p :: "'a \<Rightarrow> 'b pmf" assumes "finite A"   and "x \<notin> A" shows "Pi_pmf (insert x A) dflt p = map_pmf (\<lambda>(y, f). f(x := y)) (pair_pmf (p x) (Pi_pmf A dflt p))"
subset_spec_def: fixes X :: "'c set"   and Y :: "'c set" shows "subset_spec X Y = SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> Y)"
no_var_subterm: fixes t :: "('c, 'd) Term.term"   and v :: "'d" shows "\<not> t \<sqsubset> Var v"
1: fixes x :: "'a" assumes "x \<in> B" shows "norm x = 1"
complexes: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and Y :: "'b set set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "Y \<in> \<A>" shows "ChamberComplex Y"
Append: shows "\<turnstile> Append  \<lbrace>\<acute>Proper\<rbrace>"
lt_max_keys: fixes v :: "'x option \<Rightarrow>\<^sub>0 nat"   and p :: "('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "v \<in> keys p" shows "pp_pm.extended_ord v (pp_pm.extended_ord.lpp p)"
\<R>\<^sub>3_weak_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "weak_bisim (\<R>\<^sub>3 \<Gamma>') (\<R> \<Gamma>')"
F1: shows "R1 \<Longrightarrow>\<^sub>A P x p * F1"
f1: shows "reset_trap_val s = False \<and> get_ET (cpu_reg_val PSR s) \<noteq> 0"
inject_1: fixes x1 :: "instr"   and y1 :: "instr" shows "(StepI x1 = StepI y1) = (x1 = y1)"
less_trans: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" assumes "x \<subset># y"   and "y \<subset># z" shows "x \<subset># z"
less_TC_succ: fixes x :: "V"   and \<beta> :: "V" shows "(x \<sqsubset> ZFC_in_HOL.succ \<beta>) = (x \<sqsubset> \<beta> \<or> x = \<beta>)"
lower_semantics: fixes p :: "'a pfm"   and M :: "('a, 'b) kripke"   and w :: "'b" assumes "static p" shows "(M, w \<Turnstile> lower p) = (M, w \<Turnstile>\<^sub>! p)"
representsD2: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and r :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" assumes "pmdl.represents B r x" shows "x = pmdl.rep r"
sign_compose: fixes p :: "'d \<Rightarrow> 'd"   and q :: "'d \<Rightarrow> 'd" assumes "permutation p"   and "permutation q" shows "sign (p \<circ> q) = sign p * sign q"
over_tagged_division_lemma: fixes p :: "('c \<times> 'c set) set"   and i :: "'c set"   and d :: "'c set \<Rightarrow> 'd" assumes "p tagged_division_of i"   and "\<And>u v. box u v = {} \<Longrightarrow> d (cbox u v) = (0::'d)" shows "(\<Sum>(uu_, k)\<in>p. d k) = sum d (snd ` p)"
supp_subst_eq: fixes e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var" shows "supp e[y::=x] = supp e - {atom y} \<union> (if atom y \<in> supp e then {atom x} else {})"
simps_2: fixes e :: "'c"   and a :: "'d"   and t :: "('c, 'd) SkewBinomialTree" shows "local.insert e a [t] = [Node e a 0 [], t]"
invar_meldUniq: fixes q :: "('c, 'd) SkewBinomialTree list"   and q' :: "('c, 'd) SkewBinomialTree list" assumes "queue_invar q"   and "queue_invar q'" shows "queue_invar (meldUniq q q')"
pcj: shows "simple_proto_conjunct (proto m1) (proto m2) = Some (proto m)"
is_subset: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "ind_topology X is_open S" shows "S \<subseteq> X"
leading_coeff_0_iff: fixes p :: "'a poly" shows "(lead_coeff p = (0::'a)) = (p = 0)"
ccw_translate: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" shows "ccw p (p + q) (p + r) = ccw 0 q r"
finite_rtrancl_delta_Image1: fixes A :: "'a set"   and r :: "'a rexp" shows "finite ({(r, s). \<exists>a\<in>A. s = delta a r}\<^sup>* `` {init r})"
pj_Hom: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "pj R I \<in> rHom R (R /\<^sub>r I)"
prems_5: fixes i' :: "nat" assumes "i' < length (y # xs)"   and "\<not> tr_lookup T (i' + jj) ii" shows "(y # xs) ! i' = None"
a_star_NotL: fixes M :: "trm"   and M' :: "trm"   and a :: "coname"   and x :: "name" assumes "M \<longrightarrow>\<^sub>a* M'" shows "NotL <a>.M x \<longrightarrow>\<^sub>a* NotL <a>.M' x"
zunion_zempty_left: fixes a :: "'b zet" shows "zunion zempty a = a"
ldrop_finT: fixes t :: "'b llist"   and A :: "'b set"   and i :: "nat" assumes "t \<in> A\<^sup>\<star>" shows "t \<up> i \<in> A\<^sup>\<star>"
diff_numeral_special_2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) one = neg_numeral.sub minus plus one m num.One"
eq2: shows "adjoint (tensor_P m1 m2) * tensor_P m3 m4 = tensor_P (adjoint m1 * m3) (adjoint m2 * m4)"
prod_list_pos: fixes xs :: "real list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> 0 < x" shows "0 < prod_list xs"
N: shows "\<langle>la, a, ra\<rangle> \<in> subtrees t"
cwiseext_wf: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. cwiseext gt ys xs)"
dghm_dag_Rel_ArrMap_vdomain: fixes \<alpha> :: "V" shows "\<D>\<^sub>\<circ> (\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>) = dg_Rel \<alpha>\<lparr>Arr\<rparr>"
jordan_block_pow: fixes n :: "nat"   and a :: "'b"   and r :: "nat" shows "jordan_block n a ^\<^sub>m r = mat n n (\<lambda>(i, j). if i \<le> j then of_nat (r choose (j - i)) * a ^ (r + i - j) else (0::'b))"
simps_5: fixes f1 :: "vname \<Rightarrow> 'a"   and f2 :: "sig \<Rightarrow> 'a"   and x1 :: "vname" shows "(case fid x1 of fid x \<Rightarrow> f1 x | mid x \<Rightarrow> f2 x) = f1 x1"
ex1_iff_ex_Uniq: fixes P :: "'b \<Rightarrow> bool" shows "(\<exists>!x. P x) = ((\<exists>x. P x) \<and> (\<exists>\<^sub>\<le>\<^sub>1 x. P x))"
induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and x1 :: "'c rule list"   and x2 :: "state"   and x3 :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>x1, x2\<rangle> \<Rightarrow> x3"   and "\<And>t. P [] t t"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Accept] Undecided (Decision FinalAllow)"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Drop] Undecided (Decision FinalDeny)"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Reject] Undecided (Decision FinalDeny)"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Log] Undecided Undecided"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Empty] Undecided Undecided"   and "\<And>m a. \<not> matches \<gamma> m p \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>rs X. P rs (Decision X) (Decision X)"   and "\<And>rs\<^sub>1 t rs\<^sub>2 t'. \<lbrakk>\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t; P rs\<^sub>1 Undecided t; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'; P rs\<^sub>2 t t'; no_matching_Goto \<gamma> p rs\<^sub>1\<rbrakk> \<Longrightarrow> P (rs\<^sub>1 @ rs\<^sub>2) Undecided t'"   and "\<And>m chain rs\<^sub>1 m' rs\<^sub>2. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2); matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided; P rs\<^sub>1 Undecided Undecided; no_matching_Goto \<gamma> p rs\<^sub>1\<rbrakk> \<Longrightarrow> P [Rule m (Call chain)] Undecided Undecided"   and "\<And>m chain rs t. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t; P rs Undecided t\<rbrakk> \<Longrightarrow> P [Rule m (Call chain)] Undecided t"   and "\<And>m chain rs X rest. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision X; P rs Undecided (Decision X)\<rbrakk> \<Longrightarrow> P (Rule m (Goto chain) # rest) Undecided (Decision X)"   and "\<And>m chain rs rest. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided; P rs Undecided Undecided\<rbrakk> \<Longrightarrow> P (Rule m (Goto chain) # rest) Undecided Undecided" shows "P x1 x2 x3"
pcompose_prod: fixes f :: "'b \<Rightarrow> 'a poly"   and A :: "'b set"   and p :: "'a poly" shows "prod f A \<circ>\<^sub>p p = (\<Prod>i\<in>A. f i \<circ>\<^sub>p p)"
fbox_def: fixes x :: "'b ppath set"   and y :: "'b ppath set" shows "wp x y = pp_a (x ; pp_a y)"
fold_insert: fixes A :: "'a\<^sub>m set"   and x :: "'a\<^sub>m"   and z :: "'a\<^sub>m option" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z (insert x A) = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z A \<bind> m x"
AtWhileExpr: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and p :: "('e, 'd, 'f) ann"   and e :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and r :: "'e set"   and i :: "'e set"   and b :: "'e set" assumes "atomicsR \<Gamma> \<Theta> p e a" shows "atomicsR \<Gamma> \<Theta> (AnnWhile r i p) (While b e) a"
Lim_emeasure_decseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "decseq A"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Inter> (range A))"
induct: fixes \<Gamma> :: "('a, 'b) web_scheme" assumes "loose \<Gamma>" obtains "\<And>f. \<lbrakk>current \<Gamma> f; wave \<Gamma> f\<rbrakk> \<Longrightarrow> f = (\<lambda>_. 0)" and "\<not> hindrance \<Gamma> (\<lambda>_. 0)"
Ke0_imp_inj: assumes "carrier (Berlekamp_subspace.vs kerT) = {\<zero>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub>}" shows "inj_on (\<lambda>v. vec n (\<lambda>i. coeff (v mod m i) 0)) (carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}))"
spr_simInit: fixes iobs :: "'obs"   and a :: "'a" assumes "iobs \<in> envObs a ` set envInit" shows "spr_simAbs (local.spr_simInit a iobs) = spr_sim ` rel_ext (\<lambda>t'. t' \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jviewInit a iobs)"
distinct_1: fixes x1 :: "rat"   and x21 :: "nat"   and x22 :: "int poly \<times> rat \<times> rat" shows "Rational x1 \<noteq> Irrational x21 x22"
hyps_1: assumes "j + 2 < n" shows "d \<le> local.Dube_aux (j + 1)"
aA: shows "a \<sqinter> \<Sqinter>A \<in> carrier L"
p_2: shows "\<forall>x\<in>A. f x = p (f' x)"
intros: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and expr :: "'a expr"   and expr\<^sub>N :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile> expr \<Rrightarrow> expr\<^sub>N"   and "\<Gamma> \<turnstile>\<^sub>E expr\<^sub>N : \<tau>" shows "\<Gamma> \<turnstile> expr : \<tau>"
Quotient_dnelist: shows "Quotient (eq_onp (\<lambda>xs. distinct xs \<and> xs \<noteq> [])) Abs_dnelist list_of_dnelist cr_dnelist"
append_\<tau>exec_moves: fixes vs :: "'addr val list"   and es' :: "'addr expr1 list"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "length vs = length es'"   and "\<tau>exec_moves ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_moves ci P t (es' @ es) h (stk @ vs, loc, length (compEs2 es') + pc, xcp) (stk' @ vs, loc', length (compEs2 es') + pc', xcp')"
App_NF_D: fixes n :: "nat"   and ts :: "dB list" assumes "NF (Var n \<degree>\<degree> ts)" shows "listall NF ts"
homotopic_paths_rid: fixes p :: "real \<Rightarrow> 'b"   and s :: "'b set" assumes "path p"   and "path_image p \<subseteq> s" shows "homotopic_paths s (p +++ linepath (pathfinish p) (pathfinish p)) p"
assms_6: shows "E.Trg (Dom F) = E.Trg (Cod F)"
sum_array_def: fixes a :: "rat array"   and b :: "rat array" shows "sum_array a b = zipWithA (+) a b \<bind> (\<lambda>_. return ())"
ordinal_exp_strict_monoR: fixes x :: "ordinal"   and y :: "ordinal"   and y' :: "ordinal" assumes "oSuc 0 < x"   and "y < y'" shows "x ** y < x ** y'"
simps_2: fixes G :: "'b list_graph"   and M :: "'b SecurityInvariant list"   and accu :: "('b \<times> 'b) list"   and e :: "'b \<times> 'b"   and Es :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G M accu (e # Es) = (if TopoS_Composition_Theory_impl.all_security_requirements_fulfilled (TopoS_Composition_Theory_impl.get_IFS M) (stateful_list_policy_to_list_graph \<lparr>hostsL = nodesL G, flows_fixL = edgesL G, flows_stateL = e # accu\<rparr>) then TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G M (e # accu) Es else TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G M accu Es)"
intro: fixes bisim :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'h \<Rightarrow> bool" assumes "\<And>s1 s2. bisim s1 s2 \<Longrightarrow> final1 s1 = final2 s2" shows "bisimulation_final_axioms bisim final1 final2"
mset_neg_after_summary: fixes M :: "'t zmultiset"   and S :: "'sum antichain" assumes "mset_neg M = {#}" shows "mset_neg (after_summary M S) = {#}"
code: fixes \<omega> :: "(real \<times> 'd) stream"   and j :: "real"   and \<omega>' :: "(real \<times> 'd) stream" shows "merge_at \<omega> j \<omega>' = (case \<omega> of (t, s) ## \<omega> \<Rightarrow> if t \<le> j then (t, s) ## merge_at \<omega> j \<omega>' else \<omega>')"
simps_1: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and p :: "real \<times> real \<times> real" shows "substInfinitesimalQuadraticUni a b c d (EqUni p) = allZero' p"
arrI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "D.arr f"
Sigma_mset_plus_distrib1: fixes A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b \<Rightarrow> 'c multiset" shows "Sigma_mset (A + B) C = Sigma_mset A C + Sigma_mset B C"
mult_isol_var: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b"   and x :: "'b"   and v :: "'b"   and y :: "'b" assumes "class.pre_dioid plus times less_eq less"   and "less_eq u x"   and "less_eq v y" shows "less_eq (times u v) (times x y)"
XXX_1: shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"