hcju/isabelleps huggingface dataset:
cont_eff_ParT_BrnFT_R: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and jj :: "nat"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and s :: "'state"   and t :: "'state" assumes "length cl = length dl"   and "properL cl"   and "properL dl"   and "SbisL cl dl"   and "jj \<in> BrnFT dl cl"   and "\<And>n I i j. \<lbrakk>n < length cl; I \<in> P n; i \<in> I; j \<in> F n I\<rbrakk> \<Longrightarrow> eff (cl ! n) s i \<approx> eff (dl ! n) t j \<and> local.cont (cl ! n) s i \<approx>s local.cont (dl ! n) t j" shows "t \<approx> eff (ParT dl) t jj \<and> (ParT cl, local.cont (ParT dl) t jj) \<in> thetaZOParT"
elims: fixes x :: "('b, 'c) Combinators list"   and y :: "('b, 'c) Combinators list" assumes "removeShadowRules1 x = y" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "y = (if DenyAll \<in> set xs then removeShadowRules1 xs else xa # xs)"   | "x = []" and "y = []"
map_prod_def: fixes f :: "'h \<Rightarrow> 'j"   and g :: "'i \<Rightarrow> 'k" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
Separation_Kernel_axioms: shows "Separation_Kernel kstep output_f current cswitch kprecondition control kinvolved ifp vpeq"
not_infty_ereal: fixes x :: "ereal" shows "(\<bar>x\<bar> \<noteq> \<infinity>) = (\<exists>x'. x = ereal x')"
simps_1: fixes c :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and c' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "c \<rightarrow>\<^bsub>[]\<^esub> c' = (c = c')"
det_four_block_mat_lower_right_zero_col: fixes A1 :: "'b mat"   and n :: "nat"   and A2 :: "'b mat"   and A3 :: "'b mat"   and A4 :: "'b mat" assumes "A1 \<in> carrier_mat 1 n"   and "A2 \<in> carrier_mat 1 1"   and "A3 \<in> carrier_mat n n"   and "A4 = 0\<^sub>m n 1" shows "det (four_block_mat A1 A2 A3 A4) = (- (1::'b)) ^ n * det A2 * det A3"
Or_prop_entailment: fixes S :: "'a ltl set"   and xs :: "'a ltl list" shows "S \<Turnstile>\<^sub>P Or xs = (\<exists>x\<in>set xs. S \<Turnstile>\<^sub>P x)"
ident_integrable_on: fixes a :: "real"   and b :: "real" shows "(\<lambda>x. x) integrable_on {a..b}"
bdd_above_Iic: fixes b :: "'a" shows "bdd_above {..b}"
assms_3: shows "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"
state_simple: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" assumes "Finite_Markov_Decision_Process K S" shows "state (Finite_Markov_Decision_Process.simple K S ct s) = s"
that: fixes g :: "nat" assumes "\<forall>k>0. residue_primroot (p ^ k) g" shows "thesis"
S2_def: fixes v :: "bool"   and m :: "'grp" shows "S2 v m = sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>v = a * b mod order \<G> in sample_uniform_units (order \<G>) \<bind> (\<lambda>r0. sample_uniform_units (order \<G>) \<bind> (\<lambda>s0. let w0 = (\<^bold>g [^] a) [^] s0 \<otimes> \<^bold>g [^] r0 in sample_uniform_units (order \<G>) \<bind> (\<lambda>r1. sample_uniform_units (order \<G>) \<bind> (\<lambda>s1. let w1 = (\<^bold>g [^] a) [^] s1 \<otimes> \<^bold>g [^] r1; z' = (\<^bold>g [^] c\<^sub>v) [^] s1 \<otimes> (\<^bold>g [^] b) [^] r1 in sample_uniform (order \<G>) \<bind> (\<lambda>s'. let enc_m = \<^bold>g [^] s'; enc_m' = z' \<otimes> m in return_spmf (v, \<^bold>g [^] a, \<^bold>g [^] b, \<^bold>g [^] c\<^sub>v, w0, enc_m, w1, enc_m'))))))))"
sum_in_X: shows "(\<Sum>i\<in>U. f i * i) \<in> X"
idem: fixes a :: "'b multiset" shows "a \<inter># a = a"
Rep_zet_cases: fixes y :: "'a set" assumes "y \<in> zet" obtains x :: "'a zet" where "y = Rep_zet x"
Catch_2: fixes c2 :: "('a, 'b, 'c) com"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate" assumes "(a2 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile>a2 \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
assms_2: shows "(\<sigma>, l) \<in> oreachable A (\<lambda>s s'. I) U"
sr: shows "(\<sigma>, p) \<in> oreachable A S U"
wTimesU: fixes \<theta>\<^sub>1 :: "trm"   and \<theta>\<^sub>2 :: "trm"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([Times \<theta>\<^sub>1 \<theta>\<^sub>2]<>\<nu>) = (let (l1, u1) = [\<theta>\<^sub>1]<>\<nu>; (l2, u2) = [\<theta>\<^sub>2]<>\<nu> in (Interval_Word32.tl l1 u1 l2 u2, tu l1 u1 l2 u2))"
new_Marker_in_set_implies_nonregular_occurence: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "Marker \<notin> set (msgs c i)"   and "Marker \<in> set (msgs c' i)"   and "channel i = Some (p, q)" shows "\<not> regular_event ev"
interval_extension2D: fixes F :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and ivl1 :: "float interval"   and ivl2 :: "float interval"   and ivl :: "float interval"   and x :: "real"   and y :: "real" assumes "interval_extension2 F f"   and "F ivl1 ivl2 = Some ivl"   and "x \<in>\<^sub>r ivl1"   and "y \<in>\<^sub>r ivl2" shows "f x y \<in>\<^sub>r ivl"
LFactGroup_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" shows "G LMod H = \<lparr>carrier = lcosets\<^bsub>G\<^esub> H, mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
fun_surj_inj: fixes f :: "'c \<Rightarrow> 'b" shows "\<exists>g h. f = g \<circ> h \<and> surj_on h (range f) \<and> inj_on g (range f)"
IH: fixes y :: "V" assumes "y \<in> elts z" shows "x * lift y y = lift (x * y) (x * y)"
rsp: shows "eq_onp (\<lambda>f. finite (pdom f)) {}\<^sub>p {}\<^sub>p"
invariant: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Graph_Invariant E P"   and "P a"   and "E a b" shows "P b"
gcd_impl_main_def: fixes G1 :: "'a poly"   and G2 :: "'a poly" shows "gcd_impl_main G1 G2 = (if G1 = 0 then 0 else if G2 = 0 then normalize G1 else Polynomial.smult (gcd (content G1) (content G2)) (gcd_impl_primitive (primitive_part G1) (primitive_part G2)))"
hyps_for_subset: fixes n :: "('form, 'rule) graph_node"   and p :: "('form, 'var) antecedent" shows "hyps_for n p |\<subseteq>| outPorts n"
confluence_equiv_ChurchRosser: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "relation.is_confluent r = relation.is_ChurchRosser r"
assms_2: shows "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
not_true_and_false: fixes d :: "'dyn" shows "\<not> (is_true d \<and> is_false d)"
simps_1: fixes t :: "'b tree23" shows "hI (TI t) = height t"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "OpDyn x1 \<noteq> OpUbx1 x2"
inverse_unique: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "group f z inverse"   and "f a b = z" shows "inverse a = b"
cHeap_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "cHeap \<Gamma> e = (\<Lambda> a. pathsCard (paths (Theap \<Gamma> e\<cdot>a)))"
S1_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S1 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> S1 rmhist p$ \<or> S2 rmhist p$"
trivial_implies_finite: fixes X :: "'c set" assumes "trivial X" shows "finite X"
assms: shows "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)"
f_range_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "f_range_on A f = (\<forall>B\<subseteq>A. f B \<subseteq> B)"
scale_sum_left: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'b" shows "sum f A * x = (\<Sum>a\<in>A. f a * x)"
sup_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "\<Squnion>A \<in> carrier L"
hom_def: fixes G :: "('g, 'i) monoid_scheme"   and H :: "('h, 'j) monoid_scheme" shows "hom G H = {h \<in> carrier G \<rightarrow> carrier H. \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y}"
qfree_lift_nnf_qe: fixes qe :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)" shows "qfree (R.lift_nnf_qe qe \<phi>)"
InfinitesimalI2: fixes x :: "'a star" assumes "\<And>r. 0 < r \<Longrightarrow> hnorm x < hypreal_of_real r" shows "x \<in> Infinitesimal"
split_aux_induct: fixes P :: "pexpr \<Rightarrow> nat \<Rightarrow> pexpr \<Rightarrow> bool"   and x :: "pexpr"   and y :: "nat"   and z :: "pexpr" assumes "\<And>e1 e2 n e. \<lbrakk>P e1 n e; P e2 n (snd (snd (split_aux e1 n e)))\<rbrakk> \<Longrightarrow> P (PExpr2 (PMul e1 e2)) n e"   and "\<And>e' m n e. (m \<noteq> 0 \<Longrightarrow> P e' (m * n) e) \<Longrightarrow> P (PExpr2 (PPow e' m)) n e"   and "\<And>e' n e. P (PExpr1 e') n e" shows "P x y z"
less_contour_def: fixes b :: "contour"   and b' :: "contour" shows "(b < b') = (length (Rep_contour b) < length (Rep_contour b'))"
norm_neg_numeral: fixes w :: "num" shows "norm (- numeral w) = numeral w"
DERIV_sin: fixes x :: "'a" shows "(sin has_field_derivative cos x) (at x)"
set_sel_2: fixes a :: "('d, 'e, 'f) generat" assumes "\<not> is_Pure a" shows "generat.output a \<in> generat_outs a"
action_absc': fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" shows "action (absc cfg) = map_pmf abss (action cfg)"
norm_le_square: fixes x :: "'a"   and a :: "real" shows "(norm x \<le> a) = (0 \<le> a \<and> inner x x \<le> a\<^sup>2)"
existential_lookup_sem_R: fixes fml :: "nat fml"   and qs :: "real poly list"   and signs :: "rat list list" assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length qs"   and "set signs = consistent_sign_vectors_R qs UNIV" shows "(\<exists>x. lookup_sem fml (map (\<lambda>p. poly p x) qs)) = (find (lookup_sem fml) signs \<noteq> None)"
pmf_eq_0_set_pmf: fixes M :: "'h pmf"   and p :: "'h" shows "(pmf M p = 0) = (p \<notin> set_pmf M)"
simps_25: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x2 :: "'c" shows "map_xstate f1 f2 (Abrupt x2) = Abrupt (f1 x2)"
lead_coeff_smult: fixes c :: "'a"   and p :: "'a poly" shows "lead_coeff (Polynomial.smult c p) = c * lead_coeff p"
SN_on_Image_rtrancl: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "SN_on r A" shows "SN_on r (r\<^sup>* `` A)"
Sym_L: fixes t :: "trm"   and u :: "trm"   and H :: "fmla set"   and A :: "fmla" assumes "insert (t EQ u) H \<turnstile> A" shows "insert (u EQ t) H \<turnstile> A"
eq_snd_iff: fixes b :: "'b"   and p :: "'c \<times> 'b" shows "(b = final_b p) = (\<exists>a. p = (a, b))"
c_tr_length: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s :: "'g"   and xs :: "'e list" shows "length (c_tr step out s xs) = length xs"
DOM_cod: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Dom C (free_monoidal_category.rep C (partial_magma.cod (free_monoidal_category.comp C) f)) = monoidal_language.Cod C (free_monoidal_category.rep C f)"
C: shows "C = ''y'' ::= N 0;; ''z'' ::= N 0;; wl"
pow_mono_exp: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "(1::'a) \<le> a"   and "m \<le> n" shows "a ^ m \<le> a ^ n"
inj_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" assumes "inj f"   and "inj g" shows "inj (f \<circ> g)"
image_mset_If: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d multiset" shows "{#if P x then f x else g x. x \<in># A#} = image_mset f (filter_mset P A) + image_mset g {#x \<in># A. \<not> P x#}"
min_0_1_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq zero one = zero"
R_def: shows "R \<equiv> ((A ===> A ===> (=)) ===> A ===> A ===> (=)) ===> A ===> A ===> (=)"
rel_mono: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_option R \<le> rel_option Ra"
add_edge1_def: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar" shows "wgraph_by_map.add_edge1 M_update M_lookup \<equiv> \<lambda>(u, v) d g. M_update u ((v, d) # the_default [] (M_lookup g u)) g"
embed_iff_compat_inj_on_ofilter: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "Well_order r"   and "Well_order r'" shows "embed r r' f = (compat r r' f \<and> inj_on f (Field r) \<and> ofilter r' (f ` Field r))"
frameChainEq: fixes xvec :: "name list"   and \<Psi> :: "'d"   and yvec :: "name list"   and \<Psi>' :: "'d" assumes "\<langle>xvec, \<Psi>\<rangle> = \<langle>yvec, \<Psi>'\<rangle>"   and "xvec \<sharp>* yvec" obtains p :: "(name \<times> name) list" where "set p \<subseteq> set xvec \<times> set yvec" and "distinctPerm p" and "\<Psi>' = p \<bullet> \<Psi>"
cpu_reg_val_def: fixes reg :: "CPU_register"   and state :: "'b sparc_state" shows "cpu_reg_val reg state \<equiv> cpu_reg state reg"
1: shows "substT (rawpsubstT (Var y) (zip (map Var us) (map snd txs))) t u = rawpsubstT (Var y) (zip (map Var us) (map snd txs))"
bezout_iterate_preserves_below_n: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and i :: "'f"   and n :: "nat"   and a :: "'f"   and b :: "'e" assumes "echelon_form_upt_k A k"   and "is_bezout_ext bezout"   and "A $ i $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "n < nrows A"   and "n < mod_type_class.to_nat a"   and "k < ncols A"   and "mod_type_class.to_nat i \<le> n"   and "is_zero_row_upt_k i k A" shows "bezout_iterate A n i (mod_type_class.from_nat k) bezout $ a $ b = A $ a $ b"
lunit_char_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>PB.lunit f : trg f \<star> f \<Rightarrow> f\<guillemotright>"
empty_or_last_of_suffix: fixes q :: "'a list"   and q' :: "'a list"   and t :: "'a"   and p :: "'a list"   and ts :: "'a list" assumes "q = q' @ [t]"   and "q = p @ ts" shows "ts = [] \<or> (\<exists>ts'. q' = p @ ts' \<and> ts' @ [t] = ts)"
enum_finite_4_def: shows "enum = [a\<^sub>1, a\<^sub>2, a\<^sub>3, a\<^sub>4]"
worklist_tree_aux_def: fixes succs :: "'c \<Rightarrow> 'c list" shows "worklist_tree_aux succs = worklist_aux (\<lambda>s. succs)"
isomorphic_implies_hpar_3: fixes f :: "'d"   and f' :: "'d" assumes "D.isomorphic f f'" shows "src\<^sub>D f = src\<^sub>D f'"
load_word_mem2_mod_state: fixes s :: "'b sparc_state"   and addr :: "word32"   and asi :: "word8"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "fst (case load_word_mem s addr asi of None \<Rightarrow> (None, s) | Some w \<Rightarrow> (Some w, add_data_cache s addr w 15)) = fst (case load_word_mem (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr asi of None \<Rightarrow> (None, s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) | Some w \<Rightarrow> (Some w, add_data_cache (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr w 15))"
web_def: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "web \<Gamma> \<equiv> ((\<forall>x y. x \<in> A \<Gamma> \<longrightarrow> \<not> edge \<Gamma> y x) \<and> (\<forall>x y. x \<in> B \<Gamma> \<longrightarrow> \<not> edge \<Gamma> x y) \<and> A \<Gamma> \<subseteq> \<^bold>V\<^bsub>\<Gamma>\<^esub>) \<and> (A \<Gamma> \<inter> B \<Gamma> = {} \<and> (\<forall>x. \<not> edge \<Gamma> x x)) \<and> (\<forall>x. x \<notin> \<^bold>V\<^bsub>\<Gamma>\<^esub> \<longrightarrow> weight \<Gamma> x = 0) \<and> (\<forall>x. weight \<Gamma> x \<noteq> \<top>)"
fset_of_list_elem: fixes x :: "'a"   and xs :: "'a list" shows "(x |\<in>| fset_of_list xs) = (x \<in> set xs)"
distinct_1: fixes x1 :: "'b" shows "Fin x1 \<noteq> \<infinity>"
landau_product_preprocess_40: fixes k :: "num" shows "neg_numeral_class.dbl_inc (numeral k) = numeral (num.Bit1 k)"
MATCH1_size_mset: fixes x :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and y :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)" assumes "MATCH1 x y" shows "size_mset (fst y) < size_mset (fst x)"
is_NoMsg_tuple_def: fixes p :: "'b \<times> 'c" shows "is_NoMsg p \<equiv> is_NoMsg (fst p) \<and> is_NoMsg (snd p)"
2706: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<squnion> (y \<squnion> - - ((top \<squnion> z) \<sqinter> - x)) = y \<squnion> - - ((top \<squnion> z) \<sqinter> - x)"
elims: fixes x :: "('a \<times> 'c) list"   and xa :: "'a"   and y :: "'c" assumes "lookup_pair x xa = y" obtains xb :: "'a" where "x = []" and "xa = xb__" and "y = (0::'c)"   | k :: "'a" and v :: "'c" and xs :: "('a \<times> 'c) list" and xb :: "'a" where "x = (k, v) # xs" and "xa = xb" and "y = (case comp xb k of Eq \<Rightarrow> v | Lt \<Rightarrow> 0::'c | Gt \<Rightarrow> lookup_pair xs xb)"
pho_apply_def: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "pho_apply f x = f x"
swap_eqvt: fixes p :: "perm"   and a :: "atom"   and b :: "atom" shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
fin_cut_same_replicate: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" shows "fin_cut_same x (xs @ replicate n x) = fin_cut_same x xs"
indecomposableD: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "indecomposable \<alpha>"   and "\<beta> < \<alpha>"   and "\<gamma> < \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "\<beta> + \<gamma> < \<alpha>"
powr_less_mono': fixes x :: "real"   and a :: "real"   and b :: "real" assumes "0 < x"   and "x < 1"   and "a < b" shows "x powr b < x powr a"
coeff_Poly: fixes xs :: "'a list"   and i :: "nat" shows "poly.coeff (Poly xs) i = nth_default (0::'a) xs i"
second_derivative_componentwise: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real"   and a :: "'b"   and v :: "'b" assumes "higher_differentiable_on S f 2"   and "open S"   and "a \<in> S" shows "nth_derivative 2 f a v = (\<Sum>i\<in>Basis. (\<Sum>j\<in>Basis. frechet_derivative (\<lambda>a. frechet_derivative f (at a) j) (at a) i * (v \<bullet> j)) * (v \<bullet> i))"
update_simps_4: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "ip \<in> kD rt"   and "sqn rt ip = \<pi>\<^sub>2 r"   and "flag rt ip = Some Aodv_Basic.inv" shows "update rt ip r = rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip))))"
tMap_tLast_inv: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace"   and f' :: "'d \<Rightarrow> 'b"   and t' :: "'d Trace" assumes "tMap f t = tMap f' t'" shows "f (tLast t) = f' (tLast t')"
back_subst: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "P a"   and "a = b" shows "P b"
enc_welldef: fixes x :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "x \<in> enc (w, I)"   and "x \<in> enc (w', I')"   and "length I = length I'"   and "wf_formula (length I) \<phi>"   and "wf_interp_for_formula (w, I) \<phi>"   and "wf_interp_for_formula (w', I') \<phi>" shows "((w, I) \<Turnstile> \<phi>) = ((w', I') \<Turnstile> \<phi>)"
foldl_echelon_form_column_k_eq_2: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "k < ncols A" shows "fst (snd (foldl echelon_form_of_column_k_iarrays (matrix_to_iarray A, 0, bezout) [0..<Suc k])) = snd (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])"
pos_def: fixes rs :: "'b stream"   and r :: "'b" shows "pos rs r \<equiv> LEAST n. shd (sdrop n rs) = r"
cond_def: fixes b :: "'b upred"   and P :: "'b upred"   and Q :: "'b upred" shows "P \<triangleleft> b \<triangleright> Q = (b \<and> P \<or> \<not> b \<and> Q)"
list_graph_to_graph_def: fixes G :: "'b list_graph" shows "list_graph_to_graph G = \<lparr>nodes = set (nodesL G), edges = set (edgesL G)\<rparr>"
associatedE1: fixes a :: "'a"   and b :: "'a" assumes "normalize a = normalize b" obtains u :: "'a" where "is_unit u" and "a = u * b"
nres_order_simps_21: fixes x :: "'c"   and Y :: "'c set" shows "(RETURN x \<le> RES Y) = (x \<in> Y)"
length_remdups_eq: fixes xs :: "'a list" shows "(length (remdups xs) = length xs) = (remdups xs = xs)"
assms_1: fixes z :: "'a" assumes "z \<noteq> x" shows "\<rho> z \<sqsubseteq> \<rho>' z"
adds_triv_left: fixes a :: "'a"   and b :: "'a" shows "a adds a + b"
gderiv_def: fixes f :: "'b \<Rightarrow> real"   and x :: "'b"   and D :: "'b" shows "GDERIV f x :> D = (f has_derivative (\<lambda>h. inner h D)) (at x)"
induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'d"   and x2 :: "'d"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* x1 x2"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>r\<^sup>*\<^sup>* a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
ssufI2: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>s v"   and "\<^bold>|u\<^bold>| < \<^bold>|v\<^bold>|" shows "u <s v"
isomorphic_def: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme" shows "F \<cong> G = (\<exists>f. gbij\<^bsub>F,G\<^esub> f)"
findaugmenting_aux: fixes E1 :: "'edge set"   and E2 :: "'edge set"   and u :: "'a"   and v :: "'a" assumes "E1 \<subseteq> E"   and "E2 \<subseteq> E"   and "(u, v) \<in> connected E1"   and "(u, v) \<notin> connected E2" shows "\<exists>a b e. (a, b) \<notin> connected E2 \<and> e \<notin> E2 \<and> e \<in> E1 \<and> joins a b e"
is_Receive_def: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "stateful_strand_step.is_Receive stateful_strand_step = (\<exists>x2. stateful_strand_step = receive\<langle>x2\<rangle>)"
nn_integral_set_ennreal: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and A :: "'b set" shows "\<integral>\<^sup>+ x. ennreal (f x) * indicator A x \<partial>M = \<integral>\<^sup>+ x. ennreal (f x * indicat_real A x) \<partial>M"
card_le_sym_Diff: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "card A \<le> card B" shows "card (A - B) \<le> card (B - A)"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U"   and "a \<le>\<^sub>o\<^sub>w b" obtains "a = local.min a b"