hcju/isabelleps huggingface dataset:
basis_list_def: fixes fs :: "real vec list" shows "gso.basis_list fs = (set fs \<subseteq> carrier_vec n \<and> length fs = n \<and> carrier_vec n \<subseteq> gso.span (set fs))"
sum_count_set: fixes xs :: "'b list"   and X :: "'b set" assumes "set xs \<subseteq> X"   and "finite X" shows "sum (count_list xs) X = length xs"
in_verts_app_iso_cases: fixes v :: "'e"   and hom :: "('g, 'h, 'e, 'f) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" assumes "v \<in> verts (app_iso hom G)" obtains v0 :: "'g" where "v = iso_verts hom v0" and "v0 \<in> verts G"
g: fixes i :: "'a"   and j :: "'a" assumes "g i = g j" shows "i = j"
Cond: fixes A :: "triple set"   and P :: "assn"   and e :: "expr"   and Q :: "vassn"   and c1 :: "stmt"   and c2 :: "stmt"   and R :: "assn" assumes "A \<turnstile>\<^sub>e {P} e {Q}"   and "\<forall>v. A \<turnstile> {Q v} (if v \<noteq> Null then c1 else c2) {R}" shows "A \<turnstile> {P} If (e) c1 Else c2 {R}"
that: fixes a :: "nat \<Rightarrow> real"   and n :: "nat" assumes "l2norm {- pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>n. a k * trigonometric_set k x)) < e" shows "thesis"
p: shows "\<forall>p\<in>P. [a = b] (mod p)"
rep_eq: fixes x :: "'a" shows "cblinfun_apply (cblinfun_cinner_right x) = (\<bullet>\<^sub>C) x"
lconf_hext: fixes h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option"   and E :: "String.literal \<Rightarrow> ty option"   and h' :: "'heap" assumes "P,h \<turnstile> l (:\<le>) E"   and "h \<unlhd> h'" shows "P,h' \<turnstile> l (:\<le>) E"
simps_2: fixes x :: "'ubx1" shows "norm_unboxed (OpUbx1 x) = box_ubx1 x"
set_zmset_def: fixes M :: "'c zmultiset" shows "set_zmset M = {x. zcount M x \<noteq> 0}"
swap_swapped: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" shows "(X, X #[x \<and> y]_zs) \<in> swapped"
dvd_elements_mult_matrix_right: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat"   and Q :: "'a mat"   and x :: "'a" assumes "A \<in> carrier_mat m n"   and "Q \<in> carrier_mat n n"   and "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd A $$ (i, j)" shows "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd (A * Q) $$ (i, j)"
pfinite_converterD: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and a :: "'a" assumes "pfinite_converter \<I> \<I>' conv"   and "a \<in> outs_\<I> \<I>" shows "pfinite_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). pfinite_converter \<I> \<I>' conv')"
cbigD': fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and combine :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and collect_id :: "'c"   and o' :: "'b \<times> 'c"   and collect :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c"   and P :: "'a"   and \<sigma> :: "'b" assumes "CollectionSemantics small endset combine collect_id"   and "o' \<in> CollectionSemantics.cbig small endset collect combine collect_id P \<sigma>" shows "\<exists>n. o' \<in> CollectionSemantics.csmall_nstep small collect combine collect_id P \<sigma> n \<and> fst o' \<in> endset"
admits_1x2: shows "\<forall>A\<in>carrier_mat 1 2. admits_diagonal_reduction A"
hn_refine_frame: fixes P' :: "assn"   and c :: "'a Heap"   and Q' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres"   and P :: "assn"   and F :: "assn" assumes "hn_refine P' c Q' R m"   and "P \<Longrightarrow>\<^sub>t F * P'" shows "hn_refine P c (F * Q') R m"
sbi: shows "simple_bochner_integrable M (indicat_real A)"
contf: shows "continuous_on (path_image (circlepath z r)) f"
ordIso_equivalence_2: fixes r :: "('a \<times> 'a) set" assumes "Well_order r" shows "r =o r"
hom_0_iff: fixes x :: "'a poly" shows "(map_poly hom x = 0) = (x = 0)"
ext_inject: fixes verts :: "'c set"   and arcs :: "'d set"   and tail :: "'d \<Rightarrow> 'c"   and head :: "'d \<Rightarrow> 'c"   and more :: "'e"   and verts' :: "'c set"   and arcs' :: "'d set"   and tail' :: "'d \<Rightarrow> 'c"   and head' :: "'d \<Rightarrow> 'c"   and more' :: "'e" shows "(\<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = \<lparr>verts = verts', arcs = arcs', tail = tail', head = head', \<dots> = more'\<rparr>) = (verts = verts' \<and> arcs = arcs' \<and> tail = tail' \<and> head = head' \<and> more = more')"
avl_balL: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "avl l"   and "avl r"   and "height r - 1 \<le> height l \<and> height l \<le> height r + 2" shows "avl (balL l a r)"
M_FullStruct_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "M_FullStruct wtFsym wtPsym arOf resOf intT intF intP \<equiv> FullStruct wtFsym wtPsym arOf resOf intT intF intP"
ConcretenessSemantics2: fixes r :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool"   and x :: "\<nu>"   and w :: "i" assumes "Some r = d\<^sub>1 E!" shows "x \<in> ex1 r w \<longrightarrow> (\<exists>y. x = \<omega>\<nu> y)"
flow_initial_time: fixes x0 :: "'a" assumes "0 \<in> UNIV"   and "x0 \<in> X" shows "flow0 x0 0 = x0"
cone_mkCone: fixes e :: "'c" assumes "D.is_equalized_by e" shows "D.cone (C.dom e) (D.mkCone e)"
splitFace_f12_new_vertices: fixes f12 :: "face"   and f21 :: "face"   and newGraph :: "graph"   and g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and v :: "nat" assumes "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs"   and "v \<in> set newVs" shows "v \<in> \<V> f12"
unique_rootD_3: fixes plr :: "int poly \<times> rat \<times> rat" assumes "\<exists>!x. root_cond plr x" shows "poly (real_of_int_poly (poly_real_alg_1 plr)) (real_of_1 plr) = 0"
simps_4: fixes v :: "pol"   and va :: "nat"   and vb :: "pol" shows "isnorm (Pinj 0 (PX v va vb)) = False"
pdivides_iff_shell: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p pdivides\<^bsub>K [X]\<^esub> q = p divides\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q"
iMODb_not_empty: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<noteq> {}"
RES_refine: fixes S :: "'b set"   and S' :: "'c set"   and R :: "('b \<times> 'c) set" assumes "\<And>s. s \<in> S \<Longrightarrow> \<exists>s'\<in>S'. (s, s') \<in> R" shows "RES S \<le> \<Down> R (RES S')"
pirreducibleE_1: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "domain R"   and "subring K R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "pirreducible\<^bsub>R\<^esub> K p" shows "p \<noteq> []"
opt_group_def: shows "opt_group = push_forall \<circ> nnf \<circ> unpower 0 \<circ> groupQuantifiers \<circ> clearQuantifiers"
less_is_non_zero_p1: fixes a :: "'a word"   and k :: "'a word" assumes "a < k" shows "a + 1 \<noteq> 0"
strict_mono_on_imp_mono_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "strict_mono_on f A" shows "mono_on f A"
id: shows "root n (real_of_1 (p, l, r)) ^ n = real_of_1 (p, l, r)"
cases: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g" assumes "\<tau>trsys.\<tau>diverge trsys \<tau>move a" obtains s :: "'g" and s' :: "'g" where "a = s" and "\<tau>trsys.silent_move trsys \<tau>move s s'" and "\<tau>trsys.\<tau>diverge trsys \<tau>move s'"
uminus_sum_list: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and as :: "'a list" assumes "class.group_add minus plus zero uminus" shows "uminus (monoid_add.sum_list plus zero as) = monoid_add.sum_list plus zero (map uminus (rev as))"
oghoare_steps_Fault: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and x :: "'d"   and c' :: "('d, 'e, 'f) com"   and f :: "'f"   and P :: "('d, 'e, 'f) ann"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and Q :: "'d set"   and A :: "'d set" assumes "\<Gamma> \<turnstile> (c, Normal x) \<rightarrow>\<^sup>* (c', Fault f)"   and "x \<in> pre P"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A" shows "f \<in> F"
nonneg_Reals_def: shows "\<real>\<^sub>\<ge>\<^sub>0 = {of_real r |r. 0 \<le> r}"
le_divide_eq_1_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less_eq one (divide b a) = less_eq b a"
zero: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "A *v 0 = 0"
monad_alt_optionT': assumes "monad_alt Monad_Overloading.return Monad_Overloading.bind alt" shows "monad_alt Monad_Overloading.return Monad_Overloading.bind alt"
discrete_card: fixes X :: "'b set" assumes "discrete X" shows "card {f. CK_nf_pos f} = 1"
dirichlet_prod_def: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "dirichlet_prod f g = (\<lambda>n. \<Sum>d | d dvd n. f d * g (n div d))"
MakeNT_comp: fixes X :: "'g"   and f :: "('g, 'j, 'h, 'k, 'i, 'l) NatTrans" assumes "X \<in> obj\<^bsub>NTCatDom f\<^esub>" shows "MakeNT f $$ X = f $$ X"
of_real_sum: fixes f :: "'c \<Rightarrow> real"   and s :: "'c set" shows "of_real (sum f s) = (\<Sum>x\<in>s. of_real (f x))"
less_eq_unit_def: fixes uu :: "unit"   and uv :: "unit" shows "(uu \<le> uv) = True"
divides_primepow: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times a (power.power one times p n)" obtains m :: "nat" where "m \<le> n" and "normalize a = power.power one times p m"
BaseCl_def: shows "BaseCl = \<lparr>access = Public, cfields = [(arr, \<lparr>access = Public, static = True, type = PrimT Boolean.[]\<rparr>), (vee, \<lparr>access = Public, static = False, type = Iface HasFoo\<rparr>)], methods = [Base_foo], init = Expr (arr_viewed_from Base Base:=New PrimT Boolean[Lit (Intg 2)]), super = Object, superIfs = [HasFoo]\<rparr>"
succ_min_lvar_not_in_bounds_1: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and xr :: "nat" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "xr \<in> lvars (\<T> s)"   and "xr \<in> rvars (\<T> s')" shows "\<not> in_bounds xr \<langle>\<V> s\<rangle> (\<B> s)"
interaction_any_bounded_converter_start_1: fixes conv :: "('a, 'b, 'c, 'd) converter"   and n :: "enat"   and m :: "enat" assumes "interaction_any_bounded_converter conv n"   and "n \<le> m" shows "interaction_any_bounded_converter conv m"
assms_2: shows "theta\<inverse> \<subseteq> RetrT (theta\<inverse>)"
deg_q_cons: assumes "\<forall>q\<in>set qs. degree q = 0" shows "consistent_sign_vectors qs UNIV = {map (\<lambda>x. if poly x 0 < 0 then - 1 else if poly x 0 = 0 then 0 else 1) qs}"
tc_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.tc term_of_pair ord_term p \<equiv> lookup p (ordered_term.tt term_of_pair ord_term p)"
\<alpha>_limit_set_eq_rev: fixes x :: "'a" shows "rev.\<alpha>_limit_set x = \<omega>_limit_set x"
separable_def: fixes f :: "'a poly" shows "separable f = algebraic_semidom_class.coprime f (pderiv f)"
simps_8: fixes f :: "nat \<Rightarrow> atom \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and var :: "nat" shows "liftmap f (AllQ \<phi>) var = AllQ (liftmap f \<phi> (var + 1))"
hyps_1: shows "a \<noteq> (0::'c) \<or> p \<noteq> 0"
rem_cycles_subs: fixes i :: "'b"   and j :: "'b"   and xs :: "'b list" shows "set (rem_cycles i j xs) \<subseteq> set xs"
ctr_2: fixes xs :: "'a llist"   and P :: "'a \<Rightarrow> bool" assumes "\<not> lnull xs"   and "P (lhd xs)" shows "ltakeWhile P xs = LCons (lhd xs) (ltakeWhile P (ltl xs))"
f_Exec_Stream_Acc_LocalState_Nil: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun [] c = []"
representation_zero: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and basis :: "'e set" assumes "module scale" shows "module.representation scale basis (0::'e) = (\<lambda>b. 0::'d)"
simps_1: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" shows "prod_ord_pair f [] [] = True"
vectorize_plus: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.vectorize_poly (p + q) = local.punit.vectorize_poly p + local.punit.vectorize_poly q"
\<gamma>\<^sub>\<tau>': shows "\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')"
cases: fixes a :: "'a list" assumes "steps a" obtains x :: "'a" where "a = [xa__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "a = x # y # xs" and "E x y" and "steps (y # xs)"
div_decomp_comp: fixes m :: "nat"   and n :: "nat"   and a :: "nat" assumes "coprime m n" shows "(a dvd m * n) = (\<exists>b c. a = b * c \<and> b dvd m \<and> c dvd n)"
oarrivemsg_simps_4: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and i1 :: "nat"   and i2 :: "nat" shows "oarrivemsg P \<sigma> (connect(i1, i2)) = True"
i_6: shows "g \<in> insert x F \<rightarrow> carrier G"
simps_2: fixes i_end :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.jnf_vector_main (Suc i_end) A = (let i_start = ??.Jordan_Normal_Form_Existence.identify_block A i_end in ??.Jordan_Normal_Form_Existence.jnf_vector_main i_start A @ [(Suc i_end - i_start, A $$ (i_start, i_start))])"
base: fixes i :: "nat" assumes "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "P i"
arr_char: fixes f :: "'a" shows "Left.arr f = left (trg f) f"
B: shows "\<forall>x\<in>s. norm x \<le> B"
image_index: fixes A :: "'a set"   and xs :: "'a list"   and x :: "'a" assumes "A \<subseteq> set (xs @ [x])" shows "index (xs @ [x]) ` A = (if x \<in> A then insert (length xs) (index xs ` (A - {x})) else index xs ` A)"
seq_char': fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "seq g f = (ARR f \<and> ARR g \<and> DOM g = COD f)"
select_convs_1: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and more :: "'f" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
subseq_of_length_Suc_upt: fixes n :: "nat"   and m :: "nat"   and xs :: "nat list" shows "subseq_of_length (Suc n) [0..<m] xs = (if n = 0 then length xs = Suc 0 \<and> hd xs < m else hd xs < hd (tl xs) \<and> subseq_of_length n [0..<m] (tl xs))"
heq1: fixes t :: "real" assumes "t \<in> {0..1}" shows "h (t, 1) = g1 1"
that_3: assumes "X \<noteq> {}"   and "Y \<noteq> {}" shows "X \<inter> Y \<noteq> {}"
induction_RelS_Eg1Eg2: fixes c\<^sub>1\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mem\<^sub>1\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and c\<^sub>1\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>1\<^sub>C :: "var\<^sub>C \<Rightarrow> nat"   and c\<^sub>1\<^sub>C' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>C' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>1\<^sub>C' :: "var\<^sub>C \<Rightarrow> nat" assumes "(\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"   and "eval_abv\<^sub>C \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C" shows "\<exists>c\<^sub>1\<^sub>A' mds\<^sub>A' mem\<^sub>1\<^sub>A'. A.neval \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A (abs_steps \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A \<and> (\<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
multi_member_skip: fixes x :: "'b"   and XS :: "'b multiset"   and y :: "'b" assumes "x \<in># XS" shows "x \<in># (y, {#}) + XS"
alphaAll_sym: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and A :: "('c, 'd, 'varSort, 'var, 'e) qAbs" shows "(\<forall>X'. X #= X' \<longrightarrow> X' #= X) \<and> (\<forall>A'. A $= A' \<longrightarrow> A' $= A)"
nat_term_compD1: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a" assumes "nat_term_comp cmp"   and "snd (rep_nat_term u) = snd (rep_nat_term v)"   and "fst (rep_nat_term u) = 0" shows "cmp u v \<noteq> Gt"
cil_cons1: fixes w :: "'b list"   and wa :: "'b list"   and \<alpha> :: "'b \<Rightarrow> 'c set \<times> 'c set"   and wb :: "'b list"   and e :: "'b" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"   and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> wb) = {}" shows "e # w \<in> e # wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"
basic_monos_7: fixes A :: "'h set"   and B :: "'h set"   and x :: "'h" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
no_loop: fixes x :: "'v" shows "\<not> edge \<Gamma>' x x"
leq_trans: fixes x :: "'f llist"   and y :: "'f llist"   and z :: "'f llist" assumes "lprefix x y"   and "lprefix y z" shows "lprefix x z"
state_update_before_def: fixes sc :: "'d \<times> 'd \<Rightarrow> bool"   and Ac :: "('c, 'd) action" shows "state_update_before sc Ac = action_of (R (\<lambda>(A, A'). Pre sc (alpha_rp.more A) \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A') ;; action.relation_of Ac)"
map_scale_uminus_left: fixes k :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "- k \<cdot> p = - (k \<cdot> p)"
prv_eqv_cnj_trans: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (cnj (eqv \<phi> \<chi>) (eqv \<chi> \<psi>)) (eqv \<phi> \<psi>))"
IH: fixes y :: "V" assumes "y \<in> elts y__"   and "Transset x"   and "y \<le> x" shows "TC y \<le> x"
ev: shows "\<forall>\<^sub>F b' in at_right a. b' \<in> {a<..<b}"
le_supE: fixes a :: "'b"   and b :: "'b"   and x :: "'b" assumes "a + b \<le> x" obtains "a \<le> x" and "b \<le> x"
split_list_last: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
trans: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry"   and c :: "'b DBMEntry" assumes "a \<preceq> b"   and "b \<preceq> c" shows "a \<preceq> c"
leI: fixes x :: "'a Word.word"   and y :: "'a Word.word" assumes "\<not> x <s y" shows "y \<le>s x"
sympE: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and b :: "'g"   and a :: "'g" assumes "symp r"   and "r b a" obtains "r a b"
conservative_mpt_power: fixes n :: "nat" shows "conservative_mpt M (T ^^ n)"
FOREACHci_emp: fixes \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c" shows "FOREACH\<^sub>C\<^bsup>\<Phi>\<^esup> {} c f \<sigma> = ASSERT (\<Phi> {} \<sigma>) \<bind> (\<lambda>_. RETURN \<sigma>)"
equal_eq: fixes x :: "'b"   and y :: "'b" shows "equal_class.equal x y = (x = y)"
\<R>\<^sub>1_def: shows "\<R>\<^sub>1 \<equiv> \<lambda>\<Gamma>' \<S>' P'. {(((xa, x), xb), (xd, xc), xe). \<R>\<^sub>1p \<Gamma>' \<S>' P' xa x xb xd xc xe}"
right_total_rel_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "right_total A"   and "right_total B" shows "right_total (rel_gpv A B)"
good_idx: fixes ws :: "letter list list"   and f :: "nat \<Rightarrow> letter list" assumes "good ws"   and "is_prefix ws f" shows "\<exists>i j. emb (f i) (f j) \<and> i < j"
psiFreshVec_6: fixes xvec :: "name list"   and \<Psi> :: "'e" shows "xvec \<sharp>* \<lbrace>\<Psi>\<rbrace> = xvec \<sharp>* \<Psi>"
less_imp_not_less: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.preorder less_eq less"   and "less x y" shows "(\<not> less y x) = True"
that: fixes c :: "'b" assumes "homotopic_with_canon Q U t f (\<lambda>x. c)" shows "thesis"
n_galois1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(n x \<sqsubseteq> n y \<oplus> n z) = (n x \<cdot> t y \<sqsubseteq> n z)"
distr_or2: fixes F :: "'d \<Rightarrow> 'e decision option"   and F2 :: "'d \<Rightarrow> 'e decision option"   and F1 :: "'d \<Rightarrow> 'e decision option"   and r :: "'h \<times> 'e \<Rightarrow> 'g"   and N :: "'i \<Rightarrow> 'h decision option"   and f :: "'f \<Rightarrow> 'i \<times> 'd" assumes "F = F1 \<Oplus> F2" shows "r o\<^sub>f (N \<Otimes>\<^sub>2 F \<circ> f) = (r o\<^sub>f (N \<Otimes>\<^sub>2 F1 \<circ> f)) \<Oplus> (r o\<^sub>f (N \<Otimes>\<^sub>2 F2 \<circ> f))"