hcju/isabelleps huggingface dataset:
bij: shows "bij_betw (ordermap M VWF) M (elts (ord_of_nat (card M)))"
that: fixes Y :: "(real \<times> real pdevs) \<times> real" assumes "inverse_aform p X = Some Y"   and "inverse (aform_val e X) \<in> aform_err e Y" shows "thesis"
D_incr: fixes A :: "char list set"   and c :: "com"   and A' :: "char list set" assumes "D A c A'" shows "A \<subseteq> A'"
map_simp: fixes f :: "'a" assumes "C.arr f" shows "C.map f = f"
Conj\<^sub>\<alpha>_eq_iff: fixes tset1\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha> set['a]"   and tset2\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha> set['a]" shows "(Conj\<^sub>\<alpha> tset1\<^sub>\<alpha> = Conj\<^sub>\<alpha> tset2\<^sub>\<alpha>) = (tset1\<^sub>\<alpha> = tset2\<^sub>\<alpha>)"
repeat_Bk_no_Oc: fixes r :: "cell list"   and rn :: "nat" shows "(Oc # r = Bk \<up> rn) = False"
invertible_mult: fixes A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'd) vec, 'b) vec" assumes "invertible A"   and "invertible B" shows "invertible (A ** B)"
path_defs_3: fixes g :: "real \<Rightarrow> 'a" shows "pathfinish g = g 1"
6: shows "nth_least (lift (liset A w)) k = Suc (nth_least (liset A w) k')"
sharing_consistent_weak_sharing_consistent: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "weak_sharing_consistent \<O> sb"
ennreal_Inf_countable_INF: fixes A :: "ennreal set" assumes "A \<noteq> {}" shows "\<exists>f. decseq f \<and> range f \<subseteq> A \<and> Inf A = Inf (range f)"
Q_coarser_unfolded: shows "Q \<in> insert (insert {elem} P) (insert_into_member elem P ` P)"
assms: shows "f o\<^sub>C\<^sub>L g = id_cblinfun"
select_convs_5: fixes verts :: "'f set"   and arcs :: "'g set"   and tail :: "'g \<Rightarrow> 'f"   and head :: "'g \<Rightarrow> 'f"   and more :: "'e" shows "pre_digraph.more \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = more"
list_rel_simp_4: fixes x :: "'c"   and l :: "'c list"   and x' :: "'d"   and l' :: "'d list"   and R :: "('c \<times> 'd) set" shows "((x # l, x' # l') \<in> \<langle>R\<rangle>list_rel) = ((x, x') \<in> R \<and> (l, l') \<in> \<langle>R\<rangle>list_rel)"
set_intros_3: fixes x1 :: "'c" shows "x1 \<in> fv (Var x1)"
element_put_get: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and setter :: "('u \<Rightarrow> 'u) \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme"   and v :: "'u"   and h' :: "(_) heap"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'"   and "\<And>x. getter (setter (\<lambda>_. v) x) = v" shows "h' \<turnstile> get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter \<rightarrow>\<^sub>r v"
hyps_15: assumes "t \<in> .[X]"   and "ideal F \<div> monomial (1::'a) t = ideal (monomial (1::'a) ` S)" shows "splits_wrt (ps0, qs0) (cone (monomial (1::'a) t, U - {x})) F"
word_zero_le: fixes y :: "'b word" shows "0 \<le> y"
less_imp_le: fixes y :: "'a list"   and x :: "'a list" assumes "y <lex x" shows "y \<le>lex x"
enumerate_Suc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and n :: "nat" assumes "class.wellorder less_eq less" shows "wellorder.enumerate less_eq S (Suc n) = wellorder.enumerate less_eq (S - {ord.Least less_eq (\<lambda>n. n \<in> S)}) n"
safe_lpd_fv_regex_le: fixes fv :: "'b \<Rightarrow> 'c set"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex"   and s :: "'b regex"   and test :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d" assumes "Regex.safe_regex fv safe Futu Strict r"   and "s \<in> lpd test i r" shows "fv_regex fv r \<subseteq> fv_regex fv s"
DERIV_cong_ev: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a" assumes "x = y"   and "\<forall>\<^sub>F x in nhds x. f x = g x"   and "u = v" shows "(f has_field_derivative u) (at x) = (g has_field_derivative v) (at y)"
spr_simTrans: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "spr_simAbs ` set (local.spr_simTrans a ec) = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = SPRdet.sim_equiv_class a (t' \<leadsto> s) \<and> t' \<leadsto> s \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t)"
reduced_def: fixes \<alpha> :: "rat"   and k :: "nat" shows "gs'.reduced \<alpha> k = (gs'.weakly_reduced \<alpha> k \<and> (\<forall>i j. i < k \<longrightarrow> j < i \<longrightarrow> \<bar>gs'.\<mu> i j\<bar> \<le> 1 / 2))"
del_list_sorted3: fixes xs :: "('c \<times> 'd) list"   and a :: "'c"   and a' :: "'d"   and ys :: "('c \<times> 'd) list"   and b :: "'c"   and b' :: "'d"   and zs :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted1 (xs @ (a, a') # ys @ (b, b') # zs)"   and "x < b" shows "del_list x (xs @ (a, a') # ys @ (b, b') # zs) = del_list x (xs @ (a, a') # ys) @ (b, b') # zs"
substitution: fixes g :: "'g" assumes "redundant g" shows "isTrivialPhi g (chooseNext g) (substitution g)"
isomorphicI: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.iso f" shows "S.VV.isomorphic (S.VV.dom f) (S.VV.cod f)"
cases: fixes a1 :: "'s"   and a2 :: "'s stream" assumes "enabled a1 a2" obtains \<omega> :: "'s stream" and s :: "'s" where "a1 = s" and "a2 = \<omega>" and "enabled (shd \<omega>) (stl \<omega>)" and "shd \<omega> \<in> set_pmf (K s)"
\<alpha>'_simp: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and h :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g"   and "\<F>C.arr h" shows "\<F>C.\<alpha>' (f, g, h) = \<F>C.tensor (\<F>C.tensor f g) h \<cdot> \<F>C.assoc' (\<F>C.dom f) (\<F>C.dom g) (\<F>C.dom h)"
iexp_approx2: fixes x :: "real"   and n :: "nat" shows "cmod (iexp x - (\<Sum>k\<le>n. (\<i> * complex_of_real x) ^ k / fact k)) \<le> 2 * \<bar>x\<bar> ^ n / fact n"
InterpFn: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a"   and f :: "'b"   and C :: "'a"   and g :: "'d" assumes "Sig iS\<^bsub>I\<^esub> \<triangleright> Vx : A \<turnstile> e : B"   and "f \<in> Sig iS\<^bsub>I\<^esub> : B \<rightarrow> C"   and "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor g" shows "L\<lbrakk>Vx : A \<turnstile> f E@ e : C\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IMor (g ;;\<^bsub>iC\<^bsub>I\<^esub>\<^esub> Fn\<lbrakk>f\<rbrakk>\<^bsub>I\<^esub>)"
assms: shows "run_one_step d i (s, vs, ves, $Get_local x14) = (s', vs', res)"
simps_3: fixes x2 :: "'k" shows "Some x2 \<noteq> None"
at_trace_def: fixes t :: "'b \<Rightarrow> 'a"   and i :: "nat"   and j :: "'b" shows "(t[i ..]) j = t (AbsNat i + j)"
prems_2: fixes x :: "'a node \<times> 'a node set" shows "f x \<le> expand x"
interior_of_restrict: fixes X :: "'a topology"   and S :: "'a set" shows "X interior_of S = X interior_of (topspace X \<inter> S)"
path_valid_node_1: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n as n'" shows "While_CFG.valid_node prog n"
stmt_elist_inj_term: fixes t :: "stmt"   and w :: "expr list" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
poly_mapping_eq_zeroI: fixes s :: "'b \<Rightarrow>\<^sub>0 'c" assumes "keys s = {}" shows "s = 0"
lim: shows "(\<lambda>n. ereal (norm (f n) / norm (f (Suc n)))) \<longlonglongrightarrow> c"
vec_lambda_inverse: fixes y :: "'d \<Rightarrow> 'e" assumes "y \<in> UNIV" shows "($) (vec_lambda y) = y"
elims_2: fixes x :: "'b gexp"   and xa :: "'b gexp" assumes "gexp_same_structure x xa" obtains b :: "bool" and b' :: "bool" where "x = Bc b" and "xa = Bc b'" and "b = b'"   | a1 :: "'b aexp" and a2 :: "'b aexp" and a1' :: "'b aexp" and a2' :: "'b aexp" where "x = Eq a1__ a2__" and "xa = Eq a1'__ a2'__" and "aexp_same_structure a1__ a1'__ \<and> aexp_same_structure a2__ a2'__"   | a1 :: "'b aexp" and a2 :: "'b aexp" and a1' :: "'b aexp" and a2' :: "'b aexp" where "x = Gt a1 a2" and "xa = Gt a1' a2'" and "aexp_same_structure a1 a1' \<and> aexp_same_structure a2 a2'"   | g1 :: "'b gexp" and g2 :: "'b gexp" and g1' :: "'b gexp" and g2' :: "'b gexp" where "x = Nor g1 g2" and "xa = Nor g1' g2'" and "gexp_same_structure g1 g1' \<and> gexp_same_structure g2 g2'"   | v :: "'b" and l :: "value list" and v' :: "'b" and l' :: "value list" where "x = In v l" and "xa = In v' l'" and "v = v' \<and> l = l'"
star_inductl_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x ; y = y" shows "x\<^sup>\<star> ; z \<le> y"
ltakeWhile_all_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "(ltakeWhile P xs = xs) = (lset xs \<subseteq> {x. P x})"
representation_scale: fixes basis :: "'b set"   and v :: "'b"   and r :: "'a" assumes "vs1.independent basis"   and "v \<in> vs1.span basis" shows "vs1.representation basis (r *a v) = (\<lambda>b. r * vs1.representation basis v b)"
inv_into_f_eq: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and x :: "'f"   and y :: "'g" assumes "inj_on f A"   and "x \<in> A"   and "f x = y" shows "inv_into A f y = x"
not_is_Tensor_TensorDiagE: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "\<not> is_Tensor (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"   and "Diag t"   and "Diag u"   and "t \<noteq> \<^bold>\<I>"   and "u \<noteq> \<^bold>\<I>" shows "False"
elem_in_image2: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and A1 :: "'c set"   and x :: "'c" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A"   and "x \<in> A1" shows "f x \<in> f ` A1"
to_tree_order_node_ptrs: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "ptr' \<noteq> ptr"   and "ptr' \<in> set nodes" shows "is_node_ptr_kind ptr'"
sg_gen_def: fixes G :: "('c, 'd) Group_scheme"   and A :: "'c set" shows "sg_gen G A = \<Inter> {H. G \<guillemotright> H  \<and> A \<subseteq> H}"
pmaxsimpchainI_maxsimpchain: fixes xs :: "'a set list" assumes "folding_g.gallery xs"   and "distinct xs" shows "pgallery xs"
per_double_cong: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and C' :: "'p" assumes "Per A B C"   and "B Midpoint C C'" shows "Cong A C A C'"
neg_equal_iff_equal: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(\<ominus> x = \<ominus> y) = (x = y)"
hom_graph_dest: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b"   and k :: "'a" assumes "hom_graph f S x a"   and "k \<in> S" shows "\<exists>y z b c. x = ifte (var k) y z \<and> a = ifte (f k) b c \<and> hom_graph f (S - {k}) y b \<and> hom_graph f (S - {k}) z c"
is_Ref_def: fixes v :: "'b val" shows "is_Ref v \<equiv> v = Null \<or> is_Addr v"
hom_minus: fixes x :: "rat"   and y :: "rat" shows "of_rat (x - y) = of_rat x - of_rat y"
Pi_I: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
unit_simps_3: fixes a :: "'a" assumes "B.obj a" shows "trg\<^sub>B \<i>\<^sub>B[a] = a"
vector_derivative_const_at: fixes c :: "'a"   and a :: "real" shows "vector_derivative (\<lambda>x. c) (at a) = (0::'a)"
is_acc_reftype_def: fixes G :: "prog"   and P :: "pname"   and T :: "ref_ty" shows "is_acc_reftype G P T = (isrtype G T \<and> G \<turnstile> T accessible_in' P)"
card_image_inversion: shows "real (card (inversion ` dc_crypto)) = 2 ^ n / 2"
uf_graph_invar_preserve: fixes uf :: "('a \<times> 'a) set"   and T :: "'edge set"   and a :: "'a"   and b :: "'a"   and e :: "'edge" assumes "uf_graph_invar (uf, T)"   and "a \<in> V"   and "b \<in> V"   and "joins a b e"   and "e \<in> E"   and "T \<subseteq> E" shows "uf_graph_invar (per_union uf a b, insert e T)"
bisims1List2: fixes P :: "'a prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and es' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1"   and v :: "'addr val" assumes "P,es,h \<turnstile> (es', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)" shows "P,e # es,h \<turnstile> (Val v # es', xs) [\<leftrightarrow>] (stk @ [v], loc, length (compE2 e) + pc, xcp)"
set_plus_mono_b: fixes C :: "'a set"   and D :: "'a set"   and x :: "'a"   and a :: "'a" assumes "C \<subseteq> D"   and "x \<in> a +o C" shows "x \<in> a +o D"
FunctorComp_def: fixes F :: "('a, 'g, 'c, 'h, 'e, 'i) Functor"   and G :: "('g, 'b, 'h, 'd, 'j, 'f) Functor" shows "F ;;; G \<equiv> MakeFtor (F ;;: G)"
g_def: shows "g \<equiv> \<lambda>y. THE x. norm x < \<delta> \<and> f x = y"
path_set_mem: fixes p :: "nat list"   and G :: "graph"   and m :: "nat"   and n :: "nat" shows "(p \<in> path_set G m n) = (is_path G p \<and> hd p = m \<and> last p = n)"
set_iterator_rule_insert_P: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma>0 :: "'c"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "set_iterator it S0"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0\<rbrakk> \<Longrightarrow> I (insert x S) (f x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> c \<sigma>; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
nocross_events_neq: fixes Q :: "'a set"   and R :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "R \<in> \<P>"   and "a \<in> Q"   and "b \<in> R"   and "R \<inter> Q = {}" shows "a \<noteq> b"
B: shows "B = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) C"
rcs_eq2: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "H \<bullet> a \<inter> (H \<bullet> b) \<noteq> {}" shows "H \<bullet> a = H \<bullet> b"
PEI_2: shows "(\<lambda>i. ValTT) \<in> unlr PE.delta"
ideal_prod_la2: fixes I :: "'a set"   and J :: "'a set" assumes "ideal R I"   and "ideal R J" shows "I \<diamondsuit>\<^sub>r J \<subseteq> J"
cases: fixes x :: "prop_rule" obtains "x = andI"   | "x = andE"   | "x = impI"   | "x = impE"
f_foldl_abs_alt_def: fixes \<phi> :: "'b ltl"   and w :: "'b set list" shows "af_abs.f_foldl_abs (Abs \<phi>) w = foldl af_abs.f_abs (Abs \<phi>) w"
count_le_length: fixes xs :: "'a list"   and x :: "'a" shows "count_list xs x \<le> length xs"
set_relI: fixes A :: "'d set"   and B :: "'e set"   and R :: "('d \<times> 'e) set" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. (x, y) \<in> R"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. (x, y) \<in> R" shows "(A, B) \<in> \<langle>R\<rangle>set_rel"
fixp_induct: fixes P :: "(('c stream \<Rightarrow> bool) \<Rightarrow> 'c stream \<Rightarrow> enat) \<Rightarrow> bool" assumes "gfp.admissible (\<lambda>P. P (curry P))"   and "P (\<lambda>\<omega> y. \<Sqinter> {})"   and "\<And>f. P f \<Longrightarrow> P (\<lambda>a b. if a b then 0 else eSuc (f a (stl b)))" shows "P sfirst"
minimal_def: shows "minimal \<equiv> accessible = dfa.states M \<and> inj_on right_lang (dfa.states M)"
set_minus_imp_plus: fixes a :: "'a"   and b :: "'a"   and C :: "'a set" assumes "a - b \<in> C" shows "a \<in> b +o C"
rows_transpose: fixes A :: "'b mat" shows "rows A\<^sup>T = cols A"
mult_assoc: fixes a :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool" shows "((a \<and>* b) \<and>* c) = (a \<and>* b \<and>* c)"
f_def: shows "f = foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<Suc k]"
ext: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "\<And>x. f x = g x" shows "f = g"
fresh_var_monoinc: fixes start :: "nat"   and cs :: "('a \<times> QDelta ns_constraint) list" shows "start \<le> FirstFreshVariable (preprocess' cs start)"
sgn_mult_abs: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "times (sgn a) (abs a) = a"
B: fixes x :: "'a" assumes "x \<in> S" shows "\<bar>h x\<bar> \<le> B"
equivalence_dual: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L" shows "equivalence (inv_gorder L)"
simps_30: fixes vd :: "'a"   and v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a" shows "(SKIP {vd} \<le> IF v THEN va ELSE vb {vc}) = False"
Gateway_L6a: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and k :: "nat"   and t :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "\<forall>m\<le>k. req (t + 2 + m) \<noteq> [send]"   and "\<forall>j\<le>k. lose (t + 2 + j) = [False]"   and "ack (t + 2) = [connection_ok]"   and "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose" shows "\<forall>m\<le>k. ack (t + 2 + m) = [connection_ok]"
inject: fixes x1 :: "int"   and x2 :: "int"   and y1 :: "int"   and y2 :: "int" shows "(Gauss_Int x1 x2 = Gauss_Int y1 y2) = (x1 = y1 \<and> x2 = y2)"
summset_def: fixes G :: "('e, 'f) ring_scheme"   and M :: "'e multiset" assumes "abelian_monoid G" shows "abelian_monoid.summset G M \<equiv> fold_mset (\<oplus>\<^bsub>G\<^esub>) \<zero>\<^bsub>G\<^esub> M"
dgrad_p_set_le_args_to_set_struct: fixes d :: "'a \<Rightarrow> nat"   and sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data' :: "nat \<times> 'g" assumes "dickson_grading d"   and "struct_spec sel ap ab compl"   and "ps \<noteq> []"   and "sps = sel gs bs ps data"   and "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)" shows "dgrad_p_set_le d (args_to_set (gs, ab gs bs hs data', ap gs bs (ps -- sps) hs data')) (args_to_set (gs, bs, ps))"
order_divides: fixes a :: "'b"   and n :: "nat"   and p :: "'b poly" shows "([:- a, 1::'b:] ^ n dvd p) = (p = 0 \<or> n \<le> order a p)"
sharing_consistent_shared_exchange: fixes sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set" assumes "\<forall>a\<in>all_acquired sb. \<S>' a = \<S> a"   and "sharing_consistent \<S> \<O> sb" shows "sharing_consistent \<S>' \<O> sb"
vars_mset_subst: fixes \<rho> :: "'c \<Rightarrow> ('d, 'c) tm"   and s :: "('d, 'c) tm" shows "vars_mset (subst \<rho> s) = (\<Sum>x\<in>#vars_mset s. vars_mset (\<rho> x))"
inv_mult_group: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V" shows "\<ominus>\<^bsub>V\<^esub> (x \<oplus>\<^bsub>V\<^esub> y) = \<ominus>\<^bsub>V\<^esub> y \<oplus>\<^bsub>V\<^esub> \<ominus>\<^bsub>V\<^esub> x"
52522: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (- x \<sqinter> y) = - x \<sqinter> - y"
not_neutral_contains_not_neutral: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "F.F g A \<noteq> \<^bold>1" obtains a :: "'c" where "a \<in> A" and "g a \<noteq> \<^bold>1"
closed_vimage_Int: fixes s :: "'b set"   and t :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "closed s"   and "continuous_on t f"   and "closed t" shows "closed (f -` s \<inter> t)"
matrix_to_iarray_Hermite_reduce_above: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "n < nrows A" shows "matrix_to_iarray (Hermite_reduce_above A n i j res) = Hermite_reduce_above_iarrays (matrix_to_iarray A) n (mod_type_class.to_nat i) (mod_type_class.to_nat j) res"
m: shows "\<forall>x y. eval r_phi [the (eval m [x]), y] = eval r_phi [the (eval g [the (eval m [x]), x]), y]"
left_Nonce: fixes N :: "nat" shows "left (Nonce N) = Nonce N"
r2s_inter_hom: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R \<subseteq> Id"   and "S \<subseteq> Id" shows "r2s (R \<inter> S) = r2s R \<inter> r2s S"
simps: fixes v :: "'a" shows "\<lceil>\<lfloor>v\<rfloor>\<rceil> = v"
preserves_iso: fixes f :: "'a \<times> 'a" assumes "C.VV.iso f" shows "C.iso (fst f \<star>\<^sub>C snd f)"
list_all2': shows "list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map pos tp' a1_map) (\<lambda>ts' tp'. filter_a2_map I ts' tp' a2_map)) tl_aux (zip (linearize (tl_queue tss')) [tp - (len - 1)..<tp])"