hcju/isabelleps huggingface dataset:
simps: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'a \<times> 'c"   and a2 :: "'b \<times> 'd" shows "rel_prod R1 R2 a1 a2 = (\<exists>a b c d. a1 = (a, c) \<and> a2 = (b, d) \<and> R1 a b \<and> R2 c d)"
sel: fixes x :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf" shows "the_gpv (GPV x) = x"
BALL: shows "GEN_OP ball op_map_ball (\<langle>Rk, Rv\<rangle>Rm \<rightarrow> (Rk \<times>\<^sub>r Rv \<rightarrow> bool_rel) \<rightarrow> bool_rel)"
wf_trm_subterm: fixes t :: "('fun, 'b) Term.term"   and s :: "('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "s \<sqsubset> t" shows "wf\<^sub>t\<^sub>r\<^sub>m s"
scale_scale: fixes a :: "'a \<Rightarrow>\<^sub>0 'c"   and b :: "'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" shows "a \<odot> (b \<odot> x) = (a * b) \<odot> x"
dvd_mult_div_cancel: fixes divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times a (divide b a) = b"
f''_def: shows "f'' = fpxs_of_fls (fps_to_fls (fps_radical rt r f'))"
extract_int_def: fixes val :: "val" shows "extract_int val = (case val of IntVal x3 \<Rightarrow> x3 | _ \<Rightarrow> 0)"
lang_samequot_exec: fixes n :: "nat"   and r :: "'a atom rexp"   and as :: "'a \<times> bool list" assumes "local.wf n r"   and "as \<in> (set \<circ> \<sigma> \<Sigma>) n" shows "lang n (samequot_exec as r) = SAMEQUOT as (lang n r)"
assms_2: shows "s1 O ns1 \<subseteq> s1"
VWO_def: shows "VWO \<equiv> SOME r. {(x, y). x \<in> elts y} \<subseteq> r \<and> Well_order r \<and> Field r = UNIV"
synthetic_div_unique_lemma: fixes c :: "'a"   and p :: "'a poly"   and a :: "'a" assumes "smult c p = pCons a p" shows "p = 0"
wtPB_def: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and arOf :: "'f \<Rightarrow> 'e list"   and resOf :: "'f \<Rightarrow> 'e"   and parOf :: "'g \<Rightarrow> 'e list"   and \<Phi> :: "('f, 'g) lit list set" assumes "Signature TYPE('e) wtFsym wtPsym" shows "Signature.wtPB wtFsym wtPsym arOf resOf parOf \<Phi> \<equiv> \<forall>c\<in>\<Phi>. Signature.wtC wtFsym wtPsym arOf resOf parOf c"
circ_right_unfold_sub: fixes x :: "'a" shows "Rf (x\<^sup>\<circ>) x \<le> x\<^sup>\<circ>"
horizontal_composite: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H :: "'b \<Rightarrow> 'e"   and K :: "'b \<Rightarrow> 'e"   and \<tau> :: "'b \<Rightarrow> 'e" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation B C H K \<tau>" shows "natural_transformation A C (H \<circ> F) (K \<circ> G) (\<tau> \<circ> \<sigma>)"
lipschitz_on_le: fixes M :: "real"   and D :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and L :: "real" assumes "M-lipschitz_on D f"   and "M \<le> L" shows "L-lipschitz_on D f"
restrict_vals_nonequiv2: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant" assumes "vals_nonequiv K (Suc (Suc n)) vv" shows "vals_nonequiv K (Suc n) (compose {j. j \<le> Suc n} vv (skip 2))"
sublist_append_leftI: fixes xs :: "'b list"   and ps :: "'b list" shows "xs \<le>f ps \<cdot> xs"
expand_set: fixes g :: "'d \<Rightarrow> 'a" shows "Sum_any g = sum g {a. g a \<noteq> (0::'a)}"
return_wp_rule: fixes Q :: "'a \<Rightarrow> assn"   and x :: "'a" shows "<Q x> return x <Q>"
norm_bound_component_lt_cart: fixes x :: "(real, 'a) vec"   and e :: "real"   and i :: "'a" assumes "norm x < e" shows "\<bar>x $ i\<bar> < e"
fg_invariant2a: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and n :: "nat"   and q :: "'f"   and p :: "'f"   and m :: "'h" assumes "fg_run A rho HOs SHOs coords"   and "rcvd (rho n) q p = Some m" shows "p \<in> HOs (round (rho n) q) q"
assms: shows "\<forall>i. a $ i < b $ i \<and> u $ i < v $ i"
that: fixes m :: "'node" assumes "m \<in> set ns"   and "var g v \<in> oldDefs g m" shows "thesis"
g_bound: fixes x :: "'b" shows "g x \<le> \<alpha> (f top)"
induct: fixes g :: "('d, 'c) generator"   and P :: "'c \<Rightarrow> bool"   and a0 :: "'c" assumes "\<And>s. \<lbrakk>\<And>x2. generator g s = Skip x2 \<Longrightarrow> P x2; \<And>x31 x32. generator g s = Yield x31 x32 \<Longrightarrow> P x32\<rbrakk> \<Longrightarrow> P s" shows "P a0"
smc_SemiCAT_is_arr_iff: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<BB> = \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
KIR: fixes x :: "'a \<times> 'a" assumes "x \<notin> A \<Gamma>"   and "x \<notin> B \<Gamma>" shows "KIR f x"
ennreal_add_diff_cancel_left: fixes y :: "ennreal"   and x :: "ennreal" assumes "y \<noteq> top" shows "y + x - y = x"
branch_nominals_def: fixes branch :: "(('c, 'b) fm list \<times> 'b) list" shows "branch_nominals branch \<equiv> \<Union> (block_nominals ` set branch)"
"1_1": shows "F \<^bold>\<or> G \<in> \<Gamma>"
eq_refl: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x = y" shows "x \<subseteq># y"
lessThan_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {..<Suc n} = sum g {..<n} + g n"
finBset: shows "finite ({x. f (swap b i x) \<noteq> c} - {i, b} \<union> {i, b})"
label_incr_Exit: fixes n :: "w_node"   and i :: "nat" assumes "n \<oplus> i = (_Exit_)" shows "n = (_Exit_)"
safe_reach_steps: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and safe :: "'b \<Rightarrow> bool"   and cfg :: "'b"   and cfg' :: "'b" assumes "safe_reach step safe cfg"   and "step\<^sup>*\<^sup>* cfg cfg'" shows "safe_reach step safe cfg'"
add_right_imp_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.cancel_semigroup_add plus"   and "plus b a = plus c a" shows "b = c"
Pbij_inverse_def: fixes \<beta> :: "(Location \<times> Location) set" shows "Pbij_inverse \<beta> = {(l, ll). (ll, l) \<in> \<beta>}"
PdXXX_range: fixes Pd' :: "'b \<Rightarrow> ('a \<times> 'a) set"   and d' :: "'b"   and d :: "'b" assumes "Linear_order (Pd' d')"   and "Field (Pd' d') \<subseteq> {y. Xd y = d'}"   and "\<forall>d. d \<noteq> d' \<longrightarrow> Pd' d = Pd' d" shows "Field (Pd' d) \<subseteq> {x. Xd x = d}"
Suc_2: shows "latin_rect s (n - Suc m) n"
update_simps_21: fixes T :: "(nat \<times> linear_poly) list"   and s :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>u (\<T>_update T s) = \<B>\<^sub>i\<^sub>u s"
assms_2: shows "S \<inter> T retract_of UNIV"
rangeI: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "f x \<in> range f"
bar_idx: fixes ws :: "letter list list"   and f :: "nat \<Rightarrow> letter list" assumes "bar ws"   and "is_prefix ws f" shows "\<exists>i j. emb (f i) (f j) \<and> i < j"
u_def: shows "u = iP *\<^sub>v of_real_hom.vec_hom v"
clean_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "clean A w V \<equiv> {v \<in> V. infinite (greachable A w V v)}"
zero_le_constraint_def: shows "0 = Le_Constraint Leq_Rel 0 (0::'a)"
homf_preserves_comp: fixes f :: "'a"   and g :: "'a"   and A :: "'o" assumes "f \<in> Ar"   and "g \<in> Ar"   and "Cod f = Dom g" shows "Hom(A,_) \<^bsub>\<a>\<^esub> (g \<bullet> f) = Hom(A,_) \<^bsub>\<a>\<^esub> g \<odot> Hom(A,_) \<^bsub>\<a>\<^esub> f"
exec_move_AAss1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and e'' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e\<lfloor>e'\<rceil> := e'') h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
r_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier R"   and "a \<in> carrier R" shows "(x = a \<oplus> x) = (a = \<zero>)"
Suc_1: fixes Us' :: "'a list"   and Us :: "'a list" assumes "n = length Us'"   and "local.independent K (Us' @ Us)"   and "local.independent K Vs"   and "Span K (Us' @ Us) \<subseteq> Span K Vs" shows "\<exists>Vs'. set Vs' \<subseteq> set Vs \<and> length Vs' = length Us' \<and> local.independent K (Vs' @ Us)"
that: fixes r :: "real" assumes "0 < r"   and "ball a r \<subseteq> S"   and "inj_on f (ball a r)" shows "thesis"
results_3: shows "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)"
if_false_eval\<^sub>w\<^sub>C: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat"   and x :: "var\<^sub>C"   and c\<^sub>C_then :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_else :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set" shows "mem\<^sub>C x \<noteq> 0 \<longrightarrow> eval_abv\<^sub>C \<langle>Stmt.If (bexp\<^sub>C.Eq x 0) c\<^sub>C_then c\<^sub>C_else ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>C_else ;; c\<^sub>C_tail, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C"
psubstEnv_substEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "goodEnv rho'" shows "rho &[Y / y]_ys &[rho'] = rho &[rho' [y \<leftarrow> Y #[rho']]_ys]"
ins_non_Leaf: fixes x :: "'a"   and v :: "'b"   and t :: "('a, 'b) rbt" shows "ins x v t \<noteq> Leaf"
leq_trans: fixes x :: "'b llist"   and y :: "'b llist"   and z :: "'b llist" assumes "lprefix x y"   and "lprefix y z" shows "lprefix x z"
convolution_def: fixes M :: "'b measure"   and N :: "'b measure" shows "(M \<star> N) = distr (M \<Otimes>\<^sub>M N) borel (\<lambda>(x, y). x + y)"
chamber_in_other_half_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "SimplicialComplex.maxsimp X C"   and "C \<notin> f \<turnstile> ChamberComplex.chamber_system X" shows "C \<in> g \<turnstile> ChamberComplex.chamber_system X"
terminal_def: fixes b :: "'c" shows "C.terminal b \<equiv> C.ide b \<and> (\<forall>a. C.ide a \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>))"
incl_def: fixes f :: "'s" shows "S.incl f = (S.arr f \<and> S.Dom f \<subseteq> S.Cod f \<and> f = S.incl_of (S.dom f) (S.cod f))"
length_int_def: fixes x :: "int" shows "length_int x = (if 0 < x then Suc (length_nat (nat x)) else if x = 0 then 0 else Suc (length_nat (nat (- x - 1))))"
affine_closed: fixes S :: "'b set" assumes "affine S" shows "closed S"
mono_inc_bdd_above_has_limit_at_topI: fixes f :: "real \<Rightarrow> real"   and u :: "real" assumes "mono f"   and "\<And>x. f x \<le> u" shows "\<exists>l. (f \<longlongrightarrow> l) at_top"
sum_Un_nat: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> nat" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
irreducible_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "fixed_mat.irreducible A = (UNIV \<subseteq> (fixed_mat.G A)\<^sup>+)"
l2_step2_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "l2_step2 Rb A B Ni gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> Rb \<notin> dom (progress s) \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni \<and> sklvl1.guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny))) \<and> Confid A B Ni \<in> chan s \<and> Insec A B gnx \<in> chan s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xni, xnr, xgny, xgnx, xsk}), chan := {Confid B A (NonceF (Rb $ nr))} \<union> ({Insec B A \<langle>Exp Gen (NonceF (Rb $ ny)), hmac \<langle>Number 0, gnx, Exp Gen (NonceF (Rb $ ny)), Messages.Agent B, Messages.Agent A\<rangle> (hmac Ni (NonceF (Rb $ nr)))\<rangle>} \<union> chan s), signalsInit := if can_signal s A B then addSignal (signalsInit s) (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) else signalsInit s, signalsInit2 := if can_signal s A B then addSignal (signalsInit2 s) (Running A B (Exp gnx (NonceF (Rb $ ny)))) else signalsInit2 s\<rparr>}"
indep_in_iff_subset_basis_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier" shows "indep_in \<E> X = (\<exists>B. basis_in \<E> B \<and> X \<subseteq> B)"
conjugation_is_bij: fixes g :: "'a" assumes "g \<in> carrier G" shows "bij_betw (\<lambda>h\<in>carrier G. g \<otimes> h \<otimes> inv g) (carrier G) (carrier G)"
pfun_member_minus: fixes x :: "'a"   and y :: "'b"   and f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "((x, y) \<in>\<^sub>p f - g) = ((x, y) \<in>\<^sub>p f \<and> \<not> (x, y) \<in>\<^sub>p g)"
get_scdom_component_different_owner_documents: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr"   and ptr' :: "(_) object_ptr"   and owner_document' :: "(_) document_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_owner_document ptr \<rightarrow>\<^sub>r owner_document"   and "h \<turnstile> Shadow_DOM.get_owner_document ptr' \<rightarrow>\<^sub>r owner_document'"   and "owner_document \<noteq> owner_document'" shows "set |h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr|\<^sub>r \<inter> set |h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr'|\<^sub>r = {}"
add_mset_add_single: fixes a :: "'c"   and A :: "'c multiset" shows "add_mset a A = A + {#a#}"
length_butlast: fixes xs :: "'d list" shows "length (butlast xs) = length xs - 1"
finite_pdigraph_dg_prod_2: shows "finite_pdigraph \<alpha> (2\<^sub>\<nat>) (\<lambda>i. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>)"
is_volatile_Write\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_volatile_Write\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = False"
intros_4: fixes m :: "bool list"   and \<eta> :: "nat"   and a :: "bool list" assumes "length m = id' \<eta>" shows "S' \<eta> (return_spmf (Inl (Some (a, m)), (), Store m)) (return_spmf (Store m, None, [m \<mapsto> a]))"
hvzk_z2_r2_tuple_rewrite: fixes xb :: "nat"   and x' :: "nat"   and xa :: "nat"   and x1 :: "nat"   and x2 :: "nat" assumes "xb < order \<G>" shows "(\<^bold>g [^] (((x' + xa * x1) mod order \<G> + order \<G> * xa * x1 - xa * x1) mod order \<G>) \<otimes> g' [^] xb, xa, (x' + xa * x1) mod order \<G>, (xb + xa * x2) mod order \<G>) = (\<^bold>g [^] (((x' + xa * x1) mod order \<G> + order \<G> * xa * x1 - xa * x1) mod order \<G>) \<otimes> g' [^] (((xb + xa * x2) mod order \<G> + order \<G> * xa * x2 - xa * x2) mod order \<G>), xa, (x' + xa * x1) mod order \<G>, (xb + xa * x2) mod order \<G>)"
ACI_Times_cong: fixes r1 :: "'b rexp"   and s1 :: "'b rexp"   and r2 :: "'b rexp"   and s2 :: "'b rexp" assumes "ACI r1 s1"   and "ACI r2 s2" shows "ACI (Times r1 r2) (Times s1 s2)"
vs: fixes K :: "('a, 'b) ring_scheme"   and W :: "'c set"   and V :: "('a, 'c, 'd) module_scheme" assumes "VectorSpace.subspace K W V" shows "vectorspace K V"
in_dom_m_lookup_iff: fixes C :: "'a"   and N' :: "('a, 'b) fmap" shows "(C \<in># dom_m N') = (fmlookup N' C \<noteq> None)"
powr_nonneg_iff: fixes a :: "real"   and x :: "real" shows "(a powr x \<le> 0) = (a = 0)"
add_block_sub_sys: fixes b :: "'a set" shows "sub_incidence_system \<V> \<B> (\<V> \<union> b) (add_block b)"
cinner_add_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (plus y z) = cinner x y + cinner x z"
measure_spmf_bind: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "measure_spmf (p \<bind> f) = measure_spmf p \<bind> measure_spmf \<circ> f"
patom: fixes n :: "nat"   and m :: "nat"   and p :: "nat"   and vs :: "nat list"   and xs :: "(nat \<times> form) list"   and nfs :: "(nat \<times> form) list" assumes "(n, (m, PAtom p vs) # xs) \<in> deriv nfs"   and "\<not> is_axiom (s_of_ns ((m, PAtom p vs) # xs))" shows "(Suc n, xs @ [(0, PAtom p vs)]) \<in> deriv nfs"
uniqueness_solution_imp_independent: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "\<exists>!x. is_solution x A b" shows "vec.dim (solution_set A 0) = 0"
simps: fixes a :: "'a"   and b :: "'b"   and c :: "'c"   and d :: "'a" shows "mat_trace (a, b, c, d) = a + d"
cond_def: fixes b :: "'d upred"   and P :: "'d upred"   and Q :: "'d upred" shows "P \<triangleleft> b \<triangleright> Q = (b \<and> P \<or> \<not> b \<and> Q)"
ldrop_trans_fusion: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and n :: "enat"   and s :: "'d" shows "lunstream (ldrop_trans g) (n, s) = ldrop n (lunstream g s)"
assms_4: shows "index (mtf2 n q xs) (xs ! i) < index xs q - n"
A2b: fixes y :: "i" shows "dom cod y \<cong> cod y"
disc_14: fixes x4 :: "'d" shows "\<not> isRecv (Snapshot x4)"
dbl_simps_5: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit0 k)"
STL6: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>(F \<and> G) = (\<diamond>\<box>F \<and> \<diamond>\<box>G)"
assoc: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" shows "signed.min (signed.min a b) c = signed.min a (signed.min b c)"
genPolyTame_spec: fixes n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph" shows "generatePolygonTame n v f g = filter (\<lambda>g'. \<not> notame g') (generatePolygon n v f g)"
intersects_x_axis_positive_iff': fixes H :: "circline" assumes "is_poincare_line H" shows "intersects_x_axis_positive H = (calc_x_axis_intersection H \<in> unit_disc \<and> calc_x_axis_intersection H \<in> circline_set H \<inter> positive_x_axis)"
nset_def: fixes i :: "nat"   and j :: "nat" shows "nset i j = {k. i \<le> k \<and> k \<le> j}"
mult_1_right: fixes a :: "'c" shows "bounded_distrib_lattice_class.comp_inf.vector a"
assms: shows "x \<cdot> y = u \<cdot> v"
negative_imp_inverse_negative: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less (inverse a) zero"
prems_6: shows "set xs \<subseteq> set (ops @ [a])"
simps: fixes a :: "'a list"   and T :: "('a, 'b) TS_scheme" shows "(a \<in> beh T) = (a = [] \<or> (\<exists>s. a = [s] \<and> s \<in> init T) \<or> (\<exists>s b t. a = t # s # b \<and> s # b \<in> beh T \<and> (s, t) \<in> TS.trans T))"
bij_lens_equiv_id: fixes X :: "'b \<Longrightarrow> 'c" shows "bij_lens X = X \<approx>\<^sub>L 1\<^sub>L"
front_tickFree_single: fixes a :: "'a event" shows "front_tickFree [a]"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "plus_hom hom" shows "semigroup_add_hom hom"
lcm_gcd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "lcm a b = normalize (divide (times a b) (gcd a b))"
distinct_keys: fixes rbt :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "distinct (RBT_Set2.keys rbt)"
DERIV_cmult_Id: fixes c :: "'a"   and x :: "'a"   and s :: "'a set" shows "((*) c has_field_derivative c) (at x within s)"
add_Sup_distrib: fixes y :: "V"   and x :: "V"   and f :: "V \<Rightarrow> V" assumes "y \<noteq> 0" shows "x + \<Squnion> (f ` elts y) = (SUP z\<in>elts y. x + f z)"
prems_1: shows "\<forall>a\<in>set (x # l). case a of (v, a) \<Rightarrow> \<forall>b\<in>set (x # l). case b of (v', a') \<Rightarrow> v \<noteq> v' \<or> a = a'"
hyps: assumes "distinct (map fst ps)" shows "distinct (map fst (deleteAround g (fst (a, b)) ps))"
assms: shows "\<turnstile> A \<longrightarrow> B"
subst_lemma: fixes x :: "name"   and y :: "name"   and u :: "trm"   and v :: "trm"   and s :: "trm" assumes "x \<noteq> y"   and "x \<sharp> u" shows "s[x::=v][y::=u] = s[y::=u][x::=v[y::=u]]"
bit_d': fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom x, Atom x])" shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' BIT s qs = 0"
surjective: fixes r :: "'b heap_scheme" shows "r = \<lparr>arrays = arrays r, refs = refs r, lim = lim r, \<dots> = more r\<rparr>"
Diag_Diagonalize_1: fixes t :: "'a local.term" assumes "Arr t" shows "Diag \<^bold>\<lfloor>t\<^bold>\<rfloor>"
simps_1: fixes T1 :: "'fsym trm"   and T2 :: "'fsym trm" shows "wtA (Eq T1 T2) = (wt T1 \<and> wt T2 \<and> tpOf T1 = tpOf T2)"
mhomom3Tr1: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "surjec\<^bsub>mdl M H,mdl M (H \<minusplus> K) /\<^sub>m K\<^esub> compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (\<iota>m\<^bsub>M H,K\<^esub>)"
g_def: shows "g \<equiv> Cn 5 r_snoc [recf.Id 5 1, Cn 5 r_ifeq [recf.Id 5 0, recf.Id 5 3, recf.Id 5 4, Cn 5 r_nth [recf.Id 5 2, recf.Id 5 0]]]"
rec_by_index_def: shows "rec_by_index = (\<lambda>n1 n2. c_pair 6 (c_pair n1 n2))"
member_in_declC: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_in C" shows "G \<turnstile> m member_in declclass m"