hcju/isabelleps huggingface dataset:
upd: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "bij_betw upd {..<n} {..<n}"
med_progress_None: fixes r :: "fid_t \<Rightarrow> Runs.var set option"   and x :: "fid_t" shows "(med_progress r x = None) = (r x = None)"
HPhase1or2Write_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv4d s q"   and "HInv4a s p" shows "HInv4d s' q"
F_correct: fixes l :: "nat"   and lm :: "'a"   and tp :: "(action \<times> nat) list"   and stp :: "nat"   and m :: "nat"   and rs :: "nat"   and n :: "nat" assumes "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"   and "tm_wf0 tp"   and "0 < rs" shows "rec_exec rec_F [code tp, bl2wc (<lm>)] = rs - Suc 0"
sgn_mult: fixes a :: "'a"   and b :: "'a" shows "sgn (a * b) = sgn a * sgn b"
echelon_form_of_iarrays_def: fixes A :: "'d iarray iarray"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "echelon_form_of_iarrays A bezout = echelon_form_of_upt_k_iarrays A (ncols_iarray A - 1) bezout"
last_P_new_2: fixes v :: "'a" assumes "v \<in> set (butlast P_new)" shows "\<not> hitting_paths v"
vimage_algebra_def: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and M :: "'e measure" shows "vimage_algebra X f M = sigma X {f -` A \<inter> X |A. A \<in> sets M}"
is_Fun_Set_def: fixes t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term" shows "is_Fun_Set t \<equiv> is_Fun t \<and> args t = [] \<and> is_Set (the_Fun t)"
a_in_de: shows "a \<in> open_segment (a - complex_of_real d) (a + complex_of_real e)"
ltl_const_equiv_implies_prop_equiv_2: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>C false\<^sub>n = \<phi> \<sim>\<^sub>P false\<^sub>n"
Thm_def: fixes idx :: "idx"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" shows "Thm idx \<phi> = check_eqv idx \<phi> \<top>"
integrable_by_parts_interior_strong: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'b" assumes "bounded_bilinear prod"   and "finite s"   and "a \<le> b"   and "continuous_on {a..b} f"   and "continuous_on {a..b} g"   and "\<And>x. x \<in> {a<..<b} - s \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "\<And>x. x \<in> {a<..<b} - s \<Longrightarrow> (g has_vector_derivative g' x) (at x)"   and "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}" shows "(\<lambda>x. prod (f' x) (g x)) integrable_on {a..b}"
octo_of_real_def: fixes r :: "real" shows "octo_of_real r = r *\<^sub>R 1"
sigma_algebra_inter: fixes A :: "'b set set"   and a :: "'b set"   and b :: "'b set" assumes "sigma_algebra A"   and "a \<in> A"   and "b \<in> A" shows "a \<inter> b \<in> A"
NF_ND: fixes s :: "'a event list"   and X :: "'a event set"   and P :: "'a process" assumes "(s, X) \<notin> F P" shows "s \<notin> D P"
case_ifexi_def: fixes fti :: "bdd \<Rightarrow> 'c option"   and ffi :: "bdd \<Rightarrow> 'c option"   and fii :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bdd \<Rightarrow> 'c option"   and ni :: "nat"   and s :: "bdd" shows "mi.case_ifexi fti ffi fii ni s \<equiv> destrmi' ni s \<bind> (\<lambda>dest. case dest of TD \<Rightarrow> fti s | FD \<Rightarrow> ffi s | IFD v ti ei \<Rightarrow> fii v ti ei s)"
augFlow_resFlow: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" assumes "NPreflow c s t f"   and "NPreflow.isAugmentingPath c s t f p" shows "Flow (residualGraph c f) s t (NPreflow.augmentingFlow c f p)"
Rep_one_or_two: fixes x :: "'a one_or_two" shows "Rep_one_or_two x \<in> one_or_two"
obj_simps_3: fixes a :: "'a" assumes "B.obj a" shows "trg\<^sub>B a = a"
nonempty_ex: fixes A :: "'h set" assumes "A \<noteq> {}" shows "\<exists>x. x \<in> A"
rref_upt_condition4: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) = (0::'a))"
is_tm_dghm_axioms: shows "\<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
of_rat_rat: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "of_rat (Fract a b) = of_int a / of_int b"
Sup_upper: fixes x :: "'b \<times> 'c"   and A :: "('b \<times> 'c) set" assumes "x \<in> A" shows "less_eq_prod x (Sup_prod A)"
id1_1: shows "(\<lambda>x. x) \<in> PrimRec1"
dlist_member_insert: fixes dl :: "'b dlist"   and x :: "'b" assumes "member dl x" shows "DList.insert x dl = dl"
wave_lub: fixes Y :: "('k \<times> 'k \<Rightarrow> ennreal) set"   and \<Gamma> :: "('k, 'l) web_scheme" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "\<And>f. f \<in> Y \<Longrightarrow> wave \<Gamma> f"   and "countable (support_flow (\<Squnion> Y))" shows "wave \<Gamma> (\<Squnion> Y)"
envTrans_def: fixes eact :: "'ePubAct \<times> 'ePrivAct"   and aact :: "'a \<Rightarrow> 'pPubAct \<times> 'ps"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" shows "envTrans eact aact s \<equiv> \<lparr>es = envTransES eact (fst \<circ> aact) (es s), ps = snd \<circ> aact, pubActs = (fst eact, fst \<circ> aact)\<rparr>"
apsnd_conv: fixes f :: "'g \<Rightarrow> 'f"   and x :: "'e"   and y :: "'g" shows "apsnd f (x, y) = (x, f y)"
sum_list_addf: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.comm_monoid_add plus zero" shows "monoid_add.sum_list plus zero (map (\<lambda>x. plus (f x) (g x)) xs) = plus (monoid_add.sum_list plus zero (map f xs)) (monoid_add.sum_list plus zero (map g xs))"
bin_rsplit_aux_simp_alt: fixes n :: "nat"   and m :: "nat"   and c :: "int"   and bs :: "int list" shows "bin_rsplit_aux n m c bs = (if m = 0 \<or> n = 0 then bs else let (a, b) = bin_split n c in bin_rsplit n (m - n, a) @ b # bs)"
assms_1: fixes gs :: "(('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and bs :: "(('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and h :: "('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and ps :: "(bool \<times> (('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) \<times> ('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list" shows "set (aps flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))"
comp_converter_id_left: fixes conv :: "('y, 'z, 'aa, 'ab) converter" shows "1\<^sub>C \<odot> conv = conv"
CASE: fixes p :: "'p"   and w :: "('p, 'ba) label list"   and u' :: "'n"   and v :: "'n" assumes "eel = LOC (LCall p # w)"   and "(u, Call p, u') \<in> edges fg"   and "sh = [v, u']"   and "proc_of fg v = p"   and "mon_c fg ch = {}"   and "\<And>s. s \<in># ch \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p"   and "\<And>P. {#[entry fg p]. p \<in># P#} \<subseteq># ch \<Longrightarrow> (v, mon_w fg w, P) \<in> S_cs fg (size P)" shows "Q"
l3_inv3I: fixes x :: "l3_state" assumes "analz (ik x) \<subseteq> synth (analz (ik x \<inter> payload \<union> ik x \<inter> valid \<union> ik x \<inter> range LtK \<union> Tags))" shows "x \<in> l3_inv3"
assm: shows "tas \<in> ts_tuple_rel (set ((nt, X) # tass))"
konigDown: fixes t :: "('sequent \<times> 'rule) tree"   and S :: "'structure" assumes "local.wf t"   and "S \<in> structure"   and "\<not> sat S (fst (tree.root t))" shows "alw (\<lambda>stepSs. let ((s, r), S) = shd stepSs; ((s', uu_), S') = shd (stl stepSs) in S \<in> structure \<and> \<not> sat S s \<and> descent (s, S) r (s', S')) (konigDown t S)"
take_cols_carrier_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and inds :: "nat list" assumes "A \<in> carrier_mat nr nc" obtains n :: "nat" where "take_cols A inds \<in> carrier_mat nr n"
shift2_invI: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs"   and "\<And>d0. d0 < d \<Longrightarrow> card {q \<in> set qs. poly_deg (fst q) = d0 \<and> m < card (snd q)} \<le> 1" shows "shift2_inv qs"
inf_multiset_def: fixes A :: "'a multiset"   and B :: "'a multiset" shows "inf A B = (if A < B then A else B)"
the_elem_hom: shows "(\<lambda>X. the_elem (((+>\<^bsub>S\<^esub>) P \<circ> h) ` X)) \<in> ring_hom (R Quot a_kernel R (S Quot P) ((+>\<^bsub>S\<^esub>) P \<circ> h)) (S Quot P)"
prj_chine_assoc_1: shows "\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1 \<cdot> \<mu>_\<nu>_\<pi>\<rho>.chine_assoc = \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1\<^sub>1"
refinement_soundness: fixes R :: "('a \<times> 'b) set"   and pi :: "'c \<Rightarrow> 'd"   and Sa :: "('a, 'd) spec"   and Sc :: "('b, 'c) spec" assumes "refines R pi Sa Sc" shows "implements pi Sa Sc"
distinct_1: fixes x1 :: "lname"   and x21 :: "qtname"   and x22 :: "qtname"   and x23 :: "bool"   and x24 :: "expr"   and x25 :: "vname" shows "LVar x1 \<noteq> {x21,x22,x23}x24..x25"
val_def: shows "f''.val \<equiv> sum (augment f') (outgoing s) - sum (augment f') (incoming s)"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(In1 x1 = In1 y1) = (x1 = y1)"
lm036: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "P || X = P outside (Domain P - X)"
abelian_integer_mod_group: fixes n :: "nat" shows "comm_group (integer_mod_group n)"
compMb\<^sub>2_def: shows "compMb\<^sub>2 \<equiv> \<lambda>body. let ins = compE\<^sub>2 body @ [Return]; xt = compxE\<^sub>2 body 0 0 in (max_stack body, max_vars body, ins, xt)"
deg_minus_eq: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "deg_n R S X (-\<^sub>a p) = deg_n R S X p"
mk_mcomb1: fixes c :: "'b \<Rightarrow> 'a nres"   and x1 :: "'b" shows "c $ x1 \<equiv> (\<bind>) $ (EVAL $ x1) $ (\<lambda>x. (#SP (c $ x)#))"
dense_interval: fixes L :: "'a set"   and U :: "'a set"   and l :: "'a"   and u :: "'a"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "finite L"   and "finite U"   and "l \<in> L"   and "u \<in> U"   and "l < x"   and "x < u"   and "P x"   and "\<And>y l u. \<lbrakk>\<forall>y\<in>{l<..<x}. y \<notin> L; \<forall>y\<in>{x<..<u}. y \<notin> U; l < x; x < u; l < y; y < u\<rbrakk> \<Longrightarrow> P y" shows "\<exists>l\<in>L. \<exists>u\<in>U. l < x \<and> x < u \<and> (\<forall>y\<in>{l<..<x}. y \<notin> L) \<and> (\<forall>y\<in>{x<..<u}. y \<notin> U) \<and> (\<forall>y. l < y \<and> y < u \<longrightarrow> P y)"
LstSeqP_E: fixes s :: "tm"   and k :: "tm"   and y :: "tm"   and H :: "fm set"   and B :: "fm" assumes "insert (HDomain_Incl s (SUCC k)) (insert (OrdP k) (insert (HFun_Sigma s) (insert (HPair k y IN s) H))) \<turnstile> B" shows "insert (LstSeqP s k y) H \<turnstile> B"
deg_mult_pols1: fixes p :: "'a"   and q :: "'a" assumes "Idomain S"   and "p \<in> carrier R"   and "q \<in> carrier R" shows "deg R S X (p \<cdot>\<^sub>r q) = deg R S X p + deg R S X q"
assms_1: shows "F \<turnstile>\<^sub>F \<Phi>"
Lim_def: fixes f :: "'d \<Rightarrow> ('c, 'd) Old_Datatype.node set" shows "Old_Datatype.Lim f \<equiv> \<Union> {z. \<exists>x. z = Push_Node (Inl x) ` f x}"
outgoing'_def: fixes k :: "nat set" shows "h.cf.outgoing' k \<equiv> {(v, u) |u v. u \<notin> k \<and> v \<in> k \<and> (v, u) \<in> h.cf.E}"
fib_work_def: shows "fib_work \<equiv> fix\<cdot>(unwrapS oo fib_body oo wrapS)"
assms_7: shows "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
ltchr: shows "(\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = - 1}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) = - rat_of_int (int (card {x. poly p x = 0 \<and> (\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> poly (prod_list (retrieve_polys qs (snd I))) x < 0}))"
div_positive_int: fixes l :: "int"   and k :: "int" assumes "l \<le> k"   and "0 < l" shows "0 < k div l"
strategyproofness_1: shows "pmf (sds R27) d + pmf (sds R27) b \<le> pmf (sds R19) d + pmf (sds R19) c"
bsingleton_inj: fixes x :: "'a"   and y :: "'a" shows "(bsingleton x = bsingleton y) = (x = y)"
BSD_def: fixes \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "BSD \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ [c] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>'. \<beta> @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
Go_def: fixes a :: "'c" shows "Go a = (SOME b. \<exists>e. terminal_arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F b a e)"
rtos_rs_i_rs: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "rtos r s i = r * s - Suc 0" shows "i = r * s - Suc 0"
closed_form_1: fixes n :: "nat" shows "(fps_of_poly p / fps_of_poly q) $ n = coeff r n + (\<Sum>c | poly (reflect_poly q) c = 0. poly (rs c) (of_nat n) * c ^ n)"
simps_28: fixes f :: "'e \<Rightarrow> 'd"   and x4 :: "'e rexp" shows "map_rexp f (Star x4) = Star (map_rexp f x4)"
closedin_topspace: fixes U :: "'a topology" shows "closedin U (topspace U)"
pointfree_idE: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "f \<circ> g = id" shows "f (g x) = x"
disj_dlHPP_Bisimulation_Up_To_R'_def: fixes d :: "'d"   and PP :: "nat set"   and R' :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set" shows "disj_dlHPP_Bisimulation_Up_To_R' d PP R' R \<equiv> SdlHPPB d PP R' \<and> sym R \<and> trans R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. NDC d (V ! i) \<or> IDC d (V ! i) (htchLoc (pp (V ! i)))) \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' m2 \<alpha> p. SRabbr (V ! i) m1 \<alpha> p m2 \<and> m1 \<sim>\<^bsub>d,htchLocSet PP\<^esub> m1' \<longrightarrow> (\<exists>p' \<alpha>' m2'. SRabbr (V' ! i) m1' \<alpha>' p' m2' \<and> stepResultsinR p p' (R \<union> R') \<and> (\<alpha>, \<alpha>') \<in> R \<union> R' \<and> dhequality_alternative d PP (pp (V ! i)) m2 m2'))"
simps_1: shows "subformulae \<bottom> = [\<bottom>]"
errorT_eq_iff: fixes a :: "'e\<cdot>('f, 'g) errorT"   and b :: "'e\<cdot>('f, 'g) errorT" shows "(a = b) = (runErrorT\<cdot>a = runErrorT\<cdot>b)"
set_bit_def: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.set_bit one push_bit or n a = or a (push_bit n one)"
RSpanD_lincomb: fixes ms :: "'v list" shows "RSpan ms = {rs \<bullet>\<cdot> ms |rs. set rs \<subseteq> UNIV \<and> length rs = length ms}"
B_def: shows "B = c * 2 ^ (m + Suc j)"
hypreal_sqrt_ge_zero: fixes x :: "real star" assumes "0 \<le> x" shows "0 \<le> (*f* sqrt) x"
cong: fixes exec :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b"   and U :: "'a \<Rightarrow> 'b list \<Rightarrow> real" shows "Amortized.U_sum exec U \<equiv> Amortized.U_sum exec U"
c2dc_def: fixes \<chi> :: "nat \<Rightarrow> complex" shows "c2dc \<chi> = (\<lambda>x. \<chi> (x mod n))"
\<G>_\<F>\<G>_subset: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G> \<phi> w \<subseteq> \<F>\<G> \<phi> w"
ide_char': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C" shows "partial_magma.ide C a = (partial_magma.arr C a \<and> (partial_magma.dom C a = a \<or> partial_magma.cod C a = a))"
simps_3: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and as :: "'c negation_type list" shows "NegPos_map f (Neg a # as) = Neg (f a) # NegPos_map f as"
semiring_norm_57: fixes a :: "'a" shows "(0::'a) - a = - a"
simps: fixes c :: "'b" shows "the_In3 (In3 c) = c"
reduce_column_aux_index': fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P_init :: "'a mat"   and K_init :: "'a mat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "nat list" assumes "H \<in> carrier_mat m n"   and "P_init \<in> carrier_mat m m"   and "K_init \<in> carrier_mat m n"   and "P_init * H = K_init"   and "(P, K) = reduce_column_aux div_op xs H (P_init, K_init)"   and "0 < m"   and "invertible_mat P_init"   and "0 \<notin> set xs"   and "\<forall>x\<in>set xs. x < m"   and "distinct xs" shows "\<forall>i\<in>set xs. Matrix.row K i = Matrix.row (addrow (- div_op (H $$ (i, 0)) (H $$ (0, 0))) i 0 K_init) i"
Propositional_def: fixes F :: "\<Pi>\<^sub>1" shows "Propositional F \<equiv> \<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p)"
repr_independence: fixes G :: "('c, 'd) monoid_scheme"   and y :: "'c"   and H :: "'c set"   and x :: "'c" assumes "Group.group G"   and "y \<in> H #>\<^bsub>G\<^esub> x"   and "x \<in> carrier G"   and "subgroup H G" shows "H #>\<^bsub>G\<^esub> x = H #>\<^bsub>G\<^esub> y"
sdnets_noteq: fixes a :: "('b, 'c) Combinators"   and aa :: "('b, 'd) Combinators" assumes "onlyTwoNets a"   and "onlyTwoNets aa"   and "first_bothNet a \<noteq> first_bothNet aa"   and "\<not> member DenyAll a"   and "\<not> member DenyAll aa" shows "sdnets a \<noteq> sdnets aa"
intros: fixes r :: "'a sequent list \<times> 'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set"   and seq :: "'a sequent" assumes "r \<in> R" shows "extendRule seq r \<in> R*"
intros: fixes F1 :: "'a"   and F2 :: "'f"   and st1 :: "('fun, 'b) frame list"   and st2 :: "('fun, ('b, 'h, 'i) unboxed) frame list"   and H :: "'l" assumes "rel_fundefs (Finca_get F1) (Fubx_get F2)"   and "sp_fundefs (Fubx_get F2)"   and "rel_stacktraces (Fubx_get F2) st1 st2 None" shows "State F1 H st1 \<sim> State F2 H st2"
minSetOfComponentsTestL2p2: shows "minSetOfComponents level2 {data1, data12} = {sS2, sS4, sS5, sS6}"
that: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "pdevs_val e (pdevs_of_list xs) = pdevs_val f (pdevs_of_list (filter p xs)) + pdevs_val g (pdevs_of_list (filter (Not \<circ> p) xs))"   and "f \<in> UNIV \<rightarrow> I"   and "g \<in> UNIV \<rightarrow> I" shows "thesis"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.VV.arr f" shows "cmp f = D.null"
F1set3_Lev2: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xd :: "'e"   and xc :: "'d"   and x :: "'e" assumes "xf \<in> Lev2 s1 s2 n xd"   and "rv2 s1 s2 xf xd = Inl xc"   and "x \<in> F1set3 (s1 xc)" shows "xf @ [Inr (tobd_F13 s1 xc x)] \<in> Lev2 s1 s2 (Suc n) xd"
Limit_plus_nat_in_Limit: fixes \<alpha> :: "V"   and \<beta> :: "V"   and a :: "nat" assumes "Limit \<alpha>"   and "Limit \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<alpha> + a\<^sub>\<nat> \<in>\<^sub>\<circ> \<beta>"
injectivity: fixes x :: "int poly"   and y :: "int poly" assumes "of_int_poly x = of_int_poly y" shows "x = y"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>c. P (C c)"   and "\<And>c n p. \<lbrakk>\<And>x. P c; \<And>x. P p\<rbrakk> \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
weight_spmf_conv_pmf_None: fixes p :: "'c spmf" shows "weight_spmf p = 1 - pmf p None"
ax_flat: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.flat below"   and "below x y" shows "x = pcpo.bottom below \<or> x = y"
cases: fixes a :: "'c stream"   and A :: "'c set" assumes "a \<in> streams A" obtains aa :: "'c" and s :: "'c stream" where "a = aa ## s" and "aa \<in> A" and "s \<in> streams A"
path_append: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and r :: "(('d \<times> 'e) \<times> 'f) list"   and p :: "'f"   and s :: "(('d \<times> 'e) \<times> 'f) list" assumes "path A r p"   and "path A s (target r p)" shows "path A (r @ s) p"
kac_series_aux: fixes d :: "nat \<Rightarrow> nat \<Rightarrow> ennreal" shows "(\<Sum>n. \<Sum>i\<le>n. d i n) = (\<Sum>n. d 0 n) + (\<Sum>n i. d (i + 1) (n + 1 + i))"
GroupNH: shows "Group.group (G\<lparr>carrier := N <#> H\<rparr>)"
fdia_d_simp: fixes x :: "'a"   and y :: "'a" shows "antidomain_semiringl.fdia a (\<cdot>) x (a (a y)) = antidomain_semiringl.fdia a (\<cdot>) x y"
var_ordered_children: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and var :: "ref \<Rightarrow> nat"   and no :: "ref" assumes "Dag p low high t"   and "ordered t var"   and "no \<in> set_of t"   and "low no \<noteq> Null"   and "high no \<noteq> Null" shows "var (low no) < var no \<and> var (high no) < var no"
add_poly_l_spec: shows "(add_poly_l, uncurry (\<lambda>p q. SPEC (\<lambda>r. add_poly_p\<^sup>*\<^sup>* (p, q, {#}) ({#}, {#}, r)))) \<in> sorted_poly_rel \<times>\<^sub>r sorted_poly_rel \<rightarrow>\<^sub>f \<langle>sorted_poly_rel\<rangle>nres_rel"
ide_char: fixes a :: "'a arrow_of_spans_data" shows "V.ide a = (V.arr a \<and> V.dom a = a \<and> V.cod a = a)"
Rep_uint64_inject: fixes x :: "uint64"   and y :: "uint64" shows "(Rep_uint64 x = Rep_uint64 y) = (x = y)"
congruent_triangleD_8: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "dist a1 c1 = dist c2 a2"
iso_inv_iso: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.iso (C.VV.inv f)"
simps_3: fixes Obj :: "'e set"   and Mor :: "'d set"   and Dom :: "'d \<Rightarrow> 'e"   and Cod :: "'d \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'd"   and Comp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'f" shows "Dom \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Dom"
qfree_amin_inf: fixes a :: "atom" shows "qfree (amin_inf a)"
leq_pred_def: fixes P :: "'a Predicate.pred"   and Q :: "'a Predicate.pred" shows "local.leq_pred P Q = (\<forall>x. pred.eval P x \<longrightarrow> (\<exists>y. eq x y \<and> pred.eval Q y))"
composites_are_identities_1: shows "ide (f \<star> g \<star> h)"
rn_price_eq_ind: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat"   and matur :: "nat"   and der :: "bool stream \<Rightarrow> real"   and w :: "bool stream" assumes "N = bernoulli_stream q"   and "n < matur"   and "0 < q"   and "q < 1"   and "der \<in> borel_measurable (G matur)" shows "(1 + r) * rn_price N der matur n w = q * rn_price N der matur (Suc n) (pseudo_proj_True n w) + (1 - q) * rn_price N der matur (Suc n) (pseudo_proj_False n w)"
con_rews_2: fixes discr :: "bool discr" shows "B\<cdot>discr \<noteq> \<bottom>"
cross_ratio_def: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2" shows "cross_ratio p q r s \<equiv> proj2_Col_coeff p q s / proj2_Col_coeff p q r"