hcju/isabelleps huggingface dataset:
r_lchild_eq_l_rchild_if_well_shaped: fixes t :: "'b tree" assumes "well_shaped t"   and "l t < r t" shows "r (lchild t) = l (rchild t) - 1"
cases_scheme: fixes r :: "'c alpha_d_scheme" obtains ok :: "bool" and more :: "'c" where "r = \<lparr>ok = ok, \<dots> = more\<rparr>"
exec_Normal_elim_cases_1: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and c :: "('f, 'e, 'g) com"   and f :: "'g"   and t :: "('f, 'g) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> \<Rightarrow> t" obtains "t = Fault f"
trnr\<^sub>\<eta>_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b"   and v :: "'b"   and \<mu> :: "'b" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_bicategory.trnr\<^sub>\<eta> V H \<a> \<i> src trg f g \<eta> v \<mu> \<equiv> V (H \<mu> f) (V (bicategory.\<a>' V H \<a> src trg v g f) (V (H v \<eta>) (category.inv V (bicategory.runit V H \<a> \<i> src v))))"
subdistl_var: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<cdot> x + z \<cdot> y \<le> z \<cdot> (x + y)"
that: fixes init :: "(pat \<times> exp0) list"   and rest :: "(pat \<times> exp0) list" assumes "pes = init @ (p, e) # rest"   and "distinct (pat_bindings p [])"   and "list_all (\<lambda>(p, e). cupcake_pmatch cenv p v0 [] = No_match \<and> distinct (pat_bindings p [])) init"   and "cupcake_pmatch cenv p v0 [] = Match env'" shows "thesis"
syncWithTick_imp_NTF: fixes s :: "'c event list"   and t :: "'c event list"   and A :: "'c set"   and u :: "'c event list"   and P :: "'c process"   and Q :: "'c process" assumes "s @ [tick] setinterleaves ((t, u), insert tick (ev ` A))"   and "t \<in> T P"   and "u \<in> T Q" shows "\<exists>t1 u1. t = t1 @ [tick] \<and> u = u1 @ [tick] \<and> s setinterleaves ((t1, u1), insert tick (ev ` A))"
intro: fixes I :: "'d set" assumes "finite I" shows "finite_product_sigma_finite_axioms I"
dominators_def: fixes g :: "'g"   and n :: "'node" shows "dominators g n \<equiv> {m \<in> set (\<alpha>n g). dominates g m n}"
Abs_image: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "type_definition Rep Abs A" shows "Abs ` A = UNIV"
lemma_4_5_v: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion>1 b l\<rightarrow> c = (a l\<rightarrow> c) \<sqinter> (b l\<rightarrow> c)"
SHOmsgVectors_def: fixes A :: "('d, 'f, 'e, 'g) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'d"   and cfg :: "'d \<Rightarrow> 'f"   and HOp :: "'d set"   and SHOp :: "'d set" shows "SHOmsgVectors A r p cfg HOp SHOp \<equiv> {\<mu>. (\<forall>q. (q \<in> HOp) = (\<mu> q \<noteq> None)) \<and> (\<forall>q. q \<in> SHOp \<inter> HOp \<longrightarrow> \<mu> q = Some (sendMsg A r q p (cfg q)))}"
disc_2: fixes x21 :: "'a process"   and x22 :: "'a process" shows "\<not> isAction (Choice x21 x22)"
modelFormula_def: fixes valuation :: "Literal list"   and formula :: "Literal list list" shows "model valuation formula \<equiv> consistent valuation \<and> formulaTrue formula valuation"
coplanar_4: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and \<alpha> :: "Points" assumes "rk {A, B, C} = 3"   and "rk {B, C, \<alpha>} = 2" shows "rk {A, B, C, \<alpha>} = 3"
f_add: fixes x :: "real"   and y :: "real" shows "f (x + y + 1) = f x + f y"
normal_series_subgroups: fixes i :: "nat" assumes "i < length \<GG>" shows "subgroup (\<GG> ! i) G"
finfy: shows "finite {i \<in> I. f i (y i) \<noteq> \<one>}"
inputCases: fixes \<Psi> :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'a action \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> \<alpha> \<prec> P'"   and "\<And>K Tvec. \<lbrakk>\<Psi> \<turnstile> M \<leftrightarrow> K; set xvec \<subseteq> supp N; length xvec = length Tvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (K\<lparr>N[xvec::=Tvec]\<rparr>) (P[xvec::=Tvec])" shows "Prop \<alpha> P'"
assms_2: shows "f = moebius_pt M \<circ> conjugate"
leD: fixes y :: "'c list"   and x :: "'c list" assumes "prefix y x" shows "\<not> strict_prefix x y"
update_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "update rt ip r \<equiv> case rt ip of None \<Rightarrow> rt(ip \<mapsto> r) | Some s \<Rightarrow> if \<pi>\<^sub>2 s < \<pi>\<^sub>2 r then rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 s)) else if \<pi>\<^sub>2 s = \<pi>\<^sub>2 r \<and> (\<pi>\<^sub>5 r < \<pi>\<^sub>5 s \<or> \<pi>\<^sub>4 s = Aodv_Basic.inv) then rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 s)) else if \<pi>\<^sub>3 r = unk then rt(ip \<mapsto> (\<pi>\<^sub>2 s, snd (addpre r (\<pi>\<^sub>7 s)))) else rt(ip \<mapsto> addpre s (\<pi>\<^sub>7 r))"
simps_6: fixes r1 :: "'b rexp"   and r2 :: "'b rexp"   and c :: "'b"   and v2 :: "'b val" shows "injval (Times r1 r2) c (val.Right v2) = Seq (mkeps r1) (injval r2 c v2)"
totalValuationForFormulaDefinesAllItsClauses: fixes valuation :: "Literal list"   and formula :: "Literal list list"   and clause :: "Literal list" assumes "CNF.total valuation (vars formula)"   and "clause el formula" shows "clauseTrue clause valuation \<or> clauseFalse clause valuation"
solution_of_nat_of_nat: fixes D :: "nat"   and a :: "nat"   and b :: "nat" assumes "pell D" shows "pell.solution D (of_nat a, of_nat b) = pell.solution D (a, b)"
comp_mkarr: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Dom t = Cod u" shows "mkarr t \<cdot> mkarr u = mkarr (t \<^bold>\<cdot> u)"
skl1_init_def: shows "skl1_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, signalsInit2 = \<lambda>x. 0, signalsResp2 = \<lambda>x. 0\<rparr>}"
assms_3: shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children'"
chain_subset_trans_Union: fixes R :: "('b \<times> 'b) set set" assumes "chain\<^sub>\<subseteq> R"   and "\<forall>r\<in>R. trans r" shows "trans (\<Union> R)"
two: fixes R :: "('a \<Rightarrow> real) set"   and S :: "'a set"   and A :: "'a set"   and B :: "'a set"   and e :: "real" assumes "function_ring_on R S"   and "closed A"   and "A \<subseteq> S"   and "closed B"   and "B \<subseteq> S"   and "A \<inter> B = {}"   and "0 < e"   and "e < 1" shows "\<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>x\<in>A. f x < e) \<and> (\<forall>x\<in>B. 1 - e < f x)"
subformulas\<^sub>\<nu>_list_set: fixes \<phi> :: "'b ltln" shows "set (subformulas\<^sub>\<nu>_list \<phi>) = subformulas\<^sub>\<nu> \<phi>"
finite_theNFT: fixes cl :: "('g, 'h, 'i) cmd list" shows "finite (theNFT cl)"
intensional_simps_26: fixes y :: "'b \<Rightarrow> bool" shows "#False \<or> y \<equiv> y"
f: fixes x :: "'a" assumes "x \<in> S" shows "f differentiable at x"
arc_imp_simple_path: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" shows "simple_path g"
supp_delta0fun: fixes a :: "'a" shows "supp ((0::'b) \<delta> a) = {}"
eval_elim_cases_22: fixes G :: "prog"   and s :: "st"   and c1 :: "stmt"   and c2 :: "stmt"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (c1 Finally c2)\<succ>\<rightarrow> (x, s')" obtains x1 :: "abopt" and s1 :: "st" and s2 :: "state" where "x = \<diamondsuit>" and "s' = (if \<exists>err. x1 = Some (Error err) then (x1, s1) else abupd (abrupt_if (x1 \<noteq> None) x1) s2)" and "G\<turnstile>Norm s \<midarrow>c1\<rightarrow> (x1, s1)" and "G\<turnstile>Norm s1 \<midarrow>c2\<rightarrow> s2"
rtrancl3p_converse_induct: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and bs :: "'c list"   and a'' :: "'b"   and P :: "'b \<Rightarrow> 'c list \<Rightarrow> 'b \<Rightarrow> bool" assumes "rtrancl3p r a bs a''"   and "\<And>a. P a [] a"   and "\<And>a b a' bs a''. \<lbrakk>rtrancl3p r a' bs a''; r a b a'; P a' bs a''\<rbrakk> \<Longrightarrow> P a (b # bs) a''" shows "P a bs a''"
hupd_def: fixes a :: "loc"   and obj :: "obj"   and s :: "state" shows "hupd(a\<mapsto>obj) s \<equiv> s\<lparr>heap := heap s(a \<mapsto> obj)\<rparr>"
l_neg: fixes x :: "nat \<times> nat" assumes "x \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> x \<oplus>\<^bsub>S\<^esub> x = \<zero>\<^bsub>S\<^esub>"
asym: fixes a :: "'b"   and b :: "'b" assumes "a < b"   and "b < a" shows "False"
enf_offending_flows: fixes m :: "'b SecurityInvariant_configured"   and P :: "'b \<times> 'b \<Rightarrow> bool" assumes "configured_SecurityInvariant m"   and "\<forall>G. c_sinvar m G = (\<forall>e\<in>edges G. P e)" shows "\<forall>G. c_offending_flows m G = (if c_sinvar m G then {} else {{e \<in> edges G. \<not> P e}})"
match_fvi_trm_None: fixes f :: "nat \<Rightarrow> event_data option"   and ts :: "Formula.trm list"   and xs :: "event_data list"   and x :: "nat" assumes "Some f = match ts xs"   and "\<forall>t\<in>set ts. x \<notin> fv_trm t" shows "f x = None"
applicableUnitPropagate_def: fixes state :: "State"   and F0 :: "Literal list list" shows "applicableUnitPropagate state F0 \<equiv> \<exists>state'. appliedUnitPropagate state state' F0"
A2coprime_rsurjecTr: fixes A :: "'a set"   and B :: "'a set"   and S :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set" assumes "ideal R A"   and "ideal R B"   and "S 0 = pj R A"   and "S (Suc 0) = pj R B" shows "carrier (R /\<^sub>r A \<Oplus>\<^sub>r R /\<^sub>r B) = carr_prodag {j. j \<le> Suc 0} (prodB1 (R /\<^sub>r A) (R /\<^sub>r B))"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VoV.seq g f = (VoV.arr f \<and> VoV.arr g \<and> VxV.seq g f)"
word_set_bits_def: fixes P :: "nat \<Rightarrow> bool" shows "(BITS n. P n) = horner_sum of_bool 2 (map P [0..<LENGTH('a)])"
g: fixes n :: "nat" shows "cmod (g n) = Re (g n)"
select_convs_1: fixes E_ES :: "'c set"   and I_ES :: "'c set"   and O_ES :: "'c set"   and Tr_ES :: "'c list set"   and more :: "'d" shows "E_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = E_ES"
ccSup_least: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and z :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
less_imp_of_nat_less: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "m < n" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
flat_ord_def: fixes b :: "'b"   and x :: "'b"   and y :: "'b" shows "flat_ord b x y = (x = b \<or> x = y)"
rep_Gromov_completion_limit: fixes x :: "'a Gromov_completion" shows "(\<lambda>n. to_Gromov_completion (rep_Gromov_completion x n)) \<longlonglongrightarrow> x"
eval_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" obtains ca :: "('Var, 'AExp, 'BExp) Stmt" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "c = cxt_to_stmt E__ ca__" and "(ca__, mem) \<leadsto>\<^sub>s (c'b__, mem')" and "c'a__ = cxt_to_stmt E__ c'b__" and "mds' = mds"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "c = c\<^sub>1__ ;; c\<^sub>2" and "c'a__ = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1__, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>w"   | ca :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "c = cxt_to_stmt E (ca@[mu])" and "c'a__ = cxt_to_stmt E c'" and "\<langle>ca, update_modes mu mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w"   | b :: "'BExp" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" where "c = Await b c\<^sub>1" and "eval\<^sub>B mem b" and "no_await c\<^sub>1" and "\<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w\<^sup>+ \<langle>c'a__, mds', mem'\<rangle>\<^sub>w" and "is_final c'a__"
continuous_within_tendsto_compose': fixes a :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'e \<Rightarrow> 'c"   and F :: "'e filter" assumes "continuous (at a within s) f"   and "\<And>n. x n \<in> s"   and "(x \<longlongrightarrow> a) F" shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
k_4: fixes p :: "nat"   and X :: "'a topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_boundary (Suc p) (k p c) + k (p - Suc 0) (chain_boundary p c) = (singular_subdivision p ^^ n) c - c"
sup_commute: fixes x :: "'f multiset"   and y :: "'f multiset" shows "x \<union># y = y \<union># x"
all_Suc_conv: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>i<Suc n. P i) = (P 0 \<and> (\<forall>i<n. P (Suc i)))"
arcs_0_3: shows "f x0 \<in> arcs G"
sublist_split: fixes lo :: "nat"   and hi :: "nat"   and p :: "nat"   and xs :: "'c list" assumes "lo \<le> hi"   and "lo < p"   and "p < hi"   and "hi < length xs" shows "WB_Sort.sublist xs lo p @ WB_Sort.sublist xs (p + 1) hi = WB_Sort.sublist xs lo hi"
strip_while_eq_Cons_rec: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "(strip_while P (x # xs) = x # strip_while P xs) = (\<not> (P x \<and> (\<forall>x\<in>set xs. P x)))"
comp_inv_arr': fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.comp (VoV.inv f) f = VoV.dom f"
set_replicate_Suc: fixes n :: "nat"   and x :: "'c" shows "set (replicate (Suc n) x) = {x}"
ext_surjective: fixes x :: "'a state_t_scheme" shows "x = \<lparr>sp_impl_subj_subj = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst state_t_ext_Tuple_Iso \<circ> id))) x, sp_impl_subj_obj = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst state_t_ext_Tuple_Iso \<circ> id))) x, current = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst state_t_ext_Tuple_Iso \<circ> id))) x, obj = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst state_t_ext_Tuple_Iso \<circ> id)))) x, thread = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst state_t_ext_Tuple_Iso \<circ> id)))) x, \<dots> = (Record.iso_tuple_snd state_t_ext_Tuple_Iso \<circ> id) x\<rparr>"
emeasure_lborel_Ioo: fixes l :: "real"   and u :: "real" assumes "l \<le> u" shows "emeasure lborel {l<..<u} = ennreal (u - l)"
even_xor_iff: fixes a :: "'a"   and b :: "'a" shows "even (a XOR b) = (even a = even b)"
deg: shows "Reflected_Multivariate_Polynomial.degree p \<noteq> Reflected_Multivariate_Polynomial.degree q"
carrier_vec_induct: fixes v :: "'b vec"   and n :: "nat"   and P :: "nat \<Rightarrow> 'b vec \<Rightarrow> bool" assumes "v \<in> carrier_vec n"   and "P 0 vNil"   and "\<And>n a v. \<lbrakk>v \<in> carrier_vec n; P n v\<rbrakk> \<Longrightarrow> P (Suc n) (vCons a v)" shows "P n v"
unit_system_intro: fixes a :: "'a" assumes "UNIV = {a}" shows "OFCLASS('a, unit_system_class)"
absolutely_integrable_max: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S"   and "g absolutely_integrable_on S" shows "(\<lambda>x. \<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) absolutely_integrable_on S"
bernoulli_fps_conv_bernoulli'_fps: shows "bernoulli_fps = bernoulli'_fps - fps_X"
simps_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Inl x1 = Inl y1) = (x1 = y1)"
tagged_fun_relD_both: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'g \<Rightarrow> 'h"   and A :: "('e \<times> 'g) set"   and B :: "('f \<times> 'h) set"   and x :: "'e"   and x' :: "'g" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, Let x' f') \<in> B"
of_nat_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero n = (plus one ^^ n) zero"
nat_eq_add_iff1: fixes j :: "nat"   and i :: "nat"   and u :: "nat"   and m :: "nat"   and n :: "nat" assumes "j \<le> i" shows "(i * u + m = j * u + n) = ((i - j) * u + m = n)"
frrestrictionD_2: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<bullet> A" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
vertical_chop_empty_up: shows "\<forall>v. \<exists>u. v=u--v"
next_propagate'_def: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t" shows "next_propagate' c0 c1 loc t \<equiv> t \<in>#\<^sub>z c_work c0 loc \<and> (\<forall>t' loc'. t' \<in>#\<^sub>z c_work c0 loc' \<longrightarrow> \<not> t' < t) \<and> c1 = c0\<lparr>c_imp := (c_imp c0)(loc := c_imp c0 loc + {#t' \<in>#\<^sub>z c_work c0 loc. t' = t#}), c_work := \<lambda>loc'. if loc' = loc then {#t' \<in>#\<^sub>z c_work c0 loc'. t' \<noteq> t#} else c_work c0 loc' + after_summary (frontier_changes (c_imp c0 loc + {#t' \<in>#\<^sub>z c_work c0 loc. t' = t#}) (c_imp c0 loc)) (summary loc loc')\<rparr>"
simps_3: fixes n :: "nat"   and k :: "nat"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" shows "\<up>\<^sub>\<tau> n k (T \<rightarrow> U) = \<up>\<^sub>\<tau> n k T \<rightarrow> \<up>\<^sub>\<tau> n k U"
EWHILET_def: shows "EWHILET \<equiv> EWHILEIT (\<lambda>_. True)"
elims: fixes x :: "('c, 'd, 'd) matrix"   and y :: "('c, 'd) sqmatrix" assumes "sqmatrix_of_matrix x = y" obtains A :: "'d atMost \<Rightarrow> 'd atMost \<Rightarrow> 'c" where "x = Matrix A" and "y = SqMatrix A"
foldl_Gauss_Jordan_column_k_eq_3: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'd) vec" assumes "k < ncols A" shows "fst (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k]) \<le> nrows A"
hballI: fixes A :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "\<And>x. x \<^bold>\<in> A \<Longrightarrow> P x" shows "HBall A P"
assms_4: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "\<not> R x y"   and "P (x # xs) ys" shows "P (x # xs) (y # ys)"
vars_smult: fixes c :: "'a"   and p :: "'a mpoly" shows "vars (MPoly_Type.smult c p) \<subseteq> vars p"
c: fixes x :: "real" assumes "x \<in> T" shows "v x $ i = c"
meet_3_col_2: fixes l :: "Lines"   and A :: "Points"   and B :: "Points"   and n :: "Lines"   and P :: "Points" assumes "meet_3_in l (line A B) n P" shows "col A B P"
2: shows "strict_from_inj k g ` {0..<k} = S"
fix_clock_1: fixes s1 :: "'c state"   and s2 :: "'a state"   and x :: "'b"   and s :: "'a state" assumes "fix_clock s1 (s2, x) = (s, x)" shows "clock s \<le> clock s1"
kernel_zero: fixes G :: "('e, 'f) ring_scheme"   and H :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and i :: "'e"   and R :: "('e, 'i) ring_scheme"   and S :: "('g, 'j) ring_scheme" assumes "abelian_group_hom G H h"   and "i \<in> a_kernel R S h" shows "h i = \<zero>\<^bsub>S\<^esub>"
irs_SDG_path_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n'' :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_return_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''"   and "SDG.intra_return_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns' n'" shows "SDG.intra_return_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ ns') n'"
equal_pair_count_maximality2a: fixes ps :: "('b \<times> 'b) list" shows "equal_pair_count ps = length ps \<or> (\<forall>i\<ge>equal_pair_count ps. \<exists>j\<le>i. \<not> equal_pair (ps ! j))"
list_eq_iff_nth_eq: fixes xs :: "'e list"   and ys :: "'e list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
deg_le_sectI: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and d :: "nat" assumes "t \<in> .[X]"   and "deg_pm t \<le> d" shows "t \<in> deg_le_sect X d"
mb_class: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and CDef :: "classDef"   and m :: "nat"   and mDef :: "methodDef"   and xs :: "nat list"   and e :: "exp" assumes "CT C = Some CDef"   and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = Some mDef"   and "varDefs_names (mParams mDef) = xs"   and "mBody mDef = e" shows "mbody(CT,m,C) = xs . e"
nres_more_simps_6: fixes x :: "'a"   and y :: "'a" shows "(RETURN x = RETURN y) = (x = y)"
nhds_order: fixes x :: "'a" shows "nhds x = inf (INF a\<in>{x<..}. principal {..<a}) (INF a\<in>{..<x}. principal {a<..})"
rev_sub_is_hom: fixes g :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P" shows "rev_compose R g \<in> ring_hom P P"
dist_triangle_half_l: fixes x1 :: "'b"   and y :: "'b"   and e :: "real"   and x2 :: "'b" assumes "dist x1 y < e / 2"   and "dist x2 y < e / 2" shows "dist x1 x2 < e"
oclosed_oreachable_oreachable: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and \<zeta> :: "'d net_state"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'c) \<times> 'd, 'e seq_action) automaton"   and p :: "net_tree"   and U :: "(nat \<Rightarrow> 'c) \<Rightarrow> (nat \<Rightarrow> 'c) \<Rightarrow> bool" assumes "(\<sigma>, \<zeta>) \<in> oreachable (oclosed (opnet onp p)) (\<lambda>_ _ _. True) U" shows "(\<sigma>, \<zeta>) \<in> oreachable (opnet onp p) (\<lambda>_ _ _. True) U"
mpt_factor_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and proj :: "'b \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "mpt M T" shows "mpt.mpt_factor M T proj M2 T2 = (proj \<in> measure_preserving M M2 \<and> (AE x in M. proj (T x) = T2 (proj x)) \<and> mpt M2 T2)"
id_eq_prop_prop_4_a: fixes v :: "i"   and F :: "\<Pi>\<^sub>2" shows "[F \<^bold>= F in v]"