hcju/isabelleps huggingface dataset:
gFreshGAbs1_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gFreshGAbs1 MOD \<equiv> \<forall>ys y s X' X. isInBar (ys, s) \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> gFreshAbs MOD ys y (Abs ys y X') (gAbs MOD ys y X' X)"
Rep_IF1: fixes x :: "'d IF1" shows "Rep_IF1 x \<in> car_init1 undefined"
amod_adiv_equality: fixes a :: "int"   and b :: "int" shows "ant a = (a div b) *\<^sub>a ant b + ant (a mod b)"
Some_eq_map_of_iff: fixes xys :: "('d \<times> 'e) list"   and y :: "'e"   and x :: "'d" assumes "distinct (map fst xys)" shows "(Some y = map_of xys x) = ((x, y) \<in> set xys)"
homeomorphic_maps_id: fixes X :: "'c topology"   and Y :: "'c topology" shows "homeomorphic_maps X Y id id = (Y = X)"
red1_simulates_exec_instr: fixes h :: "'heap"   and t :: "'thread_id"   and E :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and n :: "nat"   and ST :: "ty list" assumes "wf_J1_prog P"   and "hconf h"   and "preallocated h"   and "P,h \<turnstile> t \<surd>t"   and "P,E,h \<turnstile> (e, xs) \<leftrightarrow> (stk, loc, pc, xcp)"   and "exec_move_d P t E h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"   and "n + max_vars e \<le> length xs"   and "bsok E n"   and "P,h \<turnstile> stk [:\<le>] ST"   and "conf_xcp' (compP2 P) h xcp" shows "\<exists>e'' xs''. P,E,h' \<turnstile> (e'', xs'') \<leftrightarrow> (stk', loc', pc', xcp') \<and> (if \<tau>move2 (compP2 P) h stk E pc xcp then h' = h \<and> (if xcp' = None \<longrightarrow> pc < pc' then \<tau>red1r else \<tau>red1t) P t h (e, xs) (e'', xs'') else \<exists>ta' e' xs'. \<tau>red1r P t h (e, xs) (e', xs') \<and> True,P,t \<turnstile>1 \<langle>e',(h, xs')\<rangle> -ta'\<rightarrow> \<langle>e'',(h', xs'')\<rangle> \<and> ta_bisim wbisim1 (extTA2J1 P ta') ta \<and> \<not> \<tau>move1 P h e' \<and> (call1 e = None \<or> no_call2 E pc \<or> e' = e \<and> xs' = xs))"
left_def: fixes X :: "msg" shows "left X = Abs_Msg (\<Union>U\<in>Rep_Msg X. msgrel `` {freeleft U})"
construction_uniqueness: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a"   and A :: "'a"   and X :: "'a"   and B :: "'a"   and C :: "'a"   and Y :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Q \<noteq> A"   and "Bet Q A X"   and "Cong A X B C"   and "Bet Q A Y"   and "Cong A Y B C" shows "X = Y"
add_scale_img': fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "(\<lambda>x. a + (b - a) * x) ` {0..1} = {a..b}"
distinct_343: fixes x141 :: "'a exp"   and x142 :: "'a exp"   and x17 :: "'a exp" shows "x141;; x142 \<noteq> throw x17"
ec_1: shows "\<bar>ec\<bar> \<le> snd (trunc_bound_eucl p (a * fst X + b * fst Y + c))"
RInitFailFinal_3: shows "sh' C = \<lfloor>(sfs, i)\<rfloor>"
intruder_deduct_hom_AxiomH: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set" assumes "t \<in> M" shows "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t"
remdups_sort_2: fixes xs :: "'a list" shows "set (remdups_sort xs) = set xs"
coprime_imp_mod_not_zero: fixes n :: "nat"   and x :: "nat" assumes "1 < n"   and "coprime x n" shows "0 < x mod n"
ap_spec_add_pairs_canon: shows "punit.ap_spec punit.add_pairs_canon"
Not_def: shows "Formula.Not \<equiv> (Rep_formula ---> Abs_formula) Not\<^sub>\<alpha>"
surjI: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. g (f x) = x" shows "top_pres ((`) g)"
assms_2: shows "continuous_on ({a<..<b} - s) f"
smap2_alt: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and s1 :: "'e stream"   and s2 :: "'f stream"   and s :: "'d stream" shows "(smap2 f s1 s2 = s) = (\<forall>n. f (s1 !! n) (s2 !! n) = s !! n)"
CanonicalCatCat: fixes T :: "(ZF, ZF) Axioms" assumes "ZFAxioms T" shows "Category (CanonicalCat T)"
numneg: fixes bs :: "int list"   and t :: "Cooper.num" shows "Inum bs (numneg t) = Inum bs (Neg t)"
outer_regular: fixes M :: "'b measure"   and B :: "'b set" assumes "sets M = sets borel"   and "emeasure M (space M) \<noteq> \<infinity>"   and "B \<in> sets borel" shows "emeasure M B = Inf (emeasure M ` {U. B \<subseteq> U \<and> open U})"
scale_left_commute: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "a *s (b *s x) = b *s (a *s x)"
hom: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "P x y" shows "Q (h x) (h y)"
M_def: shows "M \<equiv> image_mset \<delta> (mset (nth_drop i (dest_fun T)))"
order_decomp: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "\<exists>q. p = [:- a, 1::'a:] ^ order a p * q \<and> \<not> [:- a, 1::'a:] dvd q"
SeqWRP_unique: fixes x :: "tm"   and s :: "tm"   and y :: "tm"   and s' :: "tm"   and y' :: "tm" shows "{OrdP x, SeqWRP s x y, SeqWRP s' x y'} \<turnstile> y' EQ y"
start: shows "[], 0 \<turnstile> \<Psi> \<triangleright> [] \<hookrightarrow>\<^sup>*\<^sup>* \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> [] \<triangleright> []"
atMost_def: fixes u :: "'a" shows "{..u} \<equiv> \<down>u"
list_of_aux_def: fixes xs :: "'c list"   and ys :: "'c llist" shows "list_of_aux xs ys = (if lfinite ys then rev xs @ list_of ys else undefined)"
in_ip: fixes x :: "'b" shows "(ip x \<in> B) = (x \<in> B)"
release_constant_simp: fixes \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" assumes "is_constant \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi>"
rt_strictly_fresher_fresherD: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2" shows "the (rt1 dip) \<sqsubseteq> the (rt2 dip)"
lang\<^sub>b_def: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" shows "lang\<^sub>b idx \<phi> = L idx {\<AA>. \<AA> \<Turnstile>\<^sub>b \<phi> \<and> #\<^sub>V \<AA> = idx}"
names_def: fixes H :: "(('d, 'c) fm list \<times> 'c) set" shows "names H \<equiv> {uu_. \<exists>ps i. uu_ = i \<and> (ps, i) \<in> H}"
simps_1: fixes n0 :: "nat"   and n :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "round_robin n0 ([], n) s = None"
simps_3: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> (X\<^sub>n \<phi>) w = \<F>\<G> \<phi> w"
preserves_adjoint_pair: fixes f :: "'a"   and g :: "'a" assumes "C.adjoint_pair f g" shows "D.adjoint_pair (F f) (F g)"
in_CoCallsLubI: fixes x :: "Vars.var"   and y :: "Vars.var"   and G :: "CoCalls"   and S :: "CoCalls set" assumes "x--y\<in>G"   and "G \<in> S" shows "x--y\<in>lub S"
levelsCounter_auxSuffixContinues: fixes a :: "('b \<times> bool) list"   and b :: "('b \<times> bool) list" shows "\<forall>l. levelsCounter_aux (a @ b) l = levelsCounter_aux b (levelsCounter_aux a l)"
dom_subset_ssproj_eq_ss: fixes ss :: "('d, 'e) fmap set"   and domain :: "'d set"   and vs :: "'d set" assumes "stateSpace ss domain"   and "domain \<subseteq> vs" shows "ss_proj ss vs = ss"
Pair_Aenc: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "\<langle>X, Y\<rangle> \<noteq> Aenc X' Y'"
Un_insert_left: fixes a :: "'d"   and B :: "'d set"   and C :: "'d set" shows "insert a B \<union> C = insert a (B \<union> C)"
sqrt_ub_pos_pos_1: fixes x :: "real"   and b :: "real" assumes "sqrt x < b"   and "0 < b"   and "0 < x" shows "sqrt x < (b + x / b) / 2"
arg_def: fixes z :: "complex" shows "arg z = (if z = 0 then 0 else SOME a. sgn z = cis a \<and> - pi < a \<and> a \<le> pi)"
lipschitz_on_def: fixes C :: "real"   and U :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "C-lipschitz_on U f = (0 \<le> C \<and> (\<forall>x\<in>U. \<forall>y\<in>U. dist (f x) (f y) \<le> C * dist x y))"
sum_hom_components: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "\<Sum> (hom_components p) = p"
simps_15: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = x21 \<triangleright> set x22"
free_unlift: fixes n :: "nat"   and x :: "'a itrm"   and i :: "nat"   and j :: "nat" assumes "free (unlift' n x i) j" shows "n \<le> j \<or> i \<le> j \<and> j < i + iorder x"
size_4: fixes x2 :: "nat" shows "size (Indet x2) = 0"
Stone_Weierstrass_basic: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "continuous_on S f"   and "0 < e" shows "\<exists>g\<in>Collect real_polynomial_function. \<forall>x\<in>S. \<bar>f x - g x\<bar> < e"
ev_smap: fixes P :: "'a stream \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and s :: "'b stream" shows "ev P (smap f s) = ev (\<lambda>x. P (smap f x)) s"
map_of_map_key_inverse_fun_eq: fixes xs :: "('a \<times> 'b) list"   and g :: "'c \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'c"   and x :: "'c" assumes "\<And>x. x \<in> set xs \<Longrightarrow> g (f (fst x)) = fst x"   and "f (g x) = x" shows "map_of (map (\<lambda>(k, y). (f k, y)) xs) x = map_of xs (g x)"
pwd_spy_mult_def: fixes n :: "nat" shows "pwd_spy_mult n \<equiv> case pwd_spy_sesk n of (s, S, A, B, C, D) \<Rightarrow> (insert (Spy, Auth_PriK n \<otimes> B) s, S, A, B, C, D)"
invDeclC: shows "invDeclC = invocation_declclass G (invmode statM e) (snd s2) a statT \<lparr>name = mn, parTs = pTs'\<rparr>"
axioms_1: 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 \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "category A"
homotopic_with_imp_continuous_maps: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "homotopic_with P X Y f g" shows "continuous_map X Y f \<and> continuous_map X Y g"
take_length: fixes al :: "'a list" shows "take (length al) al = al"
greaterThanLessThan_subseteq_atLeastAtMost_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b \<subseteq> ord.atLeastAtMost less_eq c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
stable_pres_lemma: fixes step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and n :: "nat"   and ss :: "'a list"   and p :: "nat"   and w :: "nat set"   and q :: "nat" assumes "pres_type step n A"   and "bounded step n"   and "ss \<in> list n A"   and "p \<in> w"   and "\<forall>q\<in>w. q < n"   and "\<forall>q<n. q \<notin> w \<longrightarrow> stable r step ss q"   and "q < n"   and "\<forall>s'. (q, s') \<in> set (step p (ss ! p)) \<longrightarrow> s' \<squnion>\<^bsub>f\<^esub> ss ! q = ss ! q"   and "q \<notin> w \<or> q = p" shows "stable r step (merges f (step p (ss ! p)) ss) q"
hmem_self_hadd: fixes k1 :: "hf"   and k2 :: "hf" shows "(k1 \<^bold>\<in> k1 @+ k2) = (0 \<^bold>\<in> k2)"
CardinalityPrognosisSafe_def: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and cHeap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "CardinalityPrognosisSafe prognosis cHeap Aheap Aexp \<equiv> ((CardinalityPrognosisEdom prognosis \<and> CardinalityPrognosisShape prognosis) \<and> CardinalityPrognosisApp prognosis \<and> CardinalityPrognosisLam prognosis) \<and> (CardinalityPrognosisVar prognosis \<and> CardinalityPrognosisLet prognosis cHeap Aheap) \<and> CardinalityPrognosisIfThenElse prognosis \<and> CardinalityHeapSafe cHeap Aheap \<and> ArityAnalysisLetSafe Aexp Aheap"
sinc_at_0: shows "(\<lambda>x. sin x / x) \<midarrow>0\<rightarrow> 1"
L: fixes x :: "'a"   and y :: "'a" assumes "\<not> R (h x) (h y)" shows "R (h y) (h x)"
add_left_cancel: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a + b = a + c) = (b = c)"
new_Addr_SomeI: fixes h :: "loc \<Rightarrow> obj option"   and a :: "loc" assumes "h a = None" shows "\<exists>b. new_Addr h = Some b \<and> h b = None"
rel_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp R" shows "transp (rel_fun (=) R)"
ndet_assoc: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<or> Q \<or> R) = ((P \<or> Q) \<or> R)"
eSubst_simp4: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e) model"   and ys :: "'varSort"   and eY :: "'d withERR"   and y :: "'var" shows "eSubst MOD ys eY y ERR = ERR"
memb_5: fixes a :: "'a" assumes "a \<in> A" shows "b_to_a (q (a_to_b a)) \<in> A"
has_derivative_within_alt2: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> (\<forall>e>0. \<forall>\<^sub>F y in at x within s. norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)))"
if_bool_eq_conj: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "(if P then Q else R) = ((P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R))"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 f g"
size1_if_complete: fixes t :: "'b tree" assumes "complete t" shows "size1 t = 2 ^ height t"
simps_2: fixes c :: "'form"   and uv :: "nat"   and "is" :: "nat list"   and i :: "nat" shows "in_port_at (c, uv # is) i = inPorts' (iNodeOf (tree_at (it' c) is)) ! i"
coeff_abs_int_poly: fixes p :: "int poly"   and n :: "nat" shows "coeff (abs_int_poly p) n = (if lead_coeff p < 0 then - coeff p n else coeff p n)"
subsetI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
preserves_cod: fixes f :: "'j" assumes "J.arr f" shows "C.cod (E f) = E (J.cod f)"
linears_linear: fixes ts :: "term list"   and t :: "term" assumes "linears ts"   and "t \<in> set ts" shows "linear t"
pdg_dghm_proj_is_dghm': fixes i :: "V"   and \<CC> :: "V"   and \<DD> :: "V" assumes "i \<in>\<^sub>\<circ> 2\<^sub>\<nat>"   and "\<CC> = (\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>2\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>)"   and "\<DD> = (if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>)" shows "\<pi>\<^sub>D\<^sub>G (2\<^sub>\<nat>) (\<lambda>i. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>) i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
read_state_reg_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "read_state_reg_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rs1 = get_operand_w5 (op_list ! 0); rd = get_operand_w5 (op_list ! 1) in do curr_win \<leftarrow> get_curr_win (); psr_val \<leftarrow> gets (cpu_reg_val PSR); s_val \<leftarrow> gets (\<lambda>s. get_S psr_val); if (instr_name \<in> {sreg_type RDPSR, sreg_type RDWIM, sreg_type RDTBR} \<or> instr_name = sreg_type RDASR \<and> privileged_ASR rs1) \<and> ucast s_val = 0 then do raise_trap privileged_instruction; return () od else if illegal_instruction_ASR rs1 then do raise_trap illegal_instruction; return () od else if rd \<noteq> 0 then if instr_name = sreg_type RDY then do y_val \<leftarrow> gets (cpu_reg_val Y); write_reg y_val curr_win rd; return () od else if instr_name = sreg_type RDASR then do asr_val \<leftarrow> gets (cpu_reg_val (ASR rs1)); write_reg asr_val curr_win rd; return () od else if instr_name = sreg_type RDPSR then do write_reg psr_val curr_win rd; return () od else if instr_name = sreg_type RDWIM then do wim_val \<leftarrow> gets (cpu_reg_val WIM); write_reg wim_val curr_win rd; return () od else do tbr_val \<leftarrow> gets (cpu_reg_val TBR); write_reg tbr_val curr_win rd; return () od else return () od"
simps_2: fixes x :: "'b" shows "remdups_adj [x] = [x]"
simps_4: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and t :: "char list \<Rightarrow> int option" shows "defs (IF b THEN c1 ELSE c2) t = merge (defs c1 t) (defs c2 t)"
simps: fixes var :: "nat"   and sq :: "real mpoly"   and p :: "real mpoly" shows "quadratic_part_2 var sq p = (let deg = MPoly_Type.degree p var in \<Sum>i = 0..<deg + 1. isolate_variable_sparse p var i * sq ^ (i div 2) * Const (real (i mod 2)) * Var var + isolate_variable_sparse p var i * sq ^ (i div 2) * Const (1 - real (i mod 2)))"
fref_def: fixes P :: "'h \<Rightarrow> bool"   and R :: "('e \<times> 'h) set"   and S :: "('g \<times> 'i) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
class_Object: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" assumes "wf_prog wf_md P" shows "\<exists>C fs ms. class P Object = \<lfloor>(C, fs, ms)\<rfloor>"
join_equality: fixes x :: "'b"   and y :: "'b"   and sup :: "'b" assumes "is_sup x y sup" shows "x \<squnion> y = sup"
assms_12: shows "all_trms_irreducible (subst_set (trms_ecl C) \<sigma>) (\<lambda>t. trm_rep t S)"
sigma_finite_subalgebra_def: fixes M :: "'b measure"   and F :: "'b measure" shows "sigma_finite_subalgebra M F \<equiv> subalgebra M F \<and> sigma_finite_measure (restr_to_subalg M F)"
obsv_round_def: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and r_decisions :: "process \<Rightarrow> val option"   and Os :: "process set" shows "obsv_round r S v r_decisions Os = {(s, s'). r = v_state.next_round s \<and> (S \<noteq> {} \<longrightarrow> obs_safe r s v) \<and> d_guard r_decisions (const_map v S) \<and> (S \<in> Quorum \<longrightarrow> Os = UNIV) \<and> (Os \<noteq> {} \<longrightarrow> S \<noteq> {}) \<and> s' = s\<lparr>v_state.next_round := Suc r, votes := (votes s)(r := const_map v S), v_state.decisions := v_state.decisions s ++ r_decisions, obsv_state.obs := (obsv_state.obs s)(r := const_map v Os)\<rparr>}"
two_chain_valid_valid_cubes: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeII_chain i j F two_chain s" shows "\<forall>two_cube\<in>two_chain. valid_two_cube two_cube"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root_safe :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_safe_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "l_get_mode type_wf get_mode get_mode_locs"   and "l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf get_shadow_root_safe get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs" shows "l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root_safe get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs"
H'_def: shows "H' \<equiv> \<lambda>x. (embed1_blinfun o\<^sub>L fst_blinfun) + (embed2_blinfun o\<^sub>L f' x)"
sup_distr: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "times_prod' (sup_prod x y) z = sup_prod (times_prod' x z) (times_prod' y z)"
length_conv_length_it': fixes xs :: "'a list"   and acc :: "nat" shows "length xs + acc = length_it' acc xs"
u_2: shows "infdist w ({x--z} \<union> Gzt) = dist w u"
execn: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
polyn_add: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "pol_coeff S (n, f)"   and "pol_coeff S (m, g)" shows "polyn_expr R X n (n, f) \<plusminus> polyn_expr R X m (m, g) = polyn_expr R X (max n m) (add_cf S (n, f) (m, g))"
sprefD1: fixes a :: "'b list"   and b :: "'b list" assumes "a <p b" shows "a \<le>p b"
subseq_append': fixes zs :: "'c list"   and xs :: "'c list"   and ys :: "'c list" shows "subseq (zs @ xs) (zs @ ys) = subseq xs ys"
length_list_of_conv_the_enat: fixes xs :: "'b llist" assumes "lfinite xs" shows "length (list_of xs) = the_enat (llength xs)"
weak_respection_of_barbs_and_closures_5: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs (Rel\<^sup>*) CWB"
map_of_Some_fv_subset: fixes \<Gamma> :: "('b \<times> 'a) list"   and x :: "'b"   and e :: "'a" assumes "map_of \<Gamma> x = Some e" shows "fv e \<subseteq> fv \<Gamma>"
test_eq: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and p :: "'c"   and q :: "'c" assumes "dioid_tests test not" shows "(p = q) = (test p = test q)"
listsum_prod_distr_right: fixes CS :: "'b tensor list"   and ds :: "nat list"   and A :: "'b tensor" assumes "\<And>C. C \<in> set CS \<Longrightarrow> dims C = ds" shows "A \<otimes> listsum ds CS = listsum (dims A @ ds) (map ((\<otimes>) A) CS)"
15: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * e * ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g) \<le> r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * ((1::'a) \<squnion> e)"
assms: fixes f :: "'t \<Rightarrow>\<^sub>0 'b" assumes "f \<in> pmdl G" shows "(red G)\<^sup>*\<^sup>* f 0"
polynomialI: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and K :: "'c set" assumes "ring R"   and "set p \<subseteq> K"   and "lead_coeff p \<noteq> \<zero>\<^bsub>R\<^esub>" shows "polynomial\<^bsub>R\<^esub> K p"
card_eq_sum: fixes A :: "'c set" shows "card A = (\<Sum>x\<in>A. 1)"