hcju/isabelleps huggingface dataset:
process_rule_2_futures: fixes ys :: "'a list"   and ys' :: "'a list"   and Y :: "'a set"   and P :: "'a process"   and xs :: "'a list" assumes "(ys @ ys', Y) \<in> futures P xs" shows "(ys, {}) \<in> futures P xs"
wp_whileI: fixes R :: "((char list \<Rightarrow> int \<Rightarrow> int) \<times> (char list \<Rightarrow> int \<Rightarrow> int)) set"   and I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and \<ss>\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "bexp"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "wf R"   and "I \<ss>\<^sub>0"   and "\<And>\<ss>. \<lbrakk>I \<ss>; bval b \<ss>\<rbrakk> \<Longrightarrow> wp \<pi> c (\<lambda>\<ss>'. I \<ss>' \<and> (\<ss>', \<ss>) \<in> R) \<ss>"   and "\<And>\<ss>. \<lbrakk>I \<ss>; \<not> bval b \<ss>\<rbrakk> \<Longrightarrow> Q \<ss>" shows "wp \<pi> (WHILE b DO c) Q \<ss>\<^sub>0"
pqr_simps_3: shows "sturm p ! Suc (Suc i) = sturm_squarefree' p ! (i + 2) * d"
adj_sub_assign: fixes \<sigma> :: "('b, 'c, 'a) subst"   and e :: "('b, 'a) trm"   and x :: "'a" shows "(\<Union>i\<in>SIGT e. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x) \<subseteq> \<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGP (x := e)))"
basis_card: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and B\<^sub>1 :: "'b set"   and B\<^sub>2 :: "'b set" assumes "matroid carrier indep"   and "indep_system.basis carrier indep B\<^sub>1"   and "indep_system.basis carrier indep B\<^sub>2" shows "card B\<^sub>1 = card B\<^sub>2"
idx_sequence_interval: fixes idx :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "idx_sequence idx" shows "\<exists>k. n \<in> {idx k..<idx (Suc k)}"
bisims1_pc_length_compEs2: fixes P :: "'a prog"   and h :: "'heap"   and Es :: "'addr expr1 list"   and es :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,Es,h \<turnstile> (es, xs) [\<leftrightarrow>] (stk, loc, pc, xcp)" shows "pc \<le> length (compEs2 Es)"
ask_state_def: fixes f :: "'r \<Rightarrow> ('s, 'm) stateT" shows "local.ask_state f = StateT (\<lambda>s. ask (\<lambda>r. run_state (f r) s))"
ia_not_i: shows "ia \<noteq> mod_type_class.to_nat i"
echelon_form_of_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "echelon_form_of A bezout = echelon_form_of_upt_k A (ncols A - 1) bezout"
set_pair_pmf: fixes A :: "'a pmf"   and B :: "'b pmf" shows "set_pmf (pair_pmf A B) = set_pmf A \<times> set_pmf B"
rvE: fixes V :: "'var"   and S :: "'node set"   and n :: "'node" assumes "V \<in> rv S n" obtains as :: "'edge list" and n' :: "'node" where "n -as\<rightarrow>* n'" and "n' \<in> backward_slice S" and "V \<in> Use n'" and "\<forall>nx\<in>set (sourcenodes as). V \<notin> Def nx"
simps_2: fixes uv :: "'a" shows "(Val uv \<le> Bot) = False"
circuit_in_supI: fixes \<E>' :: "'a set"   and \<E> :: "'a set"   and C :: "'a set" assumes "\<E>' \<subseteq> \<E>"   and "\<E> \<subseteq> carrier"   and "C \<subseteq> \<E>'"   and "circuit_in \<E> C" shows "circuit_in \<E>' C"
okRead2_def: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and x :: "real"   and ppred :: "nat \<Rightarrow> bool" shows "okRead2 f g x ppred = (\<forall>p. ppred p \<longrightarrow> \<bar>f p - g p\<bar> \<le> x)"
l_zero: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<zero>\<^bsub>F\<^esub> \<oplus>\<^bsub>F\<^esub> x = x"
injD: fixes f :: "'h \<Rightarrow> 'i"   and x :: "'h"   and y :: "'h" assumes "inj f"   and "f x = f y" shows "x = y"
subtermeq_vars_subset: fixes M :: "('c, 'd) Term.term"   and N :: "('c, 'd) Term.term" assumes "M \<sqsubseteq> N" shows "fv M \<subseteq> fv N"
homologous_rel_diff: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and a' :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b' :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S a a'"   and "homologous_rel p X S b b'" shows "homologous_rel p X S (a - b) (a' - b')"
C: fixes x :: "hf"   and x' :: "hf"   and x'' :: "hf"   and y :: "hf"   and y' :: "hf"   and y'' :: "hf"   and z :: "hf"   and z' :: "hf"   and z'' :: "hf" assumes "C x x' x'' y y' y'' z z' z''"   and "P y y' y''"   and "P z z' z''" shows "P x x' x''"
exhaust: fixes y :: "val" obtains x1 :: "nat" where "y = VNat x1"   | x2 :: "(val \<times> val) fset" where "y = val.Fun x2"   | x3 :: "val option" where "y = Abs x3"   | "y = Wrong"
flow_initial_time: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "t0 \<in> T"   and "x0 \<in> X" shows "ll_on_open.flow T f X t0 x0 t0 = x0"
weakCongISym2: fixes P :: "ccs"   and Q :: "ccs"   and F :: "ccs \<Rightarrow> ccs" assumes "P \<cong> Q"   and "\<And>P Q. P \<cong> Q \<Longrightarrow> F P \<leadsto><weakBisimulation> F Q" shows "F P \<cong> F Q"
unit_inv_unit: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one"   and "Dmonoid.unit prod one x" shows "Dmonoid.unit prod one (Dmonoid.inv prod one x)"
lipschitz_on_le: fixes M :: "real"   and D :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and L :: "real" assumes "M-lipschitz_on D f"   and "M \<le> L" shows "L-lipschitz_on D f"
simps_3: fixes e :: "expr" shows "\<lbrace>Inverse e\<rbrace> = 1 / \<lbrace>e\<rbrace>"
Cons_2: fixes A :: "'a mat"   and j :: "nat"   and i :: "nat" assumes "A \<in> set (A__ # As)"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'a)"
ldrop_lnull: fixes xs :: "'b llist"   and n :: "enat" assumes "lnull xs" shows "ldrop n xs = <>"
rsp: shows "fractrel (0::'a, 1::'a) (0::'a, 1::'a)"
max_wp_SetDC: fixes S :: "'a \<Rightarrow> 'b set"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> S s \<Longrightarrow> maximal (wp (p a))"   and "\<And>s. S s \<noteq> {}" shows "maximal (wp (SetDC p S))"
used_Notes_rev: fixes evs :: "event list"   and A :: "agent"   and X :: "msg" shows "used (evs @ [Notes A X]) = parts {X} \<union> used evs"
matpow_1_commute: fixes n :: "nat" shows "matpow (A + mat 1) n ** A = A ** matpow (A + mat 1) n"
csum_czero2: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "czero +c r =o r"
quasi_isometry_compose: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and mu :: "real"   and D :: "real"   and Z :: "'d set"   and g :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry_between X Y f"   and "mu D -quasi_isometry_between Y Z g" shows "(lambda * mu) (C * mu + 2 * D) -quasi_isometry_between X Z (g \<circ> f)"
cont_emeasure_spmf: shows "cont lub_spmf (ord_spmf (=)) Sup (\<le>) (\<lambda>p. emeasure (measure_spmf p))"
limit_in_range: fixes r :: "nat \<Rightarrow> 'e" shows "limit r \<subseteq> range r"
adds_iff_deg_le_univariate: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t \<in> .[X]" shows "(s adds t) = (deg_pm s \<le> deg_pm t)"
moebius_pt_blaschke: fixes a :: "complex"   and z :: "complex" assumes "cmod a \<noteq> 1"   and "z \<noteq> 1 / cnj a" shows "moebius_pt (blaschke a) (of_complex z) = of_complex ((z - a) / (1 - cnj a * z))"
SystemConcl_def: shows "SystemConcl = ([], [(Geq (Var vid1) (Const 0) && Geq (f0 fid1) (Const 0)) \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) TT]]Geq (Var vid1) (Const 0)])"
add_nonneg_nonneg: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "{#} \<subseteq># a"   and "{#} \<subseteq># b" shows "{#} \<subseteq># a + b"
cp_name_name1: fixes pi1 :: "(name \<times> name) list"   and pi2 :: "(name \<times> name) list"   and x :: "'d" shows "pi1 \<bullet> pi2 \<bullet> x = (pi1 \<bullet> pi2) \<bullet> pi1 \<bullet> x"
part_def: fixes X :: "'a set" shows "part X = (K \<circ> I = I)"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "ring_functions R"
the_br: fixes R :: "('h \<times> 'i) set" assumes "single_valued R" shows "br (the_br_\<alpha> R) (\<lambda>x. x \<in> Domain R) = R"
i_take_take_eq1: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "f \<Down> n \<down> m = f \<Down> m"
prems: shows "\<not> (\<exists>ia\<in>insert i I. \<exists>j\<in>insert i I. f ia = - \<infinity> \<and> f j = \<infinity>)"
conj_disj_distribL: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "(P \<and> (Q \<or> R)) = (P \<and> Q \<or> P \<and> R)"
simps_2: fixes uv :: "String.literal" shows "is_Num (Str uv) = False"
GSMP_disjoint_subset: fixes L :: "('fun, 'var) Term.term set"   and R :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) Term.term set"   and L' :: "('fun, 'var) Term.term set"   and R' :: "('fun, 'var) Term.term set" assumes "GSMP_disjoint L R S"   and "L' \<subseteq> L"   and "R' \<subseteq> R" shows "GSMP_disjoint L' R' S"
nn_integral_disjoint_pair_countspace: fixes B :: "'b set"   and C :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "B \<inter> C = {}" shows "set_nn_integral (count_space UNIV) (B \<union> C) f = set_nn_integral (count_space UNIV) B f + set_nn_integral (count_space UNIV) C f"
us: shows "us \<equiv> getFrN (map snd txs) (Var y # map fst txs) [] (length txs)"
instrs_append_Write\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "instrs (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W]) = instrs sb @ [Write volatile a sop A L R W]"
axioms_2: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "ring R addition multiplication zero unit" shows "Group_Theory.monoid R multiplication unit"
ub_lnD: fixes prec :: "nat"   and x :: "float"   and y :: "float" assumes "ub_ln prec x = Some y" shows "ln (real_of_float x) \<le> real_of_float y \<and> 0 < real_of_float x"
with_def: shows "min.with \<equiv> \<lambda>less_eq a b. if less_eq a b then a else b"
MemReturn_enabled: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals" assumes "basevars (rtrner ch!p, rs!p)" shows "\<turnstile> Calling ch p \<and> rs!p \<noteq> #NotAResult \<longrightarrow> Enabled (<MemReturn ch rs p>_(rtrner ch!p, rs!p))"
secret_parts_agent: fixes m :: "msg"   and C :: "agent"   and evs :: "event list" assumes "m \<in> parts (knows C evs)" shows "m \<in> initState C \<or> (\<exists>A B X. Says A B X \<in> set evs \<and> m \<in> parts {X}) \<or> (\<exists>Y. Notes C Y \<in> set evs \<and> m \<in> parts {Y})"
well_order_on_Well_order: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" assumes "well_order_on A r" shows "A = Field r \<and> Well_order r"
iterm_to_fterm_subt: fixes t1 :: "(char list, char list) Term.term"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "iterm_to_fterm t1 \<cdot>\<^sub>t \<sigma> = iterm_to_fterm (t1 \<cdot> (\<lambda>x. fterm_to_iterm (\<sigma> x)))"
ewp_def: fixes x :: "'a" shows "ewp x \<equiv> \<not> (\<forall>y. y \<le> x \<cdot> y \<longrightarrow> y = (0::'a))"
less_eq_option_def: fixes x :: "'a option"   and y :: "'a option" shows "(x \<le> y) = (case x of None \<Rightarrow> True | Some x \<Rightarrow> case y of None \<Rightarrow> False | Some y \<Rightarrow> x \<le> y)"
no_marker_and_snapshotted_implies_no_more_markers_trace: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and j :: "nat"   and p :: "'d"   and cid :: "nat"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "i \<le> j"   and "j \<le> length t"   and "ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "Marker \<notin> set (msgs (distributed_system.s channel trans send recv init t i) cid)"   and "channel cid = Some (p, q)" shows "Marker \<notin> set (msgs (distributed_system.s channel trans send recv init t j) cid)"
prod_of_pdevs_def: fixes x :: "'c pdevs"   and y :: "'d pdevs" shows "prod_of_pdevs x y = binop_pdevs Pair x y"
finter_funion_distrib: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "A |\<inter>| (B |\<union>| C) = A |\<inter>| B |\<union>| (A |\<inter>| C)"
sorted_list_of_set_inject: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "class.linorder less_eq less"   and "linorder.sorted_list_of_set less_eq A = linorder.sorted_list_of_set less_eq B"   and "finite A"   and "finite B" shows "A = B"
ws_cdecl_Base: shows "ws_cdecl tprg Base Object"
cbox_sing: fixes a :: "'b" shows "cbox a a = {a}"
dom\<sigma>_mem\<^sub>2: shows "dom (to_partial mem\<^sub>2 |` differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k) = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k"
conj_zero_right: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "conj x zero = zero"
out_exprChannelSet_def: fixes sP :: "specID"   and chSet :: "chanID set"   and E :: "Expression" shows "out_exprChannelSet sP chSet E \<equiv> (\<forall>x. x \<in> chSet \<longrightarrow> x \<in> out sP \<and> exprChannel x E) \<and> (\<forall>x. x \<notin> chSet \<and> x \<in> out sP \<longrightarrow> \<not> exprChannel x E)"
LyndonI: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>"   and "\<forall>n. 0 < n \<and> n < \<^bold>|w\<^bold>| \<longrightarrow> w <lex rotate n w" shows "Lyndon w"
LBINT_I0c_sin_scale_divide: fixes T :: "real"   and \<theta> :: "real" assumes "0 \<le> T" shows "LBINT t=0..ereal T. sin (t * \<theta>) / t = sgn \<theta> * Si (T * \<bar>\<theta>\<bar>)"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and point :: "'c point" shows "P (case point of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = (\<not> (point = Infinity \<and> \<not> P f1 \<or> (\<exists>x21 x22. point = Point x21 x22 \<and> \<not> P (f2 x21 x22))))"
map: fixes x :: "String.literal"   and p :: "fol fm"   and e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and v :: "'a" assumes "\<not> free_in x p" shows "semantics e f g p = semantics (e(x := v)) f g p"
bs: fixes b :: "'a poly"   and i :: "nat"   and d :: "'a"   and fs :: "'a poly list" assumes "(b, i) \<in> set bs"   and "f b = (d, fs)" shows "b = smult d (prod_list fs) \<and> (\<forall>f\<in>set fs. 0 < degree f)"
right_ideal_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'b set" assumes "class.ring plus zero minus uminus times" shows "ring.right_ideal plus zero uminus times I = (group_add.subgroup plus zero uminus I \<and> (\<forall>x\<in>I. \<forall>r. times x r \<in> I))"
inv_closed: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier (Characters G)" shows "inv\<^bsub>Characters G\<^esub> x \<in> carrier (Characters G)"
slistmem_bottom2: fixes xs :: "[:'a:]" shows "\<not> slistmem \<bottom> xs"
bottom_edge_smooth: shows "(\<lambda>x. twoC (x, 0)) piecewise_C1_differentiable_on {0..1}"
runit_eqI: fixes f :: "'a"   and \<mu> :: "'a" assumes "ide f"   and "\<guillemotleft>\<mu> : f \<star> src f \<Rightarrow> f\<guillemotright>"   and "\<mu> \<star> src f = (f \<star> \<i>' (src f)) \<cdot> \<a>[f, src f, src f]" shows "\<mu> = PB.runit f"
proper_map_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "proper_map X Y f \<equiv> closed_map X Y f \<and> (\<forall>y\<in>topspace Y. compactin X {x \<in> topspace X. f x = y})"
incoming_def: fixes v :: "nat" shows "h.cf.incoming v \<equiv> {(u, v) |u. (u, v) \<in> h.cf.E}"
hiding_entails_def: fixes t :: "'c \<Rightarrow> 'd option"   and s :: "'c \<Rightarrow> 'e option"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(t hiding s entails R) = (\<forall>k. ! x:t k: ! y:s k: R x y)"
mult_scalar_sum_monomials: fixes q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.mult_scalar q p = (\<Sum>t\<in>keys q. local.punit.monom_mult (lookup q t) t p)"
prim_impl_correct_1: fixes \<pi>i :: "(('a \<times> 'a) \<times> color) tree" assumes "G.invar (G.from_list l)"   and "prim_impl (G.from_list l) r = \<pi>i" shows "P.invar_MST \<pi>i"
gb_schema_dummy_domI1: fixes data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "gb_schema_dummy_dom (data, D, bs, [])"
expand_superset: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "{a. g a \<noteq> (1::'c)} \<subseteq> A" shows "Prod_any g = prod g A"
simps_3: fixes x21 :: "'b"   and x22 :: "'b" shows "Point x21 x22 \<noteq> Infinity"
monotone_lmap: fixes f :: "'c \<Rightarrow> 'd" shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) (lmap f)"
tp: shows "tp = tm_of (ap [+] dummy_abc (length args))"
pfixGe_trans: fixes x :: "nat list"   and y :: "nat list"   and z :: "nat list" assumes "x pfixGe y"   and "y pfixGe z" shows "x pfixGe z"
g_has_sgnx1: assumes "g tt = 0"   and "sin tt < 0 \<or> cos tt = 1" shows "(g has_sgnx - 1) (at_left tt)"
predecessor_is_node: fixes n :: "'node"   and g :: "'g"   and n' :: "'node" assumes "n \<in> set (old.predecessors g n')"   and "invar g" shows "n \<in> set (\<alpha>n g)"
add2_Some_Some: fixes x :: "'b"   and y :: "'b" shows "total_partial.add2 (Some x) (Some y) = Sep_Eq.add2 x y"
strict_prefixE: fixes xs :: "'c list"   and ys :: "'c list" assumes "strict_prefix xs ys" obtains "prefix xs ys" and "xs \<noteq> ys"
simps: fixes y :: "name"   and v :: "tm"   and x :: "tm"   and x' :: "tm" assumes "atom y \<sharp> (v, x, x')" shows "KRP v x x' = SyntaxN.Ex y (HRP x (Var y) AND SubstFormP v (Var y) x x')"
that: fixes a :: "'a" assumes "S \<subseteq> box (- a) a" shows "thesis"
in_roofed_mono: fixes x :: "'a"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and S' :: "'a set" assumes "x \<in> roofed_gen G B S"   and "S \<subseteq> S'" shows "x \<in> roofed_gen G B S'"
cofinite_closure_finite: fixes X :: "'a cofinite set" assumes "finite X" shows "closure X = X"
HeronConf_interp_stepwise_sporadicon_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
induced_vector_apply1: fixes v :: "'v"   and x :: "('f, 'g) aezfun" assumes "v \<in> V"   and "x \<in> FH" shows "induced_vector v x = FG_proj x \<cdot> v"
assms_2: shows "set (x # xs) = set (y # ys)"
C1: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel'" shows "\<Psi>' \<rhd> S \<lessapprox><Rel'> T"
subset_1: shows "\<Union> (S ` A) \<subseteq> carrier G"
wf_jvm_prog_phi_def: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list" shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> \<equiv> wf_prog (\<lambda>P C (M, Ts, T\<^sub>r, mxs, mxl\<^sub>0, is, xt). wt_method P C Ts T\<^sub>r mxs mxl\<^sub>0 is xt (\<Phi> C M))"
assms: shows "Add fa1 fa2 = subst_floatarith (\<lambda>x. Var (s x)) fa"
eqvt: fixes p :: "perm"   and a :: "ty"   and aa :: "ty"   and ab :: "Syntax.tvar" shows "p \<bullet> subst_type a aa ab = subst_type (p \<bullet> a) (p \<bullet> aa) (p \<bullet> ab)"
abs_fun_eq2: fixes a :: "'b"   and b :: "'b"   and x :: "'a"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "a \<noteq> b"   and "[a].x = [b].y" shows "x = [(a, b)] \<bullet> y \<and> a \<sharp> y"
wf_relcomp_compatible: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "wf R"   and "R O S \<subseteq> S O R" shows "wf (S O R)"
node_lift_step_statelessassm: fixes T :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and I :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "nat"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R\<^sub>i :: "nat set" assumes "T \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg I \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). Q (\<sigma> i) (\<sigma>' i))"   and "\<And>\<xi>. Q \<xi> \<xi>" shows "\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). Q (\<sigma> i) (\<sigma>' i))"
minimal_element_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "minimal_element P A \<equiv> po_on P A \<and> wfp_on P A"
tree_chop_den: shows "tree_chop den = num + den - 2 * (num mod den)"
rev_nth_mapi_branch: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and f :: "nat \<Rightarrow> nat \<Rightarrow> ('b, 'c) fm \<Rightarrow> ('b, 'c) fm" assumes "branch !. v = Some (ps, a)" shows "(mapi (f v) ps, a) \<in>. mapi_branch f branch"
F'_4: shows "is_expansion_aux F' (\<lambda>x. f x - eval C x * b x powr e) (b # basis')"
leg0_composite: shows "cods.\<mu>\<nu>.leg0 = cods.\<nu>.leg0 \<cdot> cods.\<mu>\<nu>.prj\<^sub>0"
\<alpha>_def: fixes \<mu>\<nu>\<tau> :: "'b \<times> 'b \<times> 'b" shows "D.\<alpha> \<mu>\<nu>\<tau> \<equiv> \<a>\<^sub>D[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
f2: assumes "(incid A (line B' C') \<or> incid C' (line A B)) \<and> (incid C (line A' B') \<or> incid B' (line A C)) \<and> (incid B (line A' C') \<or> incid A' (line B C))" shows "col M N P \<or> triangle_circumscribes_triangle A B C A' B' C' \<or> triangle_circumscribes_triangle A' B' C' A B C"
IS_RIGHT_TOTAL_alt: fixes R :: "('c \<times> 'd) set" shows "IS_RIGHT_TOTAL R = (\<forall>x. \<exists>y. (y, x) \<in> R)"