hcju/isabelleps huggingface dataset:
ins_invariant: fixes succs :: "'c \<Rightarrow> 'c list"   and is_node :: "'c \<Rightarrow> bool"   and invariant :: "'d \<Rightarrow> bool"   and ins :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empt :: "'d"   and x :: "'c"   and S :: "'d" assumes "DFS succs is_node invariant ins memb empt"   and "is_node x"   and "invariant S"   and "\<not> memb x S" shows "invariant (ins x S)"
mset_size_partition_dep: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and A :: "'a multiset" shows "size {#a \<in># A. P a \<or> Q a#} = size (filter_mset P A) + size (filter_mset Q A) - size {#a \<in># A. P a \<and> Q a#}"
correctCompositionOut_prop1: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and x :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionOut PQ"   and "x \<in> out PQ" shows "x \<in> out P \<or> x \<in> out Q"
O_mono1: fixes R :: "('b \<times> 'c) set"   and R' :: "('b \<times> 'c) set"   and S :: "('d \<times> 'b) set" assumes "R \<subseteq> R'" shows "S O R \<subseteq> S O R'"
arrE: fixes f :: "'a1 \<times> 'a2" assumes "arr f" obtains "C1.arr (fst f) \<and> C2.arr (snd f)"
semiconnex: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x \<sqsubseteq> y \<or> x = y \<or> y \<sqsubseteq> x"
bij_minus_subset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and C :: "'a set" assumes "bij_betw f A B"   and "C \<subseteq> A" shows "f ` A - f ` C = f ` (A - C)"
list_ex_iff: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "list_ex P xs = Bex (set xs) P"
atm_of_def: fixes literal :: "'a literal" shows "atm_of literal = (case literal of Pos x1 \<Rightarrow> x1 | Neg x2 \<Rightarrow> x2)"
vset_diff_disj_eff_vs: fixes a :: "('c, bool) fmap \<times> ('c, bool) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'e) fmap) set"   and vs :: "'c set" assumes "varset_action a (prob_dom PROB - vs)" shows "fmdom' (snd a) \<inter> vs = {}"
id_act_3: fixes v :: "i"   and \<alpha> :: "'a"   and \<beta> :: "'a" shows "[\<alpha> \<^bold>= \<beta> \<^bold>\<equiv> \<^bold>\<A>(\<alpha> \<^bold>= \<beta>) in v]"
oth_class_taut_3_i: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" shows "[\<phi> \<^bold>\<equiv> (\<psi> \<^bold>\<equiv> \<chi>) \<^bold>\<equiv> (\<phi> \<^bold>\<equiv> \<psi> \<^bold>\<equiv> \<chi>) in v]"
isnpolyh_mono: fixes n' :: "nat"   and n :: "nat"   and p :: "'b poly" assumes "n' \<le> n"   and "isnpolyh p n" shows "isnpolyh p n'"
assms_1: shows "\<forall>\<^sub>F x in F. \<forall>a\<in>S. norm (f x a) \<le> g x a"
finite_inversions_between: fixes xs :: "'a list"   and ys :: "'a list" shows "finite (inversions_between xs ys)"
prj0_ext: fixes f :: "'a"   and g :: "'a" assumes "\<not> C.cospan f g" shows "\<p>\<^sub>0[f, g] = C.null"
weakly_preferred_by_men_strictly_preferred_by_a_man: fixes \<mu> :: "('m \<times> 'w) set"   and \<mu>' :: "('m \<times> 'w) set" assumes "match \<mu>"   and "match \<mu>'" shows "weakly_preferred_by_men \<mu> \<mu>' = (\<not> strictly_preferred_by_a_man \<mu>' \<mu>)"
list_all3_list_all2_eq: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list" shows "list_all3 P xs ys zs = (length xs = length ys \<and> list_all2 (\<lambda>(x, y). P x y) (zip xs ys) zs)"
intros: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> Sec option"   and \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<forall>x\<in>dom \<Gamma>'. \<Gamma>' x = Some High"   and "\<turnstile> \<Gamma>\<^sub>1 {c\<^sub>1} \<Gamma>'"   and "\<turnstile> \<Gamma>\<^sub>2 {c\<^sub>2} \<Gamma>'"   and "mds_consistent mds \<Gamma>\<^sub>1"   and "mds_consistent mds \<Gamma>\<^sub>2" shows "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
gb_red_aux_irredudible: fixes h :: "'t \<Rightarrow>\<^sub>0 'e"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and b :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f" assumes "h \<in> set (gb_red_aux bs ps)"   and "b \<in> set bs"   and "fst b \<noteq> 0" shows "\<not> lt (fst b) adds\<^sub>t lt h"
DF_RUN_refine_FD: fixes A :: "'b set" shows "DF A \<sqsubseteq>\<^sub>F\<^sub>D RUN A"
simps: fixes g :: "'g"   and n :: "'node"   and v :: "'var" shows "lookupDef g n v = (if n \<notin> set (\<alpha>n g) then undefined else if v \<in> defs g n then (v, n, SimpleDef) else case predecessors g n of [] \<Rightarrow> (v, n, PhiDef) | [m] \<Rightarrow> lookupDef g m v | m # aa # lista \<Rightarrow> (v, n, PhiDef))"
eq2: shows "(\<Sum>v\<in>keys p \<union> keys q. monomial (lookup q v) (decr_grading_term d n v)) = (\<Sum>v\<in>keys q. monomial (lookup q v) (decr_grading_term d n v))"
of_nat_inj: fixes x :: "nat"   and y :: "nat" assumes "x < 2 ^ LENGTH('b)"   and "y < 2 ^ LENGTH('b)" shows "(word_of_nat x = word_of_nat y) = (x = y)"
simps_1: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "shd (siterate f x) = x"
lin_independent_length_le_dim: fixes us :: "'b list" assumes "set us \<subseteq> V"   and "R_lin_independent us" shows "length us \<le> dim_R V"
S2: shows "S2 = (\<lambda>s. if s ''b'' \<le> s ''g'' then s else s(''y'' := s ''z'' * s ''b'' + s ''b'', ''g'' := s ''b''))"
Call: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and E :: "assigned"   and args :: "expr list"   and A :: "assigned"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and mn :: "mname"   and pTs :: "ty list" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E"   and "Env\<turnstile> nrm E \<guillemotright>\<langle>args\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>{accC,statT,mode}e\<cdot>mn( {pTs}args)\<rangle>\<guillemotright> A"
prems_1: shows "\<forall>p. ps c p \<noteq> None"
node_total_order_antisym: fixes e1 :: "'evt"   and i :: "nat"   and e2 :: "'evt" assumes "e1 \<sqsubset>\<^sup>i e2"   and "e2 \<sqsubset>\<^sup>i e1" shows "False"
scale_right_distrib: fixes a :: "'a"   and x :: "'c"   and y :: "'c" shows "a *b (x + y) = a *b x + a *b y"
elims_3: fixes x :: "'a list" assumes "\<not> sorted x" obtains xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "\<not> (Ball (set ys) ((\<le>) xa) \<and> sorted ys)"
union_subseteqI: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and A' :: "'a set"   and B' :: "'a set" assumes "A \<union> B \<subseteq> C"   and "A' \<subseteq> A"   and "B' \<subseteq> B" shows "A' \<union> B' \<subseteq> C"
bound: fixes t :: "'a"   and x :: "'b" assumes "t \<in> cball t0 (d / 3)"   and "x \<in> cball x0 (d / 3)" shows "norm (f (t, x)) \<le> norm (f (t0, x0)) + 1"
degree_map_poly_hom: fixes p :: "real poly" shows "degree (map_poly of_real p) = degree p"
lasso_of_prpl_param: fixes l' :: "'e list \<times> 'e list"   and l :: "'f list \<times> 'f list"   and R :: "('e \<times> 'f) set" assumes "(l', l) \<in> \<langle>R\<rangle>list_rel \<times>\<^sub>r \<langle>R\<rangle>list_rel"   and "snd l \<noteq> []" shows "(lasso_of_prpl l', lasso_of_prpl l) \<in> \<langle>unit_rel, R\<rangle>lasso_rel_ext"
returns_to_def: fixes P :: "'a set"   and x :: "'a" shows "Y.returns_to P x = ((\<forall>\<^sub>F t in at_right 0. Y.flow0 x t \<notin> P) \<and> (\<exists>t>0. t \<in> Y.existence_ivl0 x \<and> Y.flow0 x t \<in> P))"
singletonD: fixes b :: "'x"   and a :: "'x" assumes "b \<in> {a}" shows "b = a"
correctCompositionIn_prop1: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and x :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "x \<in> ins PQ" shows "x \<in> ins P \<or> x \<in> ins Q"
const_onI: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "\<And>y. y \<in> S \<Longrightarrow> f y = x" shows "const_on f S x"
IH_1: fixes inputs :: "real vec list" assumes "valid_net' m1"   and "map dim_vec inputs = input_sizes m1" shows "output_size' m1 = dim_vec (evaluate_net m1 inputs)"
p_max_eq: shows "ego.p_max = s\<^sub>e - v\<^sub>e\<^sup>2 / a\<^sub>e / 2"
euler_genus_def: fixes G :: "('b, 'c) pre_digraph"   and M :: "'c pre_map" shows "pre_digraph_map.euler_genus G M \<equiv> (int (2 * card (pre_digraph.sccs G)) - int (card (pre_digraph.isolated_verts G)) - pre_digraph_map.euler_char G M) div 2"
preservation_else: fixes x :: "'a list"   and l1 :: "'a list"   and xa :: "'a"   and l2 :: "'a list"   and \<sigma> :: "'a set" assumes "set x = carrier"   and "x = l1 @ xa # l2"   and "I_minWeightBasis (\<sigma>, set (xa # l2))"   and "indep \<sigma>"   and "\<not> indep (insert xa \<sigma>)" shows "I_minWeightBasis (\<sigma>, set l2)"
repeat_mset_cancel1: fixes a :: "nat"   and A :: "'k multiset"   and B :: "'k multiset" shows "(repeat_mset a A = repeat_mset a B) = (A = B \<or> a = 0)"
that: fixes qs :: "'a list" assumes "qs \<in> listset ss"   and "a = sum_list qs" shows "thesis"
mod_ring_mult_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (mult_p_integer pp) (*)"
bluh: fixes i :: "'sf"   and x :: "(real, 'sz) vec" shows "(Functions I i has_derivative (THE f'. \<forall>x. (Functions I i has_derivative f' x) (at x)) x) (at x)"
gWlsInp_def: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) Recursion.model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'h option" shows "gWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (gWls MOD) (arOf delta) inp"
eff_While: fixes tst :: "'test"   and c :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "eff (While tst c) s i = s"
inj_on_opp_chambers': fixes C :: "'a set"   and D :: "'a set" assumes "folding_g.chamber C"   and "C \<notin> f \<turnstile> X"   and "folding_g.chamber D"   and "D \<notin> f \<turnstile> X"   and "f ` C = f ` D" shows "C = D"
smap_streams: fixes s :: "'c stream"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "s \<in> streams A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "smap f s \<in> streams B"
tendsto_cong: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and F :: "'c filter"   and c :: "'b" assumes "class.topological_space open"   and "\<forall>\<^sub>F x in F. f x = g x" shows "filterlim f (topological_space.nhds open c) F = filterlim g (topological_space.nhds open c) F"
prems: fixes BS :: "'a tensor list" assumes "\<And>B. B \<in> set BS \<Longrightarrow> cprank_max1 B"   and "\<And>Ba. Ba \<in> set BS \<Longrightarrow> dims (A + B) = dims Ba"   and "listsum (dims (A + B)) BS = A + B"   and "length BS = Suc j" shows "thesis"
cscc_invar_ext_def: fixes G :: "('a, 'b) graph_rec_scheme"   and l :: "'a set list"   and D :: "'a set" shows "cscc_invar_ext G l D \<equiv> fr_graph G \<and> cscc_invar_ext_axioms G l D"
poly_of_vars_remdups_mset: fixes \<phi> :: "char list \<Rightarrow> nat"   and xs :: "char list multiset" assumes "poly_embed \<phi>" shows "poly_embed.poly_of_vars \<phi> (remdups_mset xs) - poly_embed.poly_of_vars \<phi> xs \<in> More_Modules.ideal polynomial_bool"
list_succ_list_succ_rev: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs" shows "list_succ (rev xs) (list_succ xs x) = x"
set_mtf: fixes x :: "'b"   and xs :: "'b list" shows "set (mtf x xs) = set xs"
unique_conjugate2: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "\<exists>!g. conjugation_pair g f"
cmp_simps_1: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "C.arr (\<Phi>\<^sub>H (f, g))"
sep_globalise: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and h :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P R h"   and "\<And>h. P h \<Longrightarrow> Q h" shows "sep_algebra.sep_conj plus sep_disj Q R h"
fbspec: fixes A :: "'a fset"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "fBall A P"   and "x |\<in>| A" shows "P x"
initCfgList_def: shows "initCfgList = map (\<lambda>ind. \<lparr>states = start, msgs = initMsg ind\<rparr>) [0..<n + 1]"
monic_0: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.monic pair_of_term term_of_pair ord_term 0 = 0"
adjust_fetch0: fixes a :: "nat"   and ap :: "(action \<times> nat) list"   and b :: "cell"   and aa :: "action" assumes "0 < a"   and "a \<le> length ap div 2"   and "fetch ap a b = (aa, 0)" shows "fetch (adjust0 ap) a b = (aa, Suc (length ap div 2))"
tensor_mat_adjoint: fixes m1 :: "'b mat"   and m2 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2" shows "adjoint (tensor_mat m1 m2) = tensor_mat (adjoint m1) (adjoint m2)"
sum_insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "\<Sum> (insert x A) = x + \<Sum> A"
S1: shows "univ_for_pr key = nat_to_pr (c_fst key) (c_snd key)"
T_drop_eq: fixes n :: "nat"   and xs :: "'a list" shows "T_drop n xs = min n (length xs) + 1"
valid_plan_mems: fixes as :: "'a list"   and PROB :: "'a set"   and a :: "'a" assumes "as \<in> valid_plans PROB"   and "ListMem a as" shows "a \<in> PROB"
strict_coboundedI2: fixes b :: "'d"   and c :: "'d"   and a :: "'d" assumes "b < c" shows "min a b < c"
construct_NofI_R_prop_helper: fixes p :: "real poly"   and I1 :: "real poly list"   and I2 :: "real poly list" assumes "p \<noteq> 0" shows "construct_NofI_R p I1 I2 = rat_of_int (int (card {x. poly (sum_list (map power2 (p # I1))) x = 0 \<and> 0 < poly (prod_list I2) x}) - int (card {x. poly (sum_list (map power2 (p # I1))) x = 0 \<and> poly (prod_list I2) x < 0}))"
square_powr_half: fixes x :: "real" shows "x\<^sup>2 powr (1 / 2) = \<bar>x\<bar>"
nn_integral_indicator: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "integral\<^sup>N M (indicator A) = emeasure M A"
assms: shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q"
divide_cancel_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide c a = divide c b) = (c = zero \<or> a = b)"
le_option_refl: fixes x :: "'h option" shows "le_option x x"
iso_char: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b"   and \<phi> :: "'b" assumes "left_hom V H \<omega>"   and "partial_magma.arr (subcategory.comp V (weak_composition.left V H \<omega>)) \<phi>" shows "category.iso (subcategory.comp V (weak_composition.left V H \<omega>)) \<phi> = category.iso V \<phi>"
dynmethd_cases: fixes G :: "prog"   and statC :: "qtname"   and sig :: "sig"   and statM :: "qtname \<times> methd"   and dynC :: "qtname" assumes "methd G statC sig = Some statM"   and "G\<turnstile>dynC\<preceq>\<^sub>C statC"   and "is_class G statC"   and "ws_prog G" obtains "dynmethd G statC dynC sig = Some statM"   | dynM :: "qtname \<times> methd" where "dynmethd G statC dynC sig = Some dynM" and "dynM \<noteq> statM" and "G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM"
assms: fixes l :: "cell list"   and r :: "cell list" assumes "P (l, r)" shows "\<exists>n. is_final (steps0 (1, l, r) p n) \<and> Q holds_for steps0 (1, l, r) p n"
planning_by_cnf_dimacs_sound: fixes \<A> :: "nat \<Rightarrow> bool"   and t :: "nat" assumes "\<A> \<Turnstile> map_formula var_to_dimacs (\<Phi>\<^sub>\<forall> \<phi> prob_with_noop abs_prob  t)" shows "valid_plan (decode_abs_plan (rem_noops (map (strips_op_to_sasp (prob_with_noop abs_prob)) (concat (\<Phi>\<inverse> \<phi> prob_with_noop abs_prob  \<A> \<circ> var_to_dimacs t)))))"
succsr_is_node: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "(x, y) \<in> (succsr (prod_succs A B))\<^sup>*"   and "prod_is_node A B x" shows "prod_is_node A B y"
augment_shared_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and W :: "nat set"   and S :: "nat set" shows "\<S> \<oplus>\<^bsub>W\<^esub> S \<equiv> \<lambda>a. if a \<in> S then Some (a \<in> W) else \<S> a"
IH: assumes "\<Union> (set As) \<subseteq> carrier"   and "is_weak_ranking As"   and "of_weak_ranking As = restrict_relation (\<Union> (set As)) le" shows "As = weak_ranking_aux (\<Union> (set As))"
heterogeneous_term_is_Fun: fixes t :: "('fun, 'var) Term.term"   and A :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and S :: "('fun, 'var) Term.term set" assumes "heterogeneous\<^sub>l\<^sub>s\<^sub>t t A S" shows "\<exists>f T. t = Fun f T"
assms: shows "\<forall>view. lossless_spmf (D2 view)"
eq_newpkt_eq: fixes newpkt :: "nat \<times> nat \<Rightarrow> 'a"   and eq_newpkt :: "'a \<Rightarrow> bool"   and d :: "nat"   and i :: "nat" assumes "class.msg newpkt eq_newpkt" shows "eq_newpkt (newpkt (d, i))"
exp_scaleR_has_vector_derivative_left': fixes A :: "'a"   and t :: "real"   and S :: "real set" shows "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative A * exp (t *\<^sub>R A)) (at t within S)"
OUTfromVCorrect1_data2: shows "OUTfromVCorrect1 data2"
snd_set_def: fixes s :: "('b \<times> 'a) set" shows "snd_set s = {a. \<exists>b. (b, a) \<in> s}"
hyps_3: assumes "\<And>a. 0 \<le> f a" shows "(\<Sum>a\<in>A. ennreal (f a)) = ennreal (sum f A)"
list_all2_reorder_right_invariance: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list"   and ys' :: "'e list" assumes "list_all2 R xs ys"   and "mset ys' = mset ys" shows "\<exists>xs'. list_all2 R xs' ys' \<and> mset xs' = mset xs"
ChamberComplexMorphism_axioms: shows "ChamberComplexMorphism X Y f"
simps_2: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and cfg :: "('a, 'c, 'b) configuration"   and fStepCfg :: "('a, 'c, 'b) configuration list \<Rightarrow> ('a, 'c) message list \<Rightarrow> ('a, 'c, 'b) configuration list"   and fStepMsg :: "('a, 'c, 'b) configuration list \<Rightarrow> ('a, 'c) message list \<Rightarrow> ('a, 'c) message list"   and n :: "nat" assumes "flpPseudoConsensus trans sends start" shows "flpPseudoConsensus.infiniteExecutionMsg cfg fStepCfg fStepMsg (Suc n) = fStepMsg (flpPseudoConsensus.infiniteExecutionCfg cfg fStepCfg fStepMsg n) (flpPseudoConsensus.infiniteExecutionMsg cfg fStepCfg fStepMsg n)"
nodeToDigit_list: fixes nd :: "('b, 'c) Node" shows "digitToList (nodeToDigit nd) = nodeToList nd"
reflP_def: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "reflP P \<phi> \<equiv> (\<forall>x y. \<phi> x y \<or> \<phi> y x \<longrightarrow> P x) \<and> (\<forall>x. P x \<longrightarrow> \<phi> x x)"
lexmax_singleton: fixes r :: "('a \<times> 'a) set"   and \<alpha> :: "'a" shows "r|[\<alpha>]| = {#\<alpha>#}"
simps: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word" shows "wgeq w1 w2 = (\<not> (w2 = NEG_INF \<and> w1 = NEG_INF \<or> w2 = POS_INF \<and> w1 = POS_INF) \<and> sint w2 \<le> sint w1)"
ins_in_Def: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and ins :: "'c list"   and outs :: "'c list"   and V :: "'c" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs"   and "V \<in> set ins" shows "V \<in> Def (targetnode a)"
has_domain_iff_arr: fixes f :: "'a" shows "(domains f \<noteq> {}) = arr f"
e''': shows "e''' \<in> UNIV \<rightarrow> I"
c': shows "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'"
bij_map_sumI: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'k" assumes "bij f"   and "bij g" shows "bij (map_sum f g)"
WHATWHERE_Secure_def: fixes V :: "'com list" shows "WHATWHERE_Secure V \<equiv> \<forall>d PP. \<exists>R. Strong_dlHPP_Bisimulation d PP R \<and> (V, V) \<in> R"
f_expand_nth_if: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs * k" shows "xs \<odot> k ! n = (if n mod k = 0 then xs ! (n div k) else NoMsg)"
a_get_attribute_locs_def: fixes element_ptr :: "(_) element_ptr" shows "a_get_attribute_locs element_ptr = {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr attrs)}"
ardens_lemma_var_equiv: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x\<^sup>\<omega> = (0::'a)" shows "(z + x \<cdot> y = y) = (x\<^sup>\<star> \<cdot> z = y)"
d\<mu>_impl: shows "d\<mu>_impl fs_init = IArray.of_fun (\<lambda>i. IArray.of_fun (fs_init.d\<mu> i) (Suc i)) (length fs_init)"
filter_FF: fixes xs :: "['b]"   and P :: "'b \<rightarrow> tr" assumes "finite_list xs"   and "\<forall>x\<in>set xs. P\<cdot>x = FF" shows "filter\<cdot>P\<cdot>xs = []"
rotate1_lemma: fixes csa :: "'a list" assumes "cycle csa" shows "cycle_of_list csa = cycle_of_list (rotate1 csa)"
EndPhase1_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "EndPhase1 s s' p = (IsMajority {d \<in> disksWritten s p. \<forall>q\<in>UNIV - {p}. hasRead s p d q} \<and> phase s p = 1 \<and> dblock s' = (dblock s)(p := dblock s p\<lparr>bal := mbal (dblock s p), inp := if nonInitBlks s p = {} then inpt s p else inp (maxBlk s p)\<rparr>) \<and> outpt s' = outpt s \<and> phase s' = (phase s)(p := phase s p + 1) \<and> InitializePhase s s' p \<and> inpt s' = inpt s \<and> disk s' = disk s)"
S_transform_Act: fixes \<alpha> :: "'b"   and x :: "('a, 'c, 'b) formula" shows "S_transform (Formula.Act \<alpha> x) = Formula.Act (S_action.Act \<alpha>) (S_transform x)"
Abs_bit0'_def: fixes x :: "int" shows "Abs_bit0' x = Abs_bit0 (x mod int CARD('b bit0))"
DynCastRedsThrow: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
lift_Suc_mono_less: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "\<And>n. f n < f (Suc n)"   and "n < n'" shows "f n < f n'"