hcju/isabelleps huggingface dataset:
elims: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int"   and xa :: "nat"   and y :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "res_seq x xa = y" obtains s :: "nat \<Rightarrow> nat \<Rightarrow> int" where "x = sa__" and "xa = 0" and "y = sa__"   | s :: "nat \<Rightarrow> nat \<Rightarrow> int" and k :: "nat" where "x = s" and "xa = Suc k" and "y = const_res_subseq (Suc k) (res_seq s k)"
szpilrajn_exists: fixes p :: "'a" assumes "order p \<and> regular p" shows "\<exists>t. linear_orderable_1 t \<and> p \<le> t"
or_negative_int_iff: fixes k :: "int"   and l :: "int" shows "(k OR l < 0) = (k < 0 \<or> l < 0)"
is_processT2: fixes P :: "'b process" shows "\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s"
map_key_plus: fixes f :: "'c \<Rightarrow> 'd"   and p :: "'d \<Rightarrow>\<^sub>0 'e"   and q :: "'d \<Rightarrow>\<^sub>0 'e" assumes "inj f" shows "Poly_Mapping.map_key f (p + q) = Poly_Mapping.map_key f p + Poly_Mapping.map_key f q"
rep_st_def: shows "rep_st \<equiv> quot_type.rep Rep_st"
bisim1CAS1: fixes P :: "'a prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and e1' :: "'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 D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1" assumes "P,e1,h \<turnstile> (e1', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3),h \<turnstile> (e1'\<bullet>compareAndSwap(D\<bullet>F, e2, e3), xs) \<leftrightarrow> (stk, loc, pc, xcp)"
in_qtableE: fixes n :: "nat"   and A :: "nat set"   and P :: "'b option list \<Rightarrow> bool"   and Q :: "'b option list \<Rightarrow> bool"   and X :: "'b option list set"   and x :: "'b option list" assumes "qtable n A P Q X"   and "x \<in> X"   and "P x" obtains "wf_tuple n A x" and "Q x"
assms_5: shows "\<C> \<turnstile> [e] : t_in _> t_out"
homotopic_loops_linear: fixes g :: "real \<Rightarrow> 'a"   and h :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "path h"   and "pathfinish g = pathstart g"   and "pathfinish h = pathstart h"   and "\<And>t x. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> S" shows "homotopic_loops S g h"
remove_vertex_def: fixes x :: "'a" shows "remove_vertex x \<equiv> G\<lparr>verts := V - {x}, arcs := Restr E (V - {x})\<rparr>"
rts1d: fixes p :: "nat"   and i :: "nat" assumes "correct p (te p (i + 1))" shows "rmin \<le> te p (i + 1) - te p i"
app_Some: fixes i :: "instr"   and P :: "'a prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list" shows "app i P mxs T\<^sub>r pc mpc xt \<lfloor>\<tau>\<rfloor> = (app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', s')\<in>set (eff i P pc xt \<lfloor>\<tau>\<rfloor>). pc' < mpc))"
preserves_cod: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "C.cod (A.map f) = A.map (J.cod f)"
DrA: shows "\<D>\<^sub>\<circ> r \<inter>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
del_block_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection" shows "incidence_system.del_block block_collection b \<equiv> remove1_mset b block_collection"
projl_def: fixes sum :: "'e + 'f" shows "projl sum = (case sum of Inl x1 \<Rightarrow> x1)"
simps_1: shows "brickmat vert = [[1, 0], [0, 1]]"
relcomppE: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and c :: "'d" assumes "(r OO s) a c" obtains b :: "'c" where "r a b" and "s b c"
mk_scheme_injective: fixes t :: "typ"   and t' :: "typ" assumes "mk_scheme t = mk_scheme t'" shows "t = t'"
inverse_distrib_swap: fixes a :: "'d"   and b :: "'d" shows "- (a + b) = - b + - a"
urel_integer_inj': fixes p :: "int"   and pp :: "integer"   and x :: "integer"   and z :: "int"   and y :: "integer" assumes "mod_ring_locale TYPE('b) p"   and "p = int_of_integer pp"   and "mod_ring_locale.urel_integer p x z"   and "mod_ring_locale.urel_integer p y z" shows "x = y"
M_def: shows "M = mat m m (\<lambda>(i, j). A $$ (if i < l then i else if i < l + n then i + k else i - n, j))"
simps_3: fixes \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and A :: "'b ptrm"   and B :: "'b ptrm" shows "ptrm_infer_type \<Gamma> (PApp A B) = (case (ptrm_infer_type \<Gamma> A, ptrm_infer_type \<Gamma> B) of (None, b) \<Rightarrow> None | (Some TUnit, b) \<Rightarrow> None | (Some (TVar xa), b) \<Rightarrow> Map.empty xa | (Some (TArr \<tau> \<sigma>), None) \<Rightarrow> None | (Some (TArr \<tau> \<sigma>), Some \<tau>') \<Rightarrow> if \<tau> = \<tau>' then Some \<sigma> else None | (Some (TPair type1 xa), b) \<Rightarrow> Map.empty xa)"
BC_wrt_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "BC_wrt \<psi> \<equiv> {U. \<exists>s. learn_bc \<psi> U s}"
numeral_Bit0_div_2: fixes n :: "num" shows "numeral (num.Bit0 n) div (2::'b) = numeral n"
deleted_ids_refs_older: fixes ops :: "('c \<times> ('c, 'd) list_op) list"   and oid :: "'c"   and oper :: "('c, 'd) list_op"   and ref :: "'c" assumes "list_ops (ops @ [(oid, oper)])"   and "ref \<in> set (deleted_ids ops)" shows "ref < oid"
simps_2: fixes b :: "bexp"   and c :: "com"   and n :: "nat"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int" shows "wpw b c (Suc n) Q s = (bval b s \<and> (\<exists>s'. (c, s) \<Rightarrow> s' \<and> wpw b c n Q s'))"
F_def: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "GAUSS.F p a = (-) (int p) ` GAUSS.E p a"
finsertCI: fixes a :: "'m"   and B :: "'m fset"   and b :: "'m" assumes "a |\<notin>| B \<Longrightarrow> a = b" shows "a |\<in>| finsert b B"
measure_space_0': fixes \<Omega> :: "'a set" shows "measure_space \<Omega> {{}, \<Omega>} (\<lambda>x. 0)"
two: shows "2\<^sub>\<nat> = set {[]\<^sub>\<circ>, 1\<^sub>\<nat>}"
closed_sum_closure_left: fixes A :: "'d set"   and B :: "'d set" shows "closure A +\<^sub>M B = A +\<^sub>M B"
ex_step1I: fixes x :: "'a"   and xs :: "'a list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" assumes "x \<in> set xs"   and "r y x" shows "\<exists>ys. step1 r ys xs \<and> y \<in> set ys"
g: shows "(g has_derivative g') (at (f x) within f ` s)"
satisfiable_def: fixes cnf :: "lit uprod set" shows "satisfiable cnf = (\<exists>\<sigma>. sat_cnf \<sigma> cnf)"
Ports_single_1: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and proto :: "8 word"   and ps :: "(16 word \<times> 16 word) list"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (Match (Src_Ports (L4Ports proto ps))) a p = (proto = p_proto p \<and> p_sport p \<in> ports_to_set ps)"
atMost_def: fixes u :: "'a" shows "{..u} \<equiv> {x. x \<le> u}"
essential_class_def: fixes C :: "'s option set" shows "K2.essential_class C = (C \<in> UNIV // K2.communicating \<and> K2.acc `` C \<subseteq> C)"
cterms_not_call: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>"   and "p \<in> cterms \<Gamma>" shows "not_call p"
affine_parallel_expl: fixes S :: "'c set"   and T :: "'c set" shows "affine_parallel S T = (\<exists>a. \<forall>x. (x \<in> S) = (a + x \<in> T))"
head_closed: fixes R :: "('c, 'd) ring_scheme"   and xs :: "'c list" assumes "cring R"   and "cring.in_carrier R xs" shows "cring.head R xs \<in> carrier R"
prems_8: fixes j :: "nat" assumes "prime j"   and "i \<le> j" shows "multiplicity j factor_pr = 0"
append_is_Nil_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<cdot> ys = \<epsilon>) = (xs = \<epsilon> \<and> ys = \<epsilon>)"
SPEC_rule_conjI: fixes A :: "'b nres"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "A \<le> SPEC P"   and "A \<le> SPEC Q" shows "A \<le> SPEC (\<lambda>v. P v \<and> Q v)"
nprv_clear2_1: fixes \<phi>2 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>1 :: "'fmla" assumes "nprv {\<phi>2} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2} \<psi>"
not_sum_le_cases: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a + 'b"   and b :: "'a + 'b" assumes "\<not> sum_le P Q a b" obtains x :: "'a" and y :: "'a" where "a = Inl x__" and "b = Inl y__" and "\<not> P x__ y__"   | x :: "'b" and y :: "'b" where "a = Inr xa__" and "b = Inr ya__" and "\<not> Q xa__ ya__"   | x :: "'a" and y :: "'b" where "a = Inl xb__" and "b = Inr yb__"   | x :: "'b" and y :: "'a" where "a = Inr x" and "b = Inl y"
valid_line_prepend_B: fixes xs :: "color list" assumes "valid_line xs" shows "valid_line (B # xs)"
bexE: fixes A :: "'b::type set"   and P :: "'b::type \<Rightarrow> bool" assumes "\<exists>x::'b::type\<in>A::'b::type set. (P::'b::type \<Rightarrow> bool) x" obtains x :: "'b::type" where "(x::'b::type) \<in> (A::'b::type set)" and "(P::'b::type \<Rightarrow> bool) (x::'b::type)"
show_rightmost_bits_def: fixes bvec :: "bit list"   and n :: "nat" shows "show_rightmost_bits bvec n = rev (take n (rev bvec))"
two_digit_le: fixes i :: "nat"   and m :: "nat"   and j :: "nat"   and n :: "nat" assumes "i < m"   and "j < n" shows "i * n + j < m * n"
simps: fixes a1 :: "real"   and a2 :: "(real \<Rightarrow> real) list" shows "is_expansion a1 a2 = (\<exists>c. a1 = c \<and> a2 = [])"
sba5_inf_commutative: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
indep_iff_subset_basis: fixes X :: "'a set" shows "indep_in \<E> X = (\<exists>B. \<E>.basis B \<and> X \<subseteq> B)"
A_least_j: shows "A $ (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> (0::'rows) \<le> n) $ j = (0::'a)"
get_dom_component_def: shows "get_dom_component \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_dom_component get_root_node to_tree_order"
that: fixes i :: "nat" assumes "\<rho> x \<^sup>@ Suc i = x" shows "thesis"
Func_def: fixes A :: "'d set"   and B :: "'e set" shows "Func A B = {f. (\<forall>a\<in>A. f a \<in> B) \<and> (\<forall>a. a \<notin> A \<longrightarrow> f a = undefined)}"
wfP_induct_rule: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "wfP r"   and "\<And>x. (\<And>y. r y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
comp_subst_assoc: fixes \<sigma> :: "'s"   and \<tau> :: "'s"   and \<gamma> :: "'s" shows "\<sigma> \<odot> (\<tau> \<odot> \<gamma>) = \<sigma> \<odot> \<tau> \<odot> \<gamma>"
k: shows "(clock_simAbs (fst k), snd k) \<in> (\<Union>a. Clock.sim_equiv_class a ` Clock.jkbpC) \<times> UNIV"
l2: shows "\<forall>t p. trace init t final \<longrightarrow> (\<exists>i. ps (S t i) p \<noteq> None \<and> i \<le> length t)"
awalk_Cons_iff: fixes u :: "'a"   and e :: "'b"   and es :: "'b list"   and w :: "'a" shows "awalk u (e # es) w = (e \<in> arcs G \<and> u = tail G e \<and> awalk (head G e) es w)"
neg_minus_divideR_le_eq: fixes c :: "real"   and b :: "'a"   and a :: "'a" assumes "c < 0" shows "(- (b /\<^sub>R c) \<le> a) = (c *\<^sub>R a \<le> - b)"
domintros: fixes h' :: "'b \<Rightarrow> 'b"   and b :: "'b"   and Q :: "'c \<Rightarrow> bool"   and x :: "'c"   and h :: "'c \<Rightarrow> 'c" assumes "recursive h' b"   and "\<not> Q x \<Longrightarrow> tailrec.recfun_aux_dom Q h (h x)" shows "tailrec.recfun_aux_dom Q h x"
fvar_def: fixes C :: "qtname"   and stat :: "bool"   and fn :: "vname"   and a' :: "val"   and s :: "state" shows "fvar C stat fn a' s = (let (oref, xf) = if stat then (Inr C, id) else (Inl (the_Addr a'), np a'); n = Inl (fn, C); f = \<lambda>v. supd (upd_gobj oref n v) in ((the (values (the (globs (snd s) oref)) n), f), abupd xf s))"
Infinitesimal_of_hypreal: fixes x :: "real star" assumes "x \<in> Infinitesimal" shows "(*f* of_real) x \<in> Infinitesimal"
i12_1: shows "i1 \<in> circline_set H \<inter> unit_circle_set"
bdia_bbox_demorgan: fixes R :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'd"   and x :: "'f" shows " \<langle>R| f x = - [R| (\<lambda>y. - f y) x"
distr_pair_swap: shows "T \<Otimes>\<^sub>M S = distr (S \<Otimes>\<^sub>M T) (T \<Otimes>\<^sub>M S) (\<lambda>(x, y). (y, x))"
elims_1: fixes x :: "'b literal"   and y :: "bool" assumes "negative_literal x = y" obtains xa :: "'b equation" where "x = Pos xa__" and "\<not> y"   | xa :: "'b equation" where "x = Neg xa" and "y"
sub: shows "\<pi> ` A \<subseteq> elts \<alpha>"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, Ghost\<^sub>s\<^sub>b A L R W # sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>', sb, \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ts m"
add2: fixes B :: "'b"   and k :: "'b"   and b :: "'b"   and a :: "'b" assumes "B \<equiv> k + b" shows "a + B \<equiv> k + (a + b)"
le_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq a (divide b c) = (if less zero c then less_eq (times a c) b else if less c zero then less_eq b (times a c) else less_eq a zero)"
one_fract_def: shows "1 \<equiv> abs_fract (1::'b, 1::'b)"
CONS_inj: fixes x :: "'x"   and \<AA> :: "'i"   and y :: "'x"   and \<BB> :: "'i" assumes "size x = #\<^sub>V \<AA>"   and "size y = #\<^sub>V \<AA>"   and "#\<^sub>V \<AA> = #\<^sub>V \<BB>" shows "(CONS x \<AA> = CONS y \<BB>) = (x = y \<and> \<AA> = \<BB>)"
step_Normal_elim_cases_10: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and r :: "('e \<times> 'e) set"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Spec r # cs, css, Normal s) \<rightarrow> u" obtains t :: "'e" where "u = (cs, css, Normal t)" and "(s, t) \<in> r"   | "u = (cs, css, Stuck)" and "\<forall>t. (s, t) \<notin> r"
Int_Collect_mono: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x" shows "A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
exec_gpv1_def: shows "exec_gpv1 \<equiv> exec_gpv"
existence_ivl_eq_rev0: fixes y :: "'a" shows "rev.existence_ivl0 y = uminus ` existence_ivl0 y"
"5_6": shows "\<forall>i<N. ps ! i = 0 \<or> ps ! i = Suc 0 \<or> ps ! i = 2 \<or> ps ! i = 3"
bbox_def: fixes R :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'd"   and y :: "'f" shows "[R| f y = \<Sqinter>{f x |x. (x, y) \<in> R}"
equiv_range_permutation: fixes A :: "'a set"   and B :: "'b set" shows "equiv (A \<rightarrow>\<^sub>E B) (range_permutation A B)"
prime_elem_dvd_prod_list: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b"   and A :: "'b list" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times p (monoid_mult.prod_list one times A)" shows "\<exists>a\<in>set A. dvd.dvd times p a"
REC_def: fixes B :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'd"   and x :: "'e" shows "REC B x \<equiv> if trimono B then lfp B x else top"
extends_tgt_out_edges: fixes e :: "'c edge"   and g :: "('c, 'd) rgraph_scheme"   and g' :: "('c, 'd) rgraph_scheme" assumes "extends g e g'" shows "out_edges g' (tgt e) = {}"
continuous_const: fixes F :: "'a filter"   and c :: "'b" shows "continuous F (\<lambda>x. c)"
sum_list_simps_2: fixes x :: "'d"   and xs :: "'d list" shows "sum_list (x # xs) = x + sum_list xs"
locked_def: fixes s :: "v_state" shows "locked s = {v. \<exists>r. locked_in s r v}"
igFreshIGOpSTR_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igFreshIGOpSTR MOD \<equiv> \<forall>ys y delta inp binp. igFreshInp MOD ys y inp \<and> igFreshBinp MOD ys y binp \<longrightarrow> igFresh MOD ys y (igOp MOD delta inp binp)"
ie: shows "te p (i + 1) < te p (i + 2)"
bernoulli_stream_sets: fixes q :: "real"   and p :: "real" assumes "0 < q"   and "q < 1"   and "0 < p"   and "p < 1" shows "sets (bernoulli_stream p) = sets (bernoulli_stream q)"
vec_carrier_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and m :: "nat" shows "(Matrix.vec n f \<in> carrier_vec m) = (n = m)"
assms_2: shows "l \<in> lists (UNIV \<times> gens)"
comp_def: fixes g :: "'a"   and f :: "'a" shows "Fop.Bop.comp g f \<equiv> Cop.comp f g"
down_set_lax_morph: fixes X :: "'b set"   and Y :: "'b set" shows "(\<down> \<circ> Sup) X \<cdot> (\<down> \<circ> Sup) Y \<subseteq> (\<down> \<circ> Sup) (X \<cdot> Y)"
mpt_factorE_3: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "mpt_factor proj M2 T2" shows "mpt M2 T2"
induct: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'b list"   and x3 :: "'a"   and P :: "'a \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> bool" assumes "rtrancl3p r x1 x2 x3"   and "\<And>a. P a [] a"   and "\<And>a bs a' b a''. \<lbrakk>rtrancl3p r a bs a'; P a bs a'; r a' b a''\<rbrakk> \<Longrightarrow> P a (bs @ [b]) a''" shows "P x1 x2 x3"
assms_1: shows "\<bar>b\<bar> + 1 \<le> norm y - a"
Ipc: fixes t :: "real" shows "piecewise_continuous_on 0 t {} I"
msb_shiftr: fixes x :: "int"   and r :: "nat" shows "msb (x >> r) = msb x"
linear_factorsE: fixes p :: "'c poly" assumes "class.alg_closed_field TYPE('c)" obtains c :: "'c" and A :: "'c set" and m :: "'c \<Rightarrow> nat" where "finite A" and "p = Polynomial.smult c (\<Prod>a\<in>A. [:- a, 1::'c:] ^ m a)" and "\<And>a. (m a = 0) = (a \<notin> A)" and "(c = (0::'c)) = (p = 0)" and "\<And>z. (poly p z = (0::'c)) = (c = (0::'c) \<or> z \<in> A)"
simps_2: fixes uu :: "'c option" shows "zip_option uu None = None"
Euler_Rule_Amicable: fixes l :: "nat"   and k :: "nat"   and f :: "nat"   and p :: "nat"   and q :: "nat"   and r :: "nat"   and m :: "nat"   and n :: "nat" assumes "l < k"   and "1 \<le> l"   and "f = 2 ^ l + 1"   and "prime p"   and "prime q"   and "prime r"   and "p = 2 ^ (k - l) * f - 1"   and "q = 2 ^ k * f - 1"   and "r = 2 ^ (2 * k - l) * f\<^sup>2 - 1"   and "m = 2 ^ k * p * q"   and "n = 2 ^ k * r" shows "m Amic n"
ord_spmf_return_spmf2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a spmf"   and y :: "'a" shows "ord_spmf R p (return_spmf y) = (\<forall>x\<in>set_spmf p. R x y)"
inf_bot_left: fixes 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 x :: "'c" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf bot x = bot"
Greater_unbounded: assumes "finite \<X>"   and "Regions.valid_region \<X> k I r"   and "c \<in> \<X>"   and "I c = Regions.intv.Greater (k c)" shows "unbounded c (Regions.region \<X> I r)"
lasso_accepted: fixes G :: "('e, 'f) igb_graph_rec_scheme"   and L :: "'e lasso" assumes "igb_graph G"   and "igb_graph.is_lasso G L" shows "igb_graph.is_acc_run G (run_of_lasso L)"
Int_absorb1: fixes B :: "'e set"   and A :: "'e set" assumes "B \<subseteq> A" shows "A \<inter> B = B"
unit_simps_2: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "trg \<i>[a] = a"
rebase_poly_id: fixes p :: "'p mod_ring poly" shows "##p = p"
keys_sum: fixes b :: "'d \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c"   and I :: "'d set" shows "keys (sum b I) \<subseteq> (\<Union>i\<in>I. keys (b i))"
complete_def: fixes n :: "nat"   and seed :: "natural \<times> natural" shows "Benchmark_Comparison.complete n seed = (let (A, seed') = build n seed; (hits, seed'') = lookup n (A, seed') in Let (test A (\<lambda>x. x \<le> of_nat n)) (Pair hits))"
abs_summable_on_def: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "(f abs_summable_on A) = integrable (count_space A) f"
that: fixes y :: "'a" assumes "y \<in> T"   and "g y = y" shows "thesis"
getParts_nonempty_elems: fixes rs :: "'b simple_rule list" shows "\<forall>w\<in>set (getParts rs). \<not> wordinterval_empty w"
hom_component_inject: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" assumes "t \<in> keys p"   and "hom_component p (deg_pm t) = hom_component p n" shows "deg_pm t = n"
not_in_closure_trivial_limitI: fixes x :: "'c"   and s :: "'c set" assumes "x \<notin> closure s" shows "at x within s = bot"
anga_not_null_lg: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "QCongAACute a"   and "QCong l"   and "a A B C"   and "l A B" shows "\<not> QCongNull l"
bisimilar_is_bisimulation: shows "S_transform.is_bisimulation (\<sim>\<cdot>\<^sub>S)"
Poly_append_replicate_zero: fixes as :: "'b list"   and n :: "nat" shows "Poly (as @ replicate n (0::'b)) = Poly as"
val_axiom4: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 \<le> v x" shows "0 \<le> v (1\<^sub>r \<plusminus> x)"
pair_one: fixes x :: "('c, 'b) boffa_mon" shows "x\<^bsub>1::'c,1::'c\<^esub> = x\<^bsub>1::'c\<^esub>"
client1_def: shows "client1 \<equiv> ''x'' ::= N 0;; WHILE Less (V ''x'') (V ''n'') DO ''x'' ::= Plus (V ''x'') (N 1);; IF BEq (V ''x'') (N 2000000) THEN ''low'' ::= V ''high'' ELSE SKIP"
elims_2: fixes x :: "char list"   and xa :: "char list" assumes "lleq_charlist x xa" obtains "xb__ = []" and "xa = []"   | v :: "char" and va :: "char list" where "xb__ = []" and "xa = v # va"   | x :: "char" and xs :: "char list" and y :: "char" and ys :: "char list" where "xb__ = x # xs" and "xa = y # ys" and "if x = y then lleq_charlist xs ys else x < y"
distinct_15: fixes x21 :: "nat set"   and x22 :: "'e"   and x31 :: "nat"   and x32 :: "'e" shows "groupcast x21 x22 \<noteq> unicast x31 x32"
trss_modify_context: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and cn :: "'a list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)"   and "mon_w fg w \<inter> mon_c fg cn = {}" shows "\<exists>csp. c' = csp + c \<and> mon_c fg csp = {} \<and> ((s, cn), w, s', csp + cn) \<in> trcl (trss fg)"
p_S1: fixes s :: "'s" assumes "s \<in> S1" shows "p s = (\<Squnion>D\<in>K s. integral\<^sup>N (measure_pmf D) p)"
prime_imp_coprime: fixes p :: "'a"   and n :: "'a" assumes "prime p"   and "\<not> p dvd n" shows "coprime p n"
elims: fixes x :: "complex \<times> complex \<times> complex \<times> complex"   and xa :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex \<times> complex \<times> complex" assumes "x +\<^sub>m\<^sub>m xa = y" obtains a1 :: "complex" and b1 :: "complex" and c1 :: "complex" and d1 :: "complex" and a2 :: "complex" and b2 :: "complex" and c2 :: "complex" and d2 :: "complex" where "x = (a1, b1, c1, d1)" and "xa = (a2, b2, c2, d2)" and "y = (a1 + a2, b1 + b2, c1 + c2, d1 + d2)"