hcju/isabelleps huggingface dataset:
enc_unique: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and i :: "nat" assumes "0 < length w \<and> (\<forall>a\<in>set w. a \<in> set \<Sigma>) \<and> (\<forall>x\<in>set I. case x of Inl p \<Rightarrow> p < length w | Inr P \<Rightarrow> \<forall>p\<in>P. p < length w)"   and "i < length I"   and "\<exists>p. I ! i = Inl p" shows "\<exists>!p. p < length (enc (w, I)) \<and> snd (enc (w, I) ! p) ! i"
fresh_1: fixes a :: "name"   and x3 :: "subject"   and x1 :: "name"   and x2 :: "pi" shows "a \<sharp> x3\<guillemotleft>x1\<guillemotright> \<prec> x2 = (a \<sharp> x3 \<and> a \<sharp> [x1].x2)"
loop_has_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "is_path \<pi>"   and "0 < i"   and "\<pi> i = \<pi> 0"   and "\<pi> 0 \<noteq> local.return" shows "\<exists>k<i. i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
x: shows "x \<in> {(last vs2, ram1), (ram1, ram2), (ram2, hd vs2)} \<union> Edges vs2"
M_MonotModel_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<Phi> :: "('a, 'b) lit list set" shows "M_MonotModel wtFsym wtPsym arOf resOf parOf intT intF intP \<Phi> \<equiv> MonotModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
\<L>\<^sub>P_split: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "a @ b \<in> CFG.\<L>\<^sub>P \<NN> \<TT> \<RR> \<SS>" shows "a \<in> CFG.\<L>\<^sub>P \<NN> \<TT> \<RR> \<SS>"
Init_totalize: fixes F :: "'b program" shows "Init (totalize F) = Init F"
simps_2: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and E :: "'b equation" shows "validate_ground_lit I (Neg E) = (\<not> validate_ground_eq I E)"
asc_sufficiency: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and m :: "nat"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM"   and V :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set"   and i :: "nat" assumes "OFSM M1"   and "OFSM M2"   and "asc_fault_domain M2 M1 m"   and "test_tools M2 M1 FAIL PM V \<Omega>"   and "final_iteration M2 M1 \<Omega> V m i" shows "M1 \<preceq>\<lbrakk>(TS M2 M1 \<Omega> V m i).\<Omega>\<rbrakk> M2 \<longrightarrow> M1 \<preceq> M2"
U_def: shows "U = \<Union> (S ` {1..m})"
explode_def: fixes z :: "ZF" shows "explode z \<equiv> {x. Elem x z}"
finter_finsert_right_ifffempty: fixes a :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "a |\<notin>| A" shows "A |\<inter>| finsert a B = A |\<inter>| B"
weak_sup_of_singleton: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<Squnion>{x} .= x"
le_imp_less_or_eq: fixes x :: "'b list"   and y :: "'b list" assumes "x \<le>s y" shows "x <s y \<or> x = y"
deg_pm_minus_le: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and s :: "'b \<Rightarrow>\<^sub>0 nat" shows "deg_pm (t - s) \<le> deg_pm t"
rSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "\<Psi> \<rhd> P \<leadsto>\<^sub>w<{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<union> weakenBisim \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"
finite_vimage_Suc_iff: fixes F :: "nat set" shows "finite (Suc -` F) = finite F"
foldr_option_def: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c option"   and xs :: "'d list"   and e :: "'c" shows "foldr_option f xs e = foldr (\<lambda>a b. Option.bind b (f a)) xs (Some e)"
list_orderE: fixes ops :: "('c \<times> 'c option) list"   and x :: "'c"   and y :: "'c" assumes "Insert_Spec.list_order ops x y" shows "\<exists>xs ys zs. interp_ins ops = xs @ [x] @ ys @ [y] @ zs"
that: fixes xr :: "'a list"   and j :: "nat" assumes "xr \<noteq> xs"   and "prefix xr xs"   and "Suc j \<le> i"   and "vs @ xr \<in> RM M2 M1 \<Omega> V m (Suc j)" shows "thesis"
signed_take_bit_eq_if_negative: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "bit a n" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = or (take_bit n a) (not (mask n))"
bhv2cnf_mono: fixes n :: "nat"   and n' :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'" shows "dynamic_component.bhv2cnf tCMP active c t n \<le> dynamic_component.bhv2cnf tCMP active c t n'"
simps_5: fixes pi :: "(name \<times> name) list"   and x :: "'d" shows "rev pi \<bullet> pi \<bullet> x = x"
higher_deriv_uminus: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and z :: "complex"   and n :: "nat" assumes "f holomorphic_on S"   and "open S"   and "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. - f w) z = - (deriv ^^ n) f z"
Cd_path_independent: fixes d :: "'d" shows "path_independent (Cd d)"
mapCommute: fixes S :: "'b sequent"   and R :: "'b sequent"   and c :: "'b sequent list" shows "map (extend S) (map (extend R) c) = map (extend R) (map (extend S) c)"
fixp_above_def: fixes a :: "'a" shows "fixp_above a = (if a \<in> Field leq then lub (iterates_above a) else a)"
lcm_nat_def: fixes x :: "nat"   and y :: "nat" shows "lcm x y = x * y div gcd x y"
Cons_eq_map_D: fixes x :: "'a"   and xs :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and ys :: "'b list" assumes "x # xs = map f ys" shows "\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs"
Inf_apply: fixes A :: "('b \<Rightarrow> 'a) set"   and x :: "'b" shows "(\<Sqinter> A) x = (\<Sqinter>f\<in>A. f x)"
stables_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list" shows "stables r step \<tau>s = (\<forall>p<length \<tau>s. stable r step \<tau>s p)"
intros_3: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<in> i_set_ind"
Auth_A_to_B_lemma: fixes evs :: "event list"   and K :: "nat"   and NB :: "nat"   and B :: "agent"   and A :: "agent" assumes "evs \<in> yahalom"   and "Key K \<notin> analz (knows Spy evs)"   and "K \<in> symKeys"   and "Crypt K (Nonce NB) \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, Nonce NB\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad" shows "\<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs"
r_lchild_le_r: fixes t :: "'b tree" assumes "well_shaped t" shows "r (lchild t) \<le> r t"
Rep_row_of_matrix: fixes A :: "'b matrix"   and r :: "nat"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (row_of_matrix A r) j i = (if j = 0 then Rep_matrix A r i else (0::'b))"
quotient_imp_surjective_map: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e" assumes "quotient_map X X' f" shows "f ` topspace X = topspace X'"
add_stalk_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and x :: "'d" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.add_stalk \<FF> \<rho> add_str I \<equiv> direct_lim.add_rel \<FF> \<rho> add_str I"
ideal_generated_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "ring.ideal_generated plus zero uminus times {one} = UNIV"
runit_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and tensor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unity :: "'a"   and lunit :: "'a \<Rightarrow> 'a"   and runit :: "'a \<Rightarrow> 'a"   and assoc :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "elementary_monoidal_category C tensor unity lunit runit assoc"   and "partial_magma.ide C a" shows "tensor (runit a) unity = C (tensor a (elementary_monoidal_category.\<iota> unity lunit)) (assoc a unity unity)"
connected_component_mono: fixes S :: "'b set"   and T :: "'b set"   and x :: "'b" assumes "S \<subseteq> T" shows "connected_component_set S x \<subseteq> connected_component_set T x"
top_on_bot: fixes c :: "com"   and X :: "char list set" shows "top_on\<^sub>c (bot c) X"
uwellformed_def: fixes G :: "nat set \<times> nat set set" shows "uwellformed G \<equiv> \<forall>e\<in>uedges G. card e = 2 \<and> (\<forall>u\<in>e. u \<in> uverts G)"
Inter_commute: fixes P :: "'a process"   and Q :: "'a process" shows "(P|||Q) = (Q|||P)"
locally_finite_on_closure: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set" assumes "locally_finite_on X I U" shows "locally_finite_on X I (\<lambda>x. closure (U x))"
simps_5: fixes Inv :: "'a"   and b :: "bexp"   and c :: "'a acom"   and P :: "'a" shows "post ({Inv} WHILE b DO c {P}) = P"
l: shows "approx_tse_form' prec t f s (fst (split_float_interval ivl)) cmp"
loxodromic_imp_not_elliptic_parabolic_2: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "\<not> parabolic_isometry f"
m\<tau>move: shows "m\<tau>move s (t, ta) s'"
j_le_\<ff>_le': fixes j :: "nat"   and s :: "'a list" assumes "0 < j"   and "j \<le> length s" shows "\<ff> s j - 1 < j"
card_ip1_to_j_is_1_in_normal_events: fixes t :: "('a, 'b, 'c) event list"   and j :: "nat"   and i :: "nat"   and n :: "nat" assumes "prerecording_event t j"   and "postrecording_event t i"   and "i < j"   and "j < length t"   and "count_violations t = Suc n"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "count_violations t = Suc n"   and "trace init t final" shows "card {k \<in> {i + 1..<j + 1}. prerecording_event t k} = 1"
upto_code: fixes i :: "int"   and j :: "int" shows "[i..j] = upto_aux i j []"
paval'_aval: fixes a :: "aexp"   and ps :: "char list \<Rightarrow> int option"   and v :: "int"   and s :: "char list \<Rightarrow> int" assumes "paval' a ps = Some v" shows "aval a (emb ps s) = v"
trivial_phi: fixes g :: "'g"   and v :: "'val" assumes "notriv.trivial g v" shows "notriv.phi g v \<noteq> None"
derivation_shift_0_shift: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and left1 :: "nat"   and left2 :: "nat"   and right2 :: "nat" shows "derivation_shift (derivation_shift D left1 0) left2 right2 = derivation_shift D (left1 + left2) right2"
chain_on_reflclp: fixes C :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "chain_on P\<^sup>=\<^sup>= A C = chain_on P A C"
assms_1: shows "insert_ops (pre @ [(oid, Some ref)] @ suf)"
eq_0_iff: fixes xs :: "nat list"   and n :: "nat" shows "(xs = zeroes n) = (length xs = n \<and> (\<forall>x\<in>set xs. x = 0))"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "compatible_ordering A less_eq less"   and "quasi_ordered_set A less_eq" shows "quasi_ordering A less_eq less"
fail_nondet_def: fixes return :: "'f \<Rightarrow> 'e"   and empty :: "'f" shows "nondetM_base.fail_nondet return empty = NondetT (return empty)"
simps: fixes x :: "'b"   and y :: "'b"   and n :: "nat" shows "dichotomous_Lazard x y n = (if n \<le> 1 then if n = 1 then x else (1::'b) else let (d, r) = divmod_nat n 2; rec = dichotomous_Lazard x y d; recsq = rec * rec div y in if r = 0 then recsq else recsq * x div y)"
assms: shows "s \<oplus> lt p \<prec>\<^sub>t v"
lambda_system_algebra: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive (lambda_system \<Omega> M f) f" shows "algebra \<Omega> (lambda_system \<Omega> (lambda_system \<Omega> M f) f)"
subringE_7: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme"   and h1 :: "'c"   and h2 :: "'c" assumes "subring H R"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> H"
nonempty_card_pos1: fixes A :: "'h set" assumes "finite A"   and "A \<noteq> {}" shows "Suc 0 \<le> card A"
add_cancl: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "D z x"   and "D z y"   and "times z x = times z y" shows "x = y"
degrp: shows "degree rp \<le> length rxs_ys - 1"
simps_2: fixes r :: "'b memref"   and rs :: "'b memref list"   and m :: "nat \<Rightarrow> nat" shows "flush (r # rs) m = (case r of Write\<^sub>s\<^sub>b volatile a x v xa xb xc xd \<Rightarrow> flush rs (m(a := v)) | _ \<Rightarrow> flush rs m)"
store_sub1_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and rd :: "word5"   and s_val :: "word1" shows "store_sub1 instr rd s_val \<equiv> do curr_win \<leftarrow> get_curr_win (); address \<leftarrow> gets (get_addr (snd instr)); asi \<leftarrow> gets (\<lambda>s. st_asi instr s_val); if (fst instr = load_store_type STH \<or> fst instr = load_store_type STHA) \<and> ucast address \<noteq> 0 then do raise_trap mem_address_not_aligned; return () od else if fst instr \<in> {load_store_type ST, load_store_type STA} \<and> ucast address \<noteq> 0 then do raise_trap mem_address_not_aligned; return () od else if fst instr \<in> {load_store_type STD, load_store_type STDA} \<and> ucast address \<noteq> 0 then do raise_trap mem_address_not_aligned; return () od else store_sub2 instr curr_win rd asi address od"
fref_to_Down_curry_left: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c nres"   and g :: "'d \<Rightarrow> 'e nres"   and P :: "'d \<Rightarrow> bool"   and A :: "(('a \<times> 'b) \<times> 'd) set"   and B :: "('c \<times> 'e) set"   and x' :: "'d"   and a :: "'a"   and b :: "'b" assumes "(uncurry f, g) \<in> [P]\<^sub>f A \<rightarrow> \<langle>B\<rangle>nres_rel"   and "P x'"   and "((a, b), x') \<in> A" shows "f a b \<le> \<Down> B (g x')"
fps_distr: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps"   and h :: "('c, 'd) fps" shows "(f + g) \<cdot> h = f \<cdot> h + g \<cdot> h"
is_singleton_the_elem: fixes A :: "'a set" shows "is_singleton A = (A = {the_elem A})"
simps_4: fixes \<iota> :: "nat"   and b :: "'a"   and c1 :: "('a, 'b) MWLsCom"   and c2 :: "('a, 'b) MWLsCom" shows "PPc (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi) = [\<iota>] @ PPc c1 @ PPc c2"
finite_range: fixes p :: "'b \<Rightarrow>\<^sub>0 'a" shows "finite (Poly_Mapping.range p)"
basis_reduction_swap_bound: fixes i :: "nat"   and fs :: "int vec list"   and upw' :: "bool"   and i' :: "nat"   and fs' :: "int vec list" assumes "LLL_bound_invariant True False i fs"   and "basis_reduction_swap i fs = (upw', i', fs')"   and "\<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 < \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) (i - 1)\<parallel>\<^sup>2"   and "i < m"   and "i \<noteq> 0" shows "LLL_bound_invariant True upw' i' fs'"
dsg1: fixes x :: "'a" shows "ad (ad x) \<cdot> x = x"
attach_callee_parallel_intercept: fixes callee1 :: "'a \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'a, 'h, 'j) gpv"   and callee2 :: "'b \<Rightarrow> 'e \<Rightarrow> ('g \<times> 'b, 'i, 'k) gpv"   and oracle1 :: "'c \<Rightarrow> 'h \<Rightarrow> ('j \<times> 'c) spmf"   and oracle2 :: "'c \<Rightarrow> 'i \<Rightarrow> ('k \<times> 'c) spmf" shows "attach_callee (callee1 \<ddagger>\<^sub>I callee2) (oracle1 \<oplus>\<^sub>O oracle2) = (rprodl ---> id ---> map_spmf (map_prod id lprodr)) (lift_state_oracle extend_state_oracle (attach_callee callee1 oracle1) \<oplus>\<^sub>O \<dagger>(attach_callee callee2 oracle2))"
opp_chamber_reverse: fixes C :: "'a set" assumes "C \<in> folding.\<C> - f \<turnstile> folding.\<C>" shows "(THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = f ` C) = C"
ncc_precond_def: shows "ncc_precond TYPE('b) = (\<forall>(Xi, X)\<in>appr_rel. ncc X)"
positive_add: fixes A :: "complex mat"   and B :: "complex mat"   and n :: "nat" assumes "positive A"   and "positive B"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "positive (A + B)"
assms_5: shows "Lfilled j lholed (ves @ [$Return]) LI"
li_le_dim_2: fixes A :: "'a vec set" assumes "fin_dim"   and "A \<subseteq> carrier_vec n"   and "lin_indpt A" shows "card A \<le> local.dim"
1: fixes l :: "int" assumes "l < int p" shows "l < int p * int q"
rsp: shows "eq_onp (\<lambda>x. x \<in> {(a, b). a \<le> b}) (1::'b, 1::'b) (1::'b, 1::'b)"
map_option_cong: fixes x :: "'a option"   and y :: "'a option"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = y"   and "\<And>a. y = pure a \<Longrightarrow> f a = g a" shows "map_option f x = map_option g y"
fun_merge_eq_nth: fixes i :: "nat"   and as :: "'c set list"   and a :: "'c"   and fs :: "('c \<Rightarrow> 'd) list" assumes "i < length as"   and "a \<in> as ! i"   and "\<And>i j a. \<lbrakk>i < length as; j < length as; a \<in> as ! i; a \<in> as ! j\<rbrakk> \<Longrightarrow> (fs ! i) a = (fs ! j) a" shows "fun_merge fs as a = (fs ! i) a"
tm_wf_t_wcode_adjust: shows "tm_wf0 t_wcode_adjust"
negate_true_guard: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "(gval (negate G) s = true) = (gval (fold gAnd G (Bc True)) s = false)"
lookup_operator_works_3: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and name' :: "char list"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat" assumes "abs_ast_prob problem"   and "is_standard_operator' (name, pres, effs, layer)"   and "(name, pres, effs, layer) \<in> set (ast_problem.ast\<delta> problem)"   and "ast_problem.lookup_action problem (abs_ast_operator (name, pres, effs, layer)) = Some (name', pres', effs', layer')" shows "effs = effs'"
emb_def: fixes f :: "'e \<Rightarrow> 'a"   and D :: "'e set"   and x :: "'e" shows "emb f D x = (if x \<in> D then f x else \<zero>\<^bsub>K\<^esub>)"
pendulum_inv: fixes r :: "real"   and G :: "(real, 2) vec \<Rightarrow> bool" shows "{s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2} \<subseteq> fb\<^sub>\<F> x\<acute>=\<lambda>t s. fball (- s $ 1) s & G on \<lambda>s. {t. 0 \<le> t} UNIV @ 0 {s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2}"
pd: shows "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"
power_decreasing_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero b"   and "less b one" shows "less_eq (power.power one times b m) (power.power one times b n) = (n \<le> m)"
concI_if_Nil2: fixes B :: "'a list set"   and xs :: "'a list"   and A :: "'a list set" assumes "[] \<in> B"   and "xs \<in> A" shows "xs \<in> A @@ B"
prems_3: fixes r :: "nat" assumes "r \<in> set rs" shows "0 < r"
simps_13: fixes va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt"   and w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b"   and ve :: "('a, 'b) rbt"   and vf :: "'a"   and vg :: "'b"   and vh :: "('a, 'b) rbt" shows "balance (Branch B va vb vc vd) w x (Branch R (Branch R b s t c) y z (Branch B ve vf vg vh)) = Branch R (Branch B (Branch B va vb vc vd) w x b) s t (Branch B c y z (Branch B ve vf vg vh))"
row_def: fixes A :: "'d mat"   and i :: "nat" shows "Matrix.row A i = Matrix.vec (dim_col A) (\<lambda>j. A $$ (i, j))"
discriminant_iff: fixes a :: "real"   and x :: "real"   and b :: "real"   and c :: "real" assumes "a \<noteq> 0" shows "(a * x\<^sup>2 + b * x + c = 0) = (0 \<le> discrim a b c \<and> (x = (- b + sqrt (discrim a b c)) / (2 * a) \<or> x = (- b - sqrt (discrim a b c)) / (2 * a)))"
a: shows "\<lbrace>p\<rbrace>if b then x else y fi\<lbrace>q\<rbrace>"
unit_disc_cmod_square_lt_1: fixes z :: "complex_homo" assumes "z \<in> unit_disc" shows "(cmod (to_complex z))\<^sup>2 < 1"
assms_1: shows "diff_invariant I\<^sub>1 f U S t\<^sub>0 G"
compactD: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a" assumes "compact A"   and "range f \<subseteq> A" shows "\<exists>h l. strict_mono h \<and> (f \<circ> h) \<longlonglongrightarrow> l"
assms_1: shows "foreach f S a\<^sub>1 a\<^sub>2"
strict_trans1: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "a \<sqsubseteq> b"   and "b \<sqsubset> c" shows "a \<sqsubset> c"
countable_datatype: fixes Rep :: "'a \<Rightarrow> ('b, unit) Old_Datatype.node set"   and Abs :: "('b, unit) Old_Datatype.node set \<Rightarrow> 'a"   and rep_set :: "('b, unit) Old_Datatype.node set \<Rightarrow> bool" assumes "type_definition Rep Abs (Collect rep_set)"   and "\<And>x. rep_set x \<Longrightarrow> Countable.finite_item x" shows "OFCLASS('a, countable_class)"
count_in_diffI: fixes N :: "'b multiset"   and x :: "'b"   and M :: "'b multiset" assumes "\<And>n. count N x = n + count M x \<Longrightarrow> False" shows "x \<in># M - N"
poly_lift_hom_X_var: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e" assumes "cring S"   and "\<phi> \<in> ring_hom R S" shows "poly_lift_hom R S \<phi> (monom (UP R) \<one> 1) = monom (UP S) \<one>\<^bsub>S\<^esub> 1"
abs_eq: fixes g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "pure g \<diamondop> pure x = pure (g x)"
from_nat_def: shows "mod_type_class.from_nat = Abs' \<circ> int"
sigma_finite_measure_axioms: shows "sigma_finite_measure (density T Py)"
cod_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ" shows "cod (mkIde A) = mkIde A"
simps_1: fixes x2 :: "'d"   and y2 :: "'d" shows "(Some x2 = Some y2) = (x2 = y2)"
that: assumes "tr r = tr r'"   and "wait r = wait r'"   and "ref r = ref r'"   and "alpha_rp.more r = alpha_rp.more r'" shows "thesis"
sound_inference_system_axioms: shows "sound_inference_system \<Gamma>"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "le b a" shows "lt b a"