hcju/isabelleps huggingface dataset:
ps_chain_to_int_chain_def: fixes G :: "diagram"   and \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain" shows "ps_chain_to_int_chain G \<Pi> \<equiv> chainmap (ps_to_int G) (case_sum (Inl \<circ> G^\<Lambda>) (Inr \<circ> snd3)) \<Pi>"
mono_neutral_cong_left: fixes T :: "'c set"   and S :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = (0::'d)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "sum g S = sum h T"
cring_fieldI2: assumes "(0::'b) \<noteq> (1::'b)"   and "\<And>a. \<lbrakk>a \<in> UNIV; a \<noteq> (0::'b)\<rbrakk> \<Longrightarrow> \<exists>b\<in>UNIV. a * b = (1::'b)" shows "field class_ring"
tendsto_power: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter"   and n :: "nat" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
linear_eq_on_span: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and x :: "'a" assumes "linear f"   and "linear g"   and "\<And>x. x \<in> B \<Longrightarrow> f x = g x"   and "x \<in> span B" shows "f x = g x"
exhaust: fixes y :: "msg" obtains x11 :: "nat" and x12 :: "nat" where "y = Pkt x11 x12"   | x21 :: "nat" and x22 :: "nat" where "y = Newpkt x21 x22"
rep2_snoc_right2: fixes x :: "'r"   and vs :: "[:'r:]"   and pat :: "[:'r:]"   and us :: "[:'r:]" assumes "prefix\<cdot>[:x:]\<cdot>vs = TT" shows "rep2\<cdot>pat\<cdot>(us :@ [:x:], stail\<cdot>vs) = right2\<cdot>pat\<cdot>(us, vs)"
cts_wp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "bd_cts (wp (Apply f))"
bounded_def: shows "bounded \<equiv> \<lambda>t. 0 \<le> t \<and> t \<le> ego.t_stop \<and> t \<le> other.t_stop"
mon_w_unconc: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and wa :: "('h, 'k) label list"   and wb :: "('h, 'k) label list" shows "mon_w fg (wa @ wb) = mon_w fg wa \<union> mon_w fg wb"
fimageD: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A" shows "x \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> (r \<restriction>\<^sup>l\<^sub>\<bullet> A)"
path_nodes: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>" shows "\<pi> k \<in> IFC_def.nodes prob"
plus_dim2: fixes A :: "'m tensor"   and B :: "'m tensor" assumes "dims A = dims B" shows "dims (A + B) = dims B"
xfB: fixes x :: "'a"   and j :: "nat" assumes "x \<in> B j" shows "1 - e / real n < xf j x"
P3: shows "Coplanar A B C X \<and> Bet P X S"
\<Gamma>'': shows "\<Gamma> = set_mset \<Gamma>''"
iso_fInf: fixes F :: "('b \<Rightarrow> 'c) set" assumes "\<forall>f\<in>F. mono f" shows "mono (\<Sqinter> F)"
id_is_homeomorphism: shows "homeomorphism S is_open S is_open (identity S)"
cases: fixes x :: "'a list" obtains "x = \<epsilon>"   | a :: "'a" and w :: "'a list" where "x = a # w"
eq_implies_ltl_equiv: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "ltl_equivalence eq"   and "eq \<phi> \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<psi>"
abs_eq: fixes xb :: "'d"   and xa :: "'c"   and x :: "('a, 'b, 'c, 'd) Tree" shows "Act\<^sub>\<alpha> xb xa (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tAct xb xa x)"
normalize_manual_orderPr_def: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" shows "normalize_manual_orderPr p l = removeAllDuplicates (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)))"
f_Exec_State_Idle_replicate_NoMsg_state: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState c)" shows "localState (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = localState c"
Units_r_cancel: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<in> Units G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes> z = y \<otimes> z) = (x = y)"
token_squats_def: fixes x :: "nat" shows "token_squats x = (\<forall>n. \<not> sink (token_run x n))"
lmap_lprefix: fixes xs :: "'a llist"   and ys :: "'a llist"   and f :: "'a \<Rightarrow> 'b" assumes "lprefix xs ys" shows "lprefix (lmap f xs) (lmap f ys)"
ide_src: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.ide (Span.src \<mu>)"
tendsto_eexp: fixes f :: "'a \<Rightarrow> ereal"   and x :: "ereal"   and F :: "'a filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. eexp (f x)) \<longlongrightarrow> eexp x) F"
preserves_lunit_2: fixes f :: "'a" assumes "C.ide f" shows "F \<l>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi> (trg\<^sub>C f, f) \<cdot>\<^sub>D (unit (trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[F f]"
bisimSubstCasePushRes: fixes x :: "name"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<Psi> :: "'b" assumes "x \<sharp> map fst Cs" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(Cases Cs) \<sim>\<^sub>s Cases map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs"
wordinterval_Union_def: fixes ws :: "'a wordinterval list" shows "wordinterval_Union ws = wordinterval_compress (foldr wordinterval_union ws Empty_WordInterval)"
fresh_quote_all_mem: fixes A :: "fm"   and p :: "perm"   and V :: "name set"   and i :: "atom" assumes "A \<in> quote_all p V"   and "finite V"   and "i \<sharp> V"   and "i \<sharp> p \<bullet> V" shows "i \<sharp> A"
eval_init_seq': fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "(\<exists>C Cs b e\<^sub>i. e = INIT C (Cs,b) \<leftarrow> e\<^sub>i) \<or> (\<exists>C e\<^sub>0 Cs e\<^sub>i. e = RI (C,e\<^sub>0) ; Cs \<leftarrow> e\<^sub>i)" shows "(\<exists>C Cs b e\<^sub>i. e = INIT C (Cs,b) \<leftarrow> e\<^sub>i \<and> P \<turnstile> \<langle>(INIT C (Cs,b) \<leftarrow> unit);; e\<^sub>i,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>) \<or> (\<exists>C e\<^sub>0 Cs e\<^sub>i. e = RI (C,e\<^sub>0) ; Cs \<leftarrow> e\<^sub>i \<and> P \<turnstile> \<langle>(RI (C,e\<^sub>0) ; Cs \<leftarrow> unit);; e\<^sub>i,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>)"
simps_7: fixes v :: "'a" shows "(dRETURN v \<le> dSUCCEEDi) = False"
connected_component_nonoverlap: fixes S :: "'c set"   and a :: "'c"   and b :: "'c" shows "(connected_component_set S a \<inter> connected_component_set S b = {}) = (a \<notin> S \<or> b \<notin> S \<or> connected_component_set S a \<noteq> connected_component_set S b)"
antisymp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "antisymp r = (\<forall>x y. r x y \<longrightarrow> r y x \<longrightarrow> x = y)"
nonzero_def: fixes R :: "('a, 'b) ring_scheme" shows "nonzero R = {a \<in> carrier R. a \<noteq> \<zero>\<^bsub>R\<^esub>}"
hyps_free_def: fixes pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'a \<times> 'b) list" shows "hyps_free pth = (\<forall>v\<^sub>1 p\<^sub>1 v\<^sub>2 p\<^sub>2. ((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> set pth \<longrightarrow> hyps (nodeOf v\<^sub>1) p\<^sub>1 = None)"
assert_bound_sat: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom"   and s' :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "assert_bound a s = s'"   and "\<not> \<U> s'" shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s' \<and> \<diamond> s'"
f: fixes x :: "'a" assumes "x \<in> affine hull S" shows "x - \<xi> = f x *\<^sub>R z"
part1_def: shows "part1 = fst \<circ> part12"
subgame_path_vmc_path: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and V' :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "V' \<subseteq> V\<^bsub>G\<^esub>"   and "lset P \<subseteq> V'" shows "vmc_path (ParityGame.subgame G V') P v0 p \<sigma>"
r_prefixes_aux: fixes j :: "nat"   and v :: "nat"   and i :: "nat" shows "eval r_prefixes_aux [j, v, i] \<down>= e_snoc v (if eval r_s [e_snoc v 0] = eval r_s [v] then 1 else 0)"
sectionE: fixes f :: "'a \<times> 'a" assumes "VV.section f" obtains g :: "'a \<times> 'a" where "VV.ide (VV.comp g f)"
flowD_finite_IN: fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and x :: "'v vertex" assumes "flow \<Psi> f"   and "x \<noteq> sink \<Psi>" shows "d_IN f x \<noteq> \<top>"
leq_refl: fixes x :: "'a" assumes "x \<in> Field leq" shows "(x, x) \<in> leq"
capacity_outside: fixes e :: "'v option \<times> 'v option" assumes "e \<notin> \<^bold>E\<^bsub>\<Psi>\<^esub>" shows "capacity \<Psi> e = 0"
lin: fixes k1 :: "'a"   and K :: "'a set"   and x :: "'a"   and k2 :: "'a" assumes "k1 \<in> simple_extension K x"   and "k2 \<in> K" shows "k1 \<otimes> x \<oplus> k2 \<in> simple_extension K x"
0: shows "Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b))) = Outside' {seller} a \<and> a \<in> maximalStrictAllocations N (set \<Omega>) b"
pcr_spmf_gpv_def: fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "pcr_spmf_gpv A = cr_spmf_gpv OO rel_gpv A (=)"
inv_free_Abelian_group: fixes x :: "'b \<Rightarrow>\<^sub>0 int"   and S :: "'b set" assumes "Poly_Mapping.keys x \<subseteq> S" shows "inv\<^bsub>free_Abelian_group S\<^esub> x = - x"
hyps_2: shows "(w, x) \<in> E \<union> U"
r_distr: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y"
case_eq_if: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and hd :: "('e, 'd) hd" shows "(case hd of Var x \<Rightarrow> f1 x | Sym x \<Rightarrow> f2 x) = (if is_Var hd then f1 (var hd) else f2 (hd.sym hd))"
cut_bits_def: fixes n :: "nat"   and m :: "nat" shows "cut_bits n m \<equiv> m mod 2 ^ n"
Op_asInp_asTerm_qOp: fixes qinp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qbinp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs option"   and delta :: "'f" assumes "qGoodInp qinp"   and "qGoodBinp qbinp" shows "Op delta (asInp qinp) (asBinp qbinp) = asTerm (qOp delta qinp qbinp)"
access_fld_simp: fixes f :: "'b \<times> 'a decl_scheme" shows "access (fld f) = accmodi f"
some_g: shows "\<G>_I \<iota> \<noteq> None"
obj_self_composable_1: fixes a :: "'c" assumes "C.obj a" shows "a \<star>\<^sub>C a \<noteq> C.null"
len: shows "e_length (e_append_zeros b k) = Suc n"
rel_flip: fixes R :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b stream"   and b :: "'c stream" shows "stream_all2 R\<inverse>\<inverse> a b = stream_all2 R b a"
assms_1: shows "fmrestrict_set vs s = fmrestrict_set vs (state_succ s a)"
list_ex_rec_eq: fixes l :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>set l. P x) = list_ex_rec P l"
seg_3: fixes z :: "'a" assumes "z \<in> {a--b}" shows "0 < f z \<bullet> rot (a - b)"
non_empty_cycle_root_equivalences_4: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "non_empty_cycle_root r x"   and "point q" shows "(r \<le> x\<^sup>\<star> ; q) = (q \<le> x ; 1)"
\<beta>_simps_5: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b"   and u :: "'b"   and w :: "'b"   and \<theta> :: "'b"   and w' :: "'b"   and \<theta>' :: "'b"   and \<beta> :: "'b" assumes "uw\<theta>w'\<theta>'\<beta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>' \<beta>" shows "partial_magma.cod V \<beta> = H g w'"
simps_1: shows "list_conj_Uni [] = TrueFUni"
set_simps_3: fixes A :: "'e set"   and B :: "'e set" shows "(A - B \<noteq> A) = (A \<inter> B \<noteq> {})"
interval_integral_Ioi: fixes a :: "ereal"   and f :: "real \<Rightarrow> 'a" assumes "\<bar>a\<bar> < \<infinity>" shows "interval_lebesgue_integral lborel a \<infinity> f = set_lebesgue_integral lborel {real_of_ereal a<..} f"
inject_1: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form"   and y11 :: "floatarith"   and y12 :: "floatarith"   and y13 :: "floatarith"   and y14 :: "form" shows "(Bound x11 x12 x13 x14 = Bound y11 y12 y13 y14) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13 \<and> x14 = y14)"
compactness: fixes S :: "'b formula set" shows "sat S = fin_sat S"
assms_3: shows "qs ! i \<in> set (fst s)"
swap_msgs_Recv_Trans: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isTrans ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
af_implies_af\<^sub>G: fixes S :: "'a ltl set"   and \<phi> :: "'a ltl"   and w :: "'a set list" assumes "S \<Turnstile>\<^sub>P af \<phi> w" shows "S \<Turnstile>\<^sub>P af\<^sub>G \<phi> w"
dom_subid_aux2'': fixes x :: "'a"   and y :: "'a" shows "x \<cdot> ad (ad y) \<le> x"
sum_part_character: fixes w :: "i"   and x :: "i"   and y :: "i" assumes "CEM" shows "P w (x \<oplus> y) = (\<forall>v. O v w \<longrightarrow> O v x \<or> O v y)"
update_convs_3: fixes vals' :: "(Location \<Rightarrow> Value) \<Rightarrow> Location \<Rightarrow> Value"   and newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a" shows "vals_update vals' \<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = \<lparr>newOID = newOID, newAID = newAID, vals = vals' vals, \<dots> = more\<rparr>"
abs_venv_map_of_rev: fixes l :: "(((nat \<times> char list) \<times> contour) \<times> d) list" shows "|map_of (rev l)| \<lessapprox> \<Union>.map (\<lambda>(v, k). |[v \<mapsto> k]|) l"
conjugate_zero_iff_vec: fixes v :: "'b vec"   and n :: "nat" shows "(conjugate v = 0\<^sub>v n) = (v = 0\<^sub>v n)"
hom_compose: fixes h :: "('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme"   and i :: "'c \<Rightarrow> 'e"   and I :: "('e, 'f) monoid_scheme" assumes "h \<in> hom (free_Abelian_group S) H"   and "i \<in> hom H I" shows "compose (carrier (free_Abelian_group S)) i h \<in> hom (free_Abelian_group S) I"
injectivity_2: fixes P :: "nat set set"   and j :: "nat"   and P' :: "nat set set"   and j' :: "nat"   and B' :: "nat set"   and k :: "nat"   and B'' :: "nat set"   and k' :: "nat"   and Q :: "nat set set"   and Q' :: "nat set set"   and f :: "nat \<Rightarrow> nat set"   and g :: "nat \<Rightarrow> nat set"   and P'' :: "nat set set"   and P''' :: "nat set set" assumes "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "Q = Q'"
card_lists_distinct_length_eq: fixes A :: "'a set"   and n :: "nat" assumes "finite A" shows "card {xs. length xs = n \<and> distinct xs \<and> set xs \<subseteq> A} = ffact n (card A)"
fail_t_inclusion: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n"   and "\<not> sink (token_run x n)"   and "sink (token_run x (Suc n))"   and "token_run x (Suc n) \<notin> F" shows "n \<in> fail_t"
groupWIs_not_empty_elems: fixes V :: "'a wordinterval list"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and w :: "'a wordinterval" assumes "V \<in> set (groupWIs c rs)"   and "w \<in> set V" shows "\<not> wordinterval_empty w"
hyps: shows "is_quantum_predicate (wlp_while_n (M 0) (M 1) WS k P)"
perm: shows "permute_list p [0..<n] = sort_key f [0..<n]"
unfold_while: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and b :: "char list exp"   and c :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" shows "P \<turnstile> \<langle>while (b) c,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> = P \<turnstile> \<langle>if (b) (c;; while (b) c) else unit,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
wf_operator_def: shows "wf_operator \<equiv> \<lambda>(name, pres, effs, cost). wf_partial_state pres \<and> distinct (map (\<lambda>(uu_, v, uu_, uu_). v) effs) \<and> (\<forall>(epres, x, vp, v)\<in>set effs. wf_partial_state epres \<and> x < numVars \<and> v < numVals x \<and> (case vp of None \<Rightarrow> True | Some v \<Rightarrow> v < numVals x))"
sep_conj_empty': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj (sep_algebra.sep_empty zero) P = P"
gcd_mult_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "gcd (c * a) (c * b) = normalize (c * gcd a b)"
d_def: fixes fs :: "int vec list"   and k :: "nat" shows "fs_init.d fs k = gs.Gramian_determinant fs k"
ipurge_tr_rev_t_invariance_2: fixes Y :: "('c, 'd) ipurge_rec"   and D :: "'c \<Rightarrow> 'd"   and I :: "('d \<times> 'd) set"   and X :: "('c, 'd) ipurge_rec"   and P :: "'c process"   and ws :: "'c list" assumes "Y \<in> ipurge_tr_rev_t_set (X\<lparr>Pol := I, Map := D\<rparr>)"   and "ipurge_tr_rev_t_inv_2 P I D ws (X\<lparr>Pol := I, Map := D\<rparr>)" shows "ipurge_tr_rev_t_inv_2 P I D ws Y"
init_approx: fixes gc :: "(cmd, local_state, global_state) Gen_Scheduler.global_config option"   and prog :: "program" assumes "gc \<in> li.init prog" shows "gc \<in> gco_approx prog"
gbinomial_code: fixes a :: "'a"   and k :: "nat" shows "a gchoose k = (if k = 0 then 1::'a else fold_atLeastAtMost_nat (\<lambda>k. (*) (a - of_nat k)) 0 (k - 1) (1::'a) / fact k)"
wens_set_imp_leadsTo: fixes A :: "'b set"   and F :: "'b program"   and B :: "'b set" assumes "A \<in> wens_set F B" shows "F \<in> A \<longmapsto> B"
le_fininf_def: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream" assumes "traces ind" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>2 \<equiv> \<exists>v\<^sub>2. v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2 \<and> traces.le_fin ind w\<^sub>1 v\<^sub>2"
rtrancl_on_trans: fixes a :: "'a"   and b :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> rtrancl_on F r"   and "(b, c) \<in> rtrancl_on F r" shows "(a, c) \<in> rtrancl_on F r"
measurement_id2: fixes d :: "nat"   and M :: "nat \<Rightarrow> complex mat" assumes "measurement d 2 M" shows "adjoint (M 0) * M 0 + adjoint (M 1) * M 1 = 1\<^sub>m d"
commute: fixes a :: "'a upred"   and b :: "'a upred" shows "(a \<and> b) = (b \<and> a)"
axioms_1: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" assumes "character G \<chi>" shows "finite_comm_group G"
graph_restrict_left_aimpl: fixes Vl :: "'a set"   and E :: "('a \<times> 'a) set" shows "op_graph_restrict_left Vl E = E_of_succ (\<lambda>v. if v \<in> Vl then E `` {v} else {})"
not_coprime: shows "\<not> comm_monoid_mult_class.coprime a1 a2"
hyps: assumes "codomain_block b \<noteq> 0"   and "is_tangle_diagram (basic (make_vert_block (nat (codomain_block b)))) \<and> is_tangle_diagram as" shows "kauff_mat (basic (make_vert_block (nat (codomain_block b))) \<otimes> as) = kauff_mat (basic (make_vert_block (nat (codomain_block b)))) \<otimes> kauff_mat as"
elims_2: fixes x :: "('i \<times> constraint) list"   and xa :: "'i set"   and xb :: "'a ns_constraint list \<times> 's" assumes "checked_cs x xa xb" obtains cs :: "('i \<times> constraint) list" and J :: "'i set" and ds :: "'a ns_constraint list" and s :: "'s" where "x = cs" and "xa = J" and "xb = (ds, s)" and "ds = map snd (to_ns cs) \<and> checked_nsc (to_ns cs) J s"
sources_cod: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "sources (cod \<mu>) = sources \<mu>"
Q1_def: shows "Q1 = (\<lambda>F X. \<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. \<exists>a. \<Gamma> (Var x) = Var a)"
I: shows "I = (\<lambda>l s. 0 \<le> s ''a'' \<and> s ''z'' \<le> s ''a'' \<and> 0 \<le> s ''z'' \<and> s ''y'' = s ''z'' * s ''b'')"
sup_inf_distrib_2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<sqinter> z) \<squnion> x = y \<squnion> x \<sqinter> z \<squnion> x"
spmf_scale_spmf: fixes r :: "real"   and p :: "'c spmf"   and x :: "'c" shows "spmf (scale_spmf r p) x = max 0 (min (inverse (weight_spmf p)) r) * spmf p x"
semiattractive_subset: fixes B :: "'a set" assumes "B \<subseteq> X" shows "semiattractive B (\<sqsubseteq>)"
False: shows "\<not> (\<exists>x\<in>M. x \<in> iterates_above (lub M'))"
onorm_def: fixes f :: "'d \<Rightarrow> 'e" shows "onorm f = (SUP x. (\<parallel>f x\<parallel>) / (\<parallel>x\<parallel>))"
simps_8: fixes x :: "octo"   and y :: "octo" shows "Im7 (x - y) = Im7 x - Im7 y"
count_inverse: fixes x :: "'a multiset" shows "Abs_multiset (count x) = x"
just_show_all_approximating_bigstep_fun_equalities_with_start_Undecided: fixes s :: "state"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs1 :: "'b rule list"   and rs2 :: "'b rule list" assumes "s = Undecided \<Longrightarrow> approximating_bigstep_fun \<gamma> p rs1 s = approximating_bigstep_fun \<gamma> p rs2 s" shows "approximating_bigstep_fun \<gamma> p rs1 s = approximating_bigstep_fun \<gamma> p rs2 s"
elts_sup_iff: fixes x :: "V"   and y :: "V" shows "elts (x \<squnion> y) = elts x \<union> elts y"
prems_4: shows "fmrestrict_set vs s = fmrestrict_set vs s'"
pderiv_add: fixes p :: "'b poly"   and q :: "'b poly" shows "pderiv (p + q) = pderiv p + pderiv q"
quasi_inverse_antipar_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalence_map V H \<a> \<i> src trg f" shows "trg (bicategory.some_quasi_inverse V H \<a> \<i> src trg f) = src f"